Henrywood and Agarwal, Equation (12)

Percentage Accurate: 67.0% → 81.4%

Time: 23.6s

Alternatives: 16

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: 67.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 81.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -1.034 \cdot 10^{+217}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{t\_0}{\sqrt{-\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{t\_0}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + h \cdot \left(-0.125 \cdot \frac{{\left(\frac{M \cdot D}{d}\right)}^{2}}{\ell}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (- d))))
   (if (<= l -1.034e+217)
     (*
      (* (sqrt (/ d h)) (/ t_0 (sqrt (- l))))
      (- 1.0 (* 0.5 (pow (* (* M (* 0.5 (/ D d))) (sqrt (/ h l))) 2.0))))
     (if (<= l -2e-310)
       (*
        (sqrt (/ d l))
        (*
         (/ t_0 (sqrt (- h)))
         (+ 1.0 (* (/ h l) (* (pow (* D (/ (/ M 2.0) d)) 2.0) -0.5)))))
       (*
        (/ d (* (sqrt l) (sqrt h)))
        (+ 1.0 (* h (* -0.125 (/ (pow (/ (* M D) d) 2.0) l)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(-d);
	double tmp;
	if (l <= -1.034e+217) {
		tmp = (sqrt((d / h)) * (t_0 / sqrt(-l))) * (1.0 - (0.5 * pow(((M * (0.5 * (D / d))) * sqrt((h / l))), 2.0)));
	} else if (l <= -2e-310) {
		tmp = sqrt((d / l)) * ((t_0 / sqrt(-h)) * (1.0 + ((h / l) * (pow((D * ((M / 2.0) / d)), 2.0) * -0.5))));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + (h * (-0.125 * (pow(((M * D) / d), 2.0) / l))));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(-d)
    if (l <= (-1.034d+217)) then
        tmp = (sqrt((d / h)) * (t_0 / sqrt(-l))) * (1.0d0 - (0.5d0 * (((m * (0.5d0 * (d_1 / d))) * sqrt((h / l))) ** 2.0d0)))
    else if (l <= (-2d-310)) then
        tmp = sqrt((d / l)) * ((t_0 / sqrt(-h)) * (1.0d0 + ((h / l) * (((d_1 * ((m / 2.0d0) / d)) ** 2.0d0) * (-0.5d0)))))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 + (h * ((-0.125d0) * ((((m * d_1) / d) ** 2.0d0) / l))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt(-d);
	double tmp;
	if (l <= -1.034e+217) {
		tmp = (Math.sqrt((d / h)) * (t_0 / Math.sqrt(-l))) * (1.0 - (0.5 * Math.pow(((M * (0.5 * (D / d))) * Math.sqrt((h / l))), 2.0)));
	} else if (l <= -2e-310) {
		tmp = Math.sqrt((d / l)) * ((t_0 / Math.sqrt(-h)) * (1.0 + ((h / l) * (Math.pow((D * ((M / 2.0) / d)), 2.0) * -0.5))));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 + (h * (-0.125 * (Math.pow(((M * D) / d), 2.0) / l))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt(-d)
	tmp = 0
	if l <= -1.034e+217:
		tmp = (math.sqrt((d / h)) * (t_0 / math.sqrt(-l))) * (1.0 - (0.5 * math.pow(((M * (0.5 * (D / d))) * math.sqrt((h / l))), 2.0)))
	elif l <= -2e-310:
		tmp = math.sqrt((d / l)) * ((t_0 / math.sqrt(-h)) * (1.0 + ((h / l) * (math.pow((D * ((M / 2.0) / d)), 2.0) * -0.5))))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 + (h * (-0.125 * (math.pow(((M * D) / d), 2.0) / l))))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(-d))
	tmp = 0.0
	if (l <= -1.034e+217)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * Float64(t_0 / sqrt(Float64(-l)))) * Float64(1.0 - Float64(0.5 * (Float64(Float64(M * Float64(0.5 * Float64(D / d))) * sqrt(Float64(h / l))) ^ 2.0))));
	elseif (l <= -2e-310)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(Float64(t_0 / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D * Float64(Float64(M / 2.0) / d)) ^ 2.0) * -0.5)))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 + Float64(h * Float64(-0.125 * Float64((Float64(Float64(M * D) / d) ^ 2.0) / l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(-d);
	tmp = 0.0;
	if (l <= -1.034e+217)
		tmp = (sqrt((d / h)) * (t_0 / sqrt(-l))) * (1.0 - (0.5 * (((M * (0.5 * (D / d))) * sqrt((h / l))) ^ 2.0)));
	elseif (l <= -2e-310)
		tmp = sqrt((d / l)) * ((t_0 / sqrt(-h)) * (1.0 + ((h / l) * (((D * ((M / 2.0) / d)) ^ 2.0) * -0.5))));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + (h * (-0.125 * ((((M * D) / d) ^ 2.0) / l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[l, -1.034e+217], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[Power[N[(N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-310], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D * N[(N[(M / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(h * N[(-0.125 * N[(N[Power[N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.034e217

   1. Initial program 44.3%

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

   3. Simplified 44.3%

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

      1. add-sqr-sqrt 44.3%

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

      2. pow2 44.3%

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

      3. sqrt-prod 44.3%

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

      4. sqrt-pow1 45.0%

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

      5. metadata-eval 45.0%

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

      6. pow1 45.0%

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

      7. frac-times 45.0%

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

      8. associate-/l* 45.0%

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

      9. *-un-lft-identity 45.0%

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

      10. times-frac 45.0%

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

      11. metadata-eval 45.0%

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

   6. Applied egg-rr 45.0%

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

      1. frac-2neg 45.0%

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

      2. sqrt-div 72.3%

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

   8. Applied egg-rr 72.3%

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

   if -1.034e217 < l < -1.999999999999994e-310

   1. Initial program 74.3%

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

   3. Simplified 74.3%

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

      1. frac-2neg 74.3%

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

      2. sqrt-div 82.7%

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

   6. Applied egg-rr 82.7%

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

   if -1.999999999999994e-310 < l

   1. Initial program 62.9%

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

   3. Simplified 61.7%

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

      1. add-sqr-sqrt 61.7%

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

      2. pow2 61.7%

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

      3. sqrt-prod 61.6%

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

      4. sqrt-pow1 62.8%

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

      5. metadata-eval 62.8%

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

      6. pow1 62.8%

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

      7. frac-times 64.1%

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

      8. associate-/l* 62.8%

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

      9. *-un-lft-identity 62.8%

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

      10. times-frac 62.8%

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

      11. metadata-eval 62.8%

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

   6. Applied egg-rr 62.8%

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

   7. Taylor expanded in h around inf 44.4%

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

      1. sub-neg 44.4%

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

      2. distribute-rgt-in 44.4%

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

      3. lft-mult-inverse 44.4%

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

      4. distribute-lft-neg-in 44.4%

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

      5. metadata-eval 44.4%

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

      6. associate-/r* 45.8%

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

      7. associate-/l* 46.5%

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

      8. unpow2 46.5%

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

      9. unpow2 46.5%

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

      10. unpow2 46.5%

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

      11. times-frac 54.9%

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

      12. swap-sqr 65.9%

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

      13. unpow2 65.9%

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

      14. associate-*r/ 65.9%

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

   9. Simplified 65.9%

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

       1. *-commutative 65.9%

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

       2. sqrt-div 74.5%

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

       3. sqrt-div 87.7%

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

       4. frac-times 87.9%

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

       5. add-sqr-sqrt 88.0%

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

   11. Applied egg-rr 88.0%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 + \left(-0.125 \cdot \frac{{\left(\frac{D \cdot M}{d}\right)}^{2}}{\ell}\right) \cdot h\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.034 \cdot 10^{+217}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + h \cdot \left(-0.125 \cdot \frac{{\left(\frac{M \cdot D}{d}\right)}^{2}}{\ell}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.3% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -2e-310)
   (*
    (sqrt (/ d l))
    (*
     (/ (sqrt (- d)) (sqrt (- h)))
     (+ 1.0 (* (/ h l) (* (pow (* D (/ (/ M 2.0) d)) 2.0) -0.5)))))
   (*
    (/ d (* (sqrt l) (sqrt h)))
    (+ 1.0 (* h (* -0.125 (/ (pow (/ (* M D) d) 2.0) l)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -2e-310) {
		tmp = sqrt((d / l)) * ((sqrt(-d) / sqrt(-h)) * (1.0 + ((h / l) * (pow((D * ((M / 2.0) / d)), 2.0) * -0.5))));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + (h * (-0.125 * (pow(((M * D) / d), 2.0) / l))));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-2d-310)) then
        tmp = sqrt((d / l)) * ((sqrt(-d) / sqrt(-h)) * (1.0d0 + ((h / l) * (((d_1 * ((m / 2.0d0) / d)) ** 2.0d0) * (-0.5d0)))))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 + (h * ((-0.125d0) * ((((m * d_1) / d) ** 2.0d0) / l))))
    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 <= -2e-310) {
		tmp = Math.sqrt((d / l)) * ((Math.sqrt(-d) / Math.sqrt(-h)) * (1.0 + ((h / l) * (Math.pow((D * ((M / 2.0) / d)), 2.0) * -0.5))));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 + (h * (-0.125 * (Math.pow(((M * D) / d), 2.0) / l))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -2e-310:
		tmp = math.sqrt((d / l)) * ((math.sqrt(-d) / math.sqrt(-h)) * (1.0 + ((h / l) * (math.pow((D * ((M / 2.0) / d)), 2.0) * -0.5))))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 + (h * (-0.125 * (math.pow(((M * D) / d), 2.0) / l))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -2e-310)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D * Float64(Float64(M / 2.0) / d)) ^ 2.0) * -0.5)))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 + Float64(h * Float64(-0.125 * Float64((Float64(Float64(M * D) / d) ^ 2.0) / l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -2e-310)
		tmp = sqrt((d / l)) * ((sqrt(-d) / sqrt(-h)) * (1.0 + ((h / l) * (((D * ((M / 2.0) / d)) ^ 2.0) * -0.5))));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + (h * (-0.125 * ((((M * D) / d) ^ 2.0) / l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -2e-310], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D * N[(N[(M / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(h * N[(-0.125 * N[(N[Power[N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.999999999999994e-310

   1. Initial program 69.4%

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

   3. Simplified 69.4%

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

      1. frac-2neg 69.4%

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

      2. sqrt-div 77.3%

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

   6. Applied egg-rr 77.3%

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

   if -1.999999999999994e-310 < l

   1. Initial program 62.9%

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

   3. Simplified 61.7%

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

      1. add-sqr-sqrt 61.7%

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

      2. pow2 61.7%

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

      3. sqrt-prod 61.6%

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

      4. sqrt-pow1 62.8%

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

      5. metadata-eval 62.8%

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

      6. pow1 62.8%

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

      7. frac-times 64.1%

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

      8. associate-/l* 62.8%

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

      9. *-un-lft-identity 62.8%

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

      10. times-frac 62.8%

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

      11. metadata-eval 62.8%

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

   6. Applied egg-rr 62.8%

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

   7. Taylor expanded in h around inf 44.4%

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

      1. sub-neg 44.4%

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

      2. distribute-rgt-in 44.4%

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

      3. lft-mult-inverse 44.4%

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

      4. distribute-lft-neg-in 44.4%

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

      5. metadata-eval 44.4%

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

      6. associate-/r* 45.8%

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

      7. associate-/l* 46.5%

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

      8. unpow2 46.5%

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

      9. unpow2 46.5%

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

      10. unpow2 46.5%

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

      11. times-frac 54.9%

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

      12. swap-sqr 65.9%

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

      13. unpow2 65.9%

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

      14. associate-*r/ 65.9%

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

   9. Simplified 65.9%

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

       1. *-commutative 65.9%

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

       2. sqrt-div 74.5%

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

       3. sqrt-div 87.7%

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

       4. frac-times 87.9%

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

       5. add-sqr-sqrt 88.0%

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

   11. Applied egg-rr 88.0%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 + \left(-0.125 \cdot \frac{{\left(\frac{D \cdot M}{d}\right)}^{2}}{\ell}\right) \cdot h\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.4%

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

Alternative 3: 77.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + h \cdot \left(-0.125 \cdot \frac{{\left(\frac{M \cdot D}{d}\right)}^{2}}{\ell}\right)\\ \mathbf{if}\;d \leq 2 \cdot 10^{-285}:\\ \;\;\;\;t\_0 \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{-224}:\\ \;\;\;\;\left(-0.125 \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}\right) \cdot {\left(D \cdot \frac{M}{\sqrt{d}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* h (* -0.125 (/ (pow (/ (* M D) d) 2.0) l))))))
   (if (<= d 2e-285)
     (* t_0 (* (sqrt (/ d h)) (sqrt (/ d l))))
     (if (<= d 2.8e-224)
       (* (* -0.125 (/ (sqrt h) (pow l 1.5))) (pow (* D (/ M (sqrt d))) 2.0))
       (* (/ d (* (sqrt l) (sqrt h))) t_0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 + (h * (-0.125 * (pow(((M * D) / d), 2.0) / l)));
	double tmp;
	if (d <= 2e-285) {
		tmp = t_0 * (sqrt((d / h)) * sqrt((d / l)));
	} else if (d <= 2.8e-224) {
		tmp = (-0.125 * (sqrt(h) / pow(l, 1.5))) * pow((D * (M / sqrt(d))), 2.0);
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * t_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) :: tmp
    t_0 = 1.0d0 + (h * ((-0.125d0) * ((((m * d_1) / d) ** 2.0d0) / l)))
    if (d <= 2d-285) then
        tmp = t_0 * (sqrt((d / h)) * sqrt((d / l)))
    else if (d <= 2.8d-224) then
        tmp = ((-0.125d0) * (sqrt(h) / (l ** 1.5d0))) * ((d_1 * (m / sqrt(d))) ** 2.0d0)
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * t_0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 + (h * (-0.125 * (Math.pow(((M * D) / d), 2.0) / l)));
	double tmp;
	if (d <= 2e-285) {
		tmp = t_0 * (Math.sqrt((d / h)) * Math.sqrt((d / l)));
	} else if (d <= 2.8e-224) {
		tmp = (-0.125 * (Math.sqrt(h) / Math.pow(l, 1.5))) * Math.pow((D * (M / Math.sqrt(d))), 2.0);
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = 1.0 + (h * (-0.125 * (math.pow(((M * D) / d), 2.0) / l)))
	tmp = 0
	if d <= 2e-285:
		tmp = t_0 * (math.sqrt((d / h)) * math.sqrt((d / l)))
	elif d <= 2.8e-224:
		tmp = (-0.125 * (math.sqrt(h) / math.pow(l, 1.5))) * math.pow((D * (M / math.sqrt(d))), 2.0)
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(1.0 + Float64(h * Float64(-0.125 * Float64((Float64(Float64(M * D) / d) ^ 2.0) / l))))
	tmp = 0.0
	if (d <= 2e-285)
		tmp = Float64(t_0 * Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))));
	elseif (d <= 2.8e-224)
		tmp = Float64(Float64(-0.125 * Float64(sqrt(h) / (l ^ 1.5))) * (Float64(D * Float64(M / sqrt(d))) ^ 2.0));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = 1.0 + (h * (-0.125 * ((((M * D) / d) ^ 2.0) / l)));
	tmp = 0.0;
	if (d <= 2e-285)
		tmp = t_0 * (sqrt((d / h)) * sqrt((d / l)));
	elseif (d <= 2.8e-224)
		tmp = (-0.125 * (sqrt(h) / (l ^ 1.5))) * ((D * (M / sqrt(d))) ^ 2.0);
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(1.0 + N[(h * N[(-0.125 * N[(N[Power[N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, 2e-285], N[(t$95$0 * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.8e-224], N[(N[(-0.125 * N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(D * N[(M / N[Sqrt[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < 2.00000000000000015e-285

   1. Initial program 69.6%

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

   3. Simplified 67.0%

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

      1. add-sqr-sqrt 67.0%

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

      2. pow2 67.0%

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

      3. sqrt-prod 67.1%

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

      4. sqrt-pow1 67.9%

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

      5. metadata-eval 67.9%

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

      6. pow1 67.9%

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

      7. frac-times 70.4%

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

      8. associate-/l* 67.9%

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

      9. *-un-lft-identity 67.9%

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

      10. times-frac 67.9%

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

      11. metadata-eval 67.9%

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

   6. Applied egg-rr 67.9%

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

   7. Taylor expanded in h around inf 45.2%

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

      1. sub-neg 45.2%

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

      2. distribute-rgt-in 45.2%

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

      3. lft-mult-inverse 45.2%

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

      4. distribute-lft-neg-in 45.2%

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

      5. metadata-eval 45.2%

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

      6. associate-/r* 46.1%

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

      7. associate-/l* 43.5%

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

      8. unpow2 43.5%

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

      9. unpow2 43.5%

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

      10. unpow2 43.5%

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

      11. times-frac 54.8%

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

      12. swap-sqr 73.1%

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

      13. unpow2 73.1%

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

      14. associate-*r/ 73.1%

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

   9. Simplified 73.1%

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

   if 2.00000000000000015e-285 < d < 2.7999999999999998e-224

   1. Initial program 33.3%

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

   3. Simplified 33.2%

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

   5. Taylor expanded in M around inf 11.1%

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

      1. associate-*r/ 11.1%

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

      2. associate-*r* 11.1%

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

      3. associate-*r* 11.1%

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

      4. associate-*l/ 11.5%

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

      5. associate-*r/ 11.5%

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

      6. *-commutative 11.5%

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

      7. associate-*l* 11.5%

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

   7. Simplified 22.5%

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

   8. Taylor expanded in d around 0 24.1%

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

      1. *-commutative 24.1%

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

      2. associate-*l* 24.1%

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

      3. associate-/l* 24.1%

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

   10. Simplified 24.1%

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

       1. add-sqr-sqrt 1.5%

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

       2. pow2 1.5%

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

   12. Applied egg-rr 0.0%

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

       1. unpow2 0.0%

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

       2. swap-sqr 0.0%

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

       3. rem-square-sqrt 99.8%

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

       4. *-commutative 99.8%

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

       5. unpow2 99.8%

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

   14. Simplified 99.8%

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

   if 2.7999999999999998e-224 < d

   1. Initial program 64.6%

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

   3. Simplified 63.9%

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

      1. add-sqr-sqrt 63.9%

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

      2. pow2 63.9%

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

      3. sqrt-prod 63.9%

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

      4. sqrt-pow1 65.2%

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

      5. metadata-eval 65.2%

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

      6. pow1 65.2%

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

      7. frac-times 65.9%

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

      8. associate-/l* 65.2%

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

      9. *-un-lft-identity 65.2%

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

      10. times-frac 65.2%

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

      11. metadata-eval 65.2%

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

   6. Applied egg-rr 65.2%

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

   7. Taylor expanded in h around inf 47.2%

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

      1. sub-neg 47.2%

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

      2. distribute-rgt-in 47.2%

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

      3. lft-mult-inverse 47.2%

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

      4. distribute-lft-neg-in 47.2%

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

      5. metadata-eval 47.2%

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

      6. associate-/r* 48.8%

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

      7. associate-/l* 49.5%

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

      8. unpow2 49.5%

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

      9. unpow2 49.5%

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

      10. unpow2 49.5%

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

      11. times-frac 56.5%

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

      12. swap-sqr 68.6%

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

      13. unpow2 68.6%

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

      14. associate-*r/ 68.5%

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

   9. Simplified 68.5%

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

       1. *-commutative 68.5%

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

       2. sqrt-div 77.2%

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

       3. sqrt-div 91.0%

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

       4. frac-times 91.2%

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

       5. add-sqr-sqrt 91.3%

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

   11. Applied egg-rr 91.3%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 + \left(-0.125 \cdot \frac{{\left(\frac{D \cdot M}{d}\right)}^{2}}{\ell}\right) \cdot h\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 2 \cdot 10^{-285}:\\ \;\;\;\;\left(1 + h \cdot \left(-0.125 \cdot \frac{{\left(\frac{M \cdot D}{d}\right)}^{2}}{\ell}\right)\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{-224}:\\ \;\;\;\;\left(-0.125 \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}\right) \cdot {\left(D \cdot \frac{M}{\sqrt{d}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + h \cdot \left(-0.125 \cdot \frac{{\left(\frac{M \cdot D}{d}\right)}^{2}}{\ell}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + h \cdot \left(-0.125 \cdot \frac{{\left(\frac{M \cdot D}{d}\right)}^{2}}{\ell}\right)\\ t_1 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{+174}:\\ \;\;\;\;t\_1 \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* h (* -0.125 (/ (pow (/ (* M D) d) 2.0) l)))))
        (t_1 (sqrt (/ d h))))
   (if (<= l -5e+174)
     (* t_1 (/ (sqrt (- d)) (sqrt (- l))))
     (if (<= l -2e-310)
       (* t_0 (* t_1 (sqrt (/ d l))))
       (* (/ d (* (sqrt l) (sqrt h))) t_0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 + (h * (-0.125 * (pow(((M * D) / d), 2.0) / l)));
	double t_1 = sqrt((d / h));
	double tmp;
	if (l <= -5e+174) {
		tmp = t_1 * (sqrt(-d) / sqrt(-l));
	} else if (l <= -2e-310) {
		tmp = t_0 * (t_1 * sqrt((d / l)));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * t_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 = 1.0d0 + (h * ((-0.125d0) * ((((m * d_1) / d) ** 2.0d0) / l)))
    t_1 = sqrt((d / h))
    if (l <= (-5d+174)) then
        tmp = t_1 * (sqrt(-d) / sqrt(-l))
    else if (l <= (-2d-310)) then
        tmp = t_0 * (t_1 * sqrt((d / l)))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * t_0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 + (h * (-0.125 * (Math.pow(((M * D) / d), 2.0) / l)));
	double t_1 = Math.sqrt((d / h));
	double tmp;
	if (l <= -5e+174) {
		tmp = t_1 * (Math.sqrt(-d) / Math.sqrt(-l));
	} else if (l <= -2e-310) {
		tmp = t_0 * (t_1 * Math.sqrt((d / l)));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = 1.0 + (h * (-0.125 * (math.pow(((M * D) / d), 2.0) / l)))
	t_1 = math.sqrt((d / h))
	tmp = 0
	if l <= -5e+174:
		tmp = t_1 * (math.sqrt(-d) / math.sqrt(-l))
	elif l <= -2e-310:
		tmp = t_0 * (t_1 * math.sqrt((d / l)))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(1.0 + Float64(h * Float64(-0.125 * Float64((Float64(Float64(M * D) / d) ^ 2.0) / l))))
	t_1 = sqrt(Float64(d / h))
	tmp = 0.0
	if (l <= -5e+174)
		tmp = Float64(t_1 * Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))));
	elseif (l <= -2e-310)
		tmp = Float64(t_0 * Float64(t_1 * sqrt(Float64(d / l))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = 1.0 + (h * (-0.125 * ((((M * D) / d) ^ 2.0) / l)));
	t_1 = sqrt((d / h));
	tmp = 0.0;
	if (l <= -5e+174)
		tmp = t_1 * (sqrt(-d) / sqrt(-l));
	elseif (l <= -2e-310)
		tmp = t_0 * (t_1 * sqrt((d / l)));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(1.0 + N[(h * N[(-0.125 * N[(N[Power[N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -5e+174], N[(t$95$1 * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-310], N[(t$95$0 * N[(t$95$1 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.9999999999999997e174

   1. Initial program 39.3%

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

   3. Simplified 39.3%

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

   5. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 3.0%

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

      4. mul-1-neg 3.0%

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

   7. Simplified 3.0%

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

      1. add-sqr-sqrt 2.0%

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

      2. sqrt-unprod 40.6%

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

      3. sqr-neg 40.6%

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

      4. add-sqr-sqrt 40.6%

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

      5. frac-2neg 40.6%

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

      6. sqrt-div 57.9%

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

   9. Applied egg-rr 57.9%

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

   if -4.9999999999999997e174 < l < -1.999999999999994e-310

   1. Initial program 77.4%

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

   3. Simplified 75.1%

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

      1. add-sqr-sqrt 75.1%

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

      2. pow2 75.1%

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

      3. sqrt-prod 75.1%

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

      4. sqrt-pow1 76.0%

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

      5. metadata-eval 76.0%

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

      6. pow1 76.0%

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

      7. frac-times 78.2%

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

      8. associate-/l* 76.0%

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

      9. *-un-lft-identity 76.0%

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

      10. times-frac 76.0%

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

      11. metadata-eval 76.0%

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

   6. Applied egg-rr 76.0%

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

   7. Taylor expanded in h around inf 51.8%

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

      1. sub-neg 51.8%

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

      2. distribute-rgt-in 51.8%

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

      3. lft-mult-inverse 51.8%

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

      4. distribute-lft-neg-in 51.8%

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

      5. metadata-eval 51.8%

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

      6. associate-/r* 53.0%

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

      7. associate-/l* 49.6%

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

      8. unpow2 49.6%

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

      9. unpow2 49.6%

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

      10. unpow2 49.6%

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

      11. times-frac 62.1%

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

      12. swap-sqr 82.0%

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

      13. unpow2 82.0%

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

      14. associate-*r/ 82.0%

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

   9. Simplified 82.0%

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

   if -1.999999999999994e-310 < l

   1. Initial program 62.9%

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

   3. Simplified 61.7%

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

      1. add-sqr-sqrt 61.7%

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

      2. pow2 61.7%

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

      3. sqrt-prod 61.6%

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

      4. sqrt-pow1 62.8%

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

      5. metadata-eval 62.8%

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

      6. pow1 62.8%

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

      7. frac-times 64.1%

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

      8. associate-/l* 62.8%

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

      9. *-un-lft-identity 62.8%

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

      10. times-frac 62.8%

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

      11. metadata-eval 62.8%

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

   6. Applied egg-rr 62.8%

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

   7. Taylor expanded in h around inf 44.4%

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

      1. sub-neg 44.4%

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

      2. distribute-rgt-in 44.4%

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

      3. lft-mult-inverse 44.4%

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

      4. distribute-lft-neg-in 44.4%

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

      5. metadata-eval 44.4%

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

      6. associate-/r* 45.8%

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

      7. associate-/l* 46.5%

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

      8. unpow2 46.5%

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

      9. unpow2 46.5%

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

      10. unpow2 46.5%

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

      11. times-frac 54.9%

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

      12. swap-sqr 65.9%

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

      13. unpow2 65.9%

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

      14. associate-*r/ 65.9%

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

   9. Simplified 65.9%

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

       1. *-commutative 65.9%

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

       2. sqrt-div 74.5%

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

       3. sqrt-div 87.7%

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

       4. frac-times 87.9%

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

       5. add-sqr-sqrt 88.0%

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

   11. Applied egg-rr 88.0%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 + \left(-0.125 \cdot \frac{{\left(\frac{D \cdot M}{d}\right)}^{2}}{\ell}\right) \cdot h\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{+174}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(1 + h \cdot \left(-0.125 \cdot \frac{{\left(\frac{M \cdot D}{d}\right)}^{2}}{\ell}\right)\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + h \cdot \left(-0.125 \cdot \frac{{\left(\frac{M \cdot D}{d}\right)}^{2}}{\ell}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := 1 + h \cdot \left(-0.125 \cdot \frac{{\left(\frac{M \cdot D}{d}\right)}^{2}}{\ell}\right)\\ \mathbf{if}\;\ell \leq -1.3 \cdot 10^{+172}:\\ \;\;\;\;t\_0 \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(t\_0 \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h)))
        (t_1 (+ 1.0 (* h (* -0.125 (/ (pow (/ (* M D) d) 2.0) l))))))
   (if (<= l -1.3e+172)
     (* t_0 (/ (sqrt (- d)) (sqrt (- l))))
     (if (<= l -2e-310)
       (* (sqrt (/ d l)) (* t_0 t_1))
       (* (/ d (* (sqrt l) (sqrt h))) t_1)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h));
	double t_1 = 1.0 + (h * (-0.125 * (pow(((M * D) / d), 2.0) / l)));
	double tmp;
	if (l <= -1.3e+172) {
		tmp = t_0 * (sqrt(-d) / sqrt(-l));
	} else if (l <= -2e-310) {
		tmp = sqrt((d / l)) * (t_0 * t_1);
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * t_1;
	}
	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 / h))
    t_1 = 1.0d0 + (h * ((-0.125d0) * ((((m * d_1) / d) ** 2.0d0) / l)))
    if (l <= (-1.3d+172)) then
        tmp = t_0 * (sqrt(-d) / sqrt(-l))
    else if (l <= (-2d-310)) then
        tmp = sqrt((d / l)) * (t_0 * t_1)
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * t_1
    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 / h));
	double t_1 = 1.0 + (h * (-0.125 * (Math.pow(((M * D) / d), 2.0) / l)));
	double tmp;
	if (l <= -1.3e+172) {
		tmp = t_0 * (Math.sqrt(-d) / Math.sqrt(-l));
	} else if (l <= -2e-310) {
		tmp = Math.sqrt((d / l)) * (t_0 * t_1);
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * t_1;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / h))
	t_1 = 1.0 + (h * (-0.125 * (math.pow(((M * D) / d), 2.0) / l)))
	tmp = 0
	if l <= -1.3e+172:
		tmp = t_0 * (math.sqrt(-d) / math.sqrt(-l))
	elif l <= -2e-310:
		tmp = math.sqrt((d / l)) * (t_0 * t_1)
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * t_1
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / h))
	t_1 = Float64(1.0 + Float64(h * Float64(-0.125 * Float64((Float64(Float64(M * D) / d) ^ 2.0) / l))))
	tmp = 0.0
	if (l <= -1.3e+172)
		tmp = Float64(t_0 * Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))));
	elseif (l <= -2e-310)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(t_0 * t_1));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / h));
	t_1 = 1.0 + (h * (-0.125 * ((((M * D) / d) ^ 2.0) / l)));
	tmp = 0.0;
	if (l <= -1.3e+172)
		tmp = t_0 * (sqrt(-d) / sqrt(-l));
	elseif (l <= -2e-310)
		tmp = sqrt((d / l)) * (t_0 * t_1);
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(h * N[(-0.125 * N[(N[Power[N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.3e+172], N[(t$95$0 * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-310], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.3e172

   1. Initial program 39.3%

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

   3. Simplified 39.3%

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

   5. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 3.0%

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

      4. mul-1-neg 3.0%

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

   7. Simplified 3.0%

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

      1. add-sqr-sqrt 2.0%

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

      2. sqrt-unprod 40.6%

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

      3. sqr-neg 40.6%

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

      4. add-sqr-sqrt 40.6%

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

      5. frac-2neg 40.6%

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

      6. sqrt-div 57.9%

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

   9. Applied egg-rr 57.9%

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

   if -1.3e172 < l < -1.999999999999994e-310

   1. Initial program 77.4%

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

   3. Simplified 77.3%

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

      1. associate-*l/ 83.0%

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

      2. *-commutative 83.0%

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

      3. associate-*r/ 83.0%

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

      4. *-un-lft-identity 83.0%

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

      5. times-frac 83.0%

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

      6. associate-/l/ 83.0%

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

      7. *-commutative 83.0%

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

      8. times-frac 83.0%

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

      9. *-un-lft-identity 83.0%

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

      10. *-commutative 83.0%

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

      11. associate-/l* 80.8%

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

      12. *-un-lft-identity 80.8%

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

      13. times-frac 80.8%

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

      14. metadata-eval 80.8%

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

   6. Applied egg-rr 80.8%

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

      1. associate-/l* 79.7%

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

      2. *-commutative 79.7%

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

      3. associate-/l* 79.7%

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

      4. *-commutative 79.7%

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

      5. metadata-eval 79.7%

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

      6. times-frac 79.7%

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

      7. *-rgt-identity 79.7%

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

      8. associate-/l* 82.0%

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

      9. *-commutative 82.0%

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

      10. associate-/l* 82.0%

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

      11. *-commutative 82.0%

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

   8. Simplified 82.0%

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

   9. Taylor expanded in D around 0 51.8%

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

       1. associate-/r* 53.0%

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

       2. associate-/l* 49.6%

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

       3. unpow2 49.6%

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

       4. unpow2 49.6%

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

       5. unpow2 49.6%

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

       6. times-frac 62.1%

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

       7. swap-sqr 82.0%

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

       8. unpow2 82.0%

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

       9. associate-*r/ 82.0%

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

   11. Simplified 82.0%

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

   if -1.999999999999994e-310 < l

   1. Initial program 62.9%

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

   3. Simplified 61.7%

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

      1. add-sqr-sqrt 61.7%

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

      2. pow2 61.7%

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

      3. sqrt-prod 61.6%

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

      4. sqrt-pow1 62.8%

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

      5. metadata-eval 62.8%

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

      6. pow1 62.8%

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

      7. frac-times 64.1%

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

      8. associate-/l* 62.8%

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

      9. *-un-lft-identity 62.8%

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

      10. times-frac 62.8%

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

      11. metadata-eval 62.8%

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

   6. Applied egg-rr 62.8%

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

   7. Taylor expanded in h around inf 44.4%

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

      1. sub-neg 44.4%

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

      2. distribute-rgt-in 44.4%

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

      3. lft-mult-inverse 44.4%

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

      4. distribute-lft-neg-in 44.4%

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

      5. metadata-eval 44.4%

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

      6. associate-/r* 45.8%

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

      7. associate-/l* 46.5%

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

      8. unpow2 46.5%

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

      9. unpow2 46.5%

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

      10. unpow2 46.5%

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

      11. times-frac 54.9%

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

      12. swap-sqr 65.9%

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

      13. unpow2 65.9%

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

      14. associate-*r/ 65.9%

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

   9. Simplified 65.9%

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

       1. *-commutative 65.9%

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

       2. sqrt-div 74.5%

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

       3. sqrt-div 87.7%

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

       4. frac-times 87.9%

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

       5. add-sqr-sqrt 88.0%

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

   11. Applied egg-rr 88.0%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 + \left(-0.125 \cdot \frac{{\left(\frac{D \cdot M}{d}\right)}^{2}}{\ell}\right) \cdot h\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.3 \cdot 10^{+172}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(-0.125 \cdot \frac{{\left(\frac{M \cdot D}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + h \cdot \left(-0.125 \cdot \frac{{\left(\frac{M \cdot D}{d}\right)}^{2}}{\ell}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + h \cdot \left(-0.125 \cdot \frac{{\left(\frac{M \cdot D}{d}\right)}^{2}}{\ell}\right)\\ \mathbf{if}\;\ell \leq -2.9 \cdot 10^{+72}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t\_0 \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* h (* -0.125 (/ (pow (/ (* M D) d) 2.0) l))))))
   (if (<= l -2.9e+72)
     (* (sqrt (/ d h)) (/ (sqrt (- d)) (sqrt (- l))))
     (if (<= l -2e-310)
       (* t_0 (sqrt (* (/ d h) (/ d l))))
       (* (/ d (* (sqrt l) (sqrt h))) t_0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 + (h * (-0.125 * (pow(((M * D) / d), 2.0) / l)));
	double tmp;
	if (l <= -2.9e+72) {
		tmp = sqrt((d / h)) * (sqrt(-d) / sqrt(-l));
	} else if (l <= -2e-310) {
		tmp = t_0 * sqrt(((d / h) * (d / l)));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * t_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) :: tmp
    t_0 = 1.0d0 + (h * ((-0.125d0) * ((((m * d_1) / d) ** 2.0d0) / l)))
    if (l <= (-2.9d+72)) then
        tmp = sqrt((d / h)) * (sqrt(-d) / sqrt(-l))
    else if (l <= (-2d-310)) then
        tmp = t_0 * sqrt(((d / h) * (d / l)))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * t_0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 + (h * (-0.125 * (Math.pow(((M * D) / d), 2.0) / l)));
	double tmp;
	if (l <= -2.9e+72) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt(-d) / Math.sqrt(-l));
	} else if (l <= -2e-310) {
		tmp = t_0 * Math.sqrt(((d / h) * (d / l)));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = 1.0 + (h * (-0.125 * (math.pow(((M * D) / d), 2.0) / l)))
	tmp = 0
	if l <= -2.9e+72:
		tmp = math.sqrt((d / h)) * (math.sqrt(-d) / math.sqrt(-l))
	elif l <= -2e-310:
		tmp = t_0 * math.sqrt(((d / h) * (d / l)))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(1.0 + Float64(h * Float64(-0.125 * Float64((Float64(Float64(M * D) / d) ^ 2.0) / l))))
	tmp = 0.0
	if (l <= -2.9e+72)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))));
	elseif (l <= -2e-310)
		tmp = Float64(t_0 * sqrt(Float64(Float64(d / h) * Float64(d / l))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = 1.0 + (h * (-0.125 * ((((M * D) / d) ^ 2.0) / l)));
	tmp = 0.0;
	if (l <= -2.9e+72)
		tmp = sqrt((d / h)) * (sqrt(-d) / sqrt(-l));
	elseif (l <= -2e-310)
		tmp = t_0 * sqrt(((d / h) * (d / l)));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(1.0 + N[(h * N[(-0.125 * N[(N[Power[N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.9e+72], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-310], N[(t$95$0 * N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.90000000000000017e72

   1. Initial program 53.8%

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

   3. Simplified 51.5%

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

   5. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 2.9%

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

      4. mul-1-neg 2.9%

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

   7. Simplified 2.9%

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

      1. add-sqr-sqrt 1.3%

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

      2. sqrt-unprod 46.3%

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

      3. sqr-neg 46.3%

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

      4. add-sqr-sqrt 46.3%

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

      5. frac-2neg 46.3%

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

      6. sqrt-div 57.6%

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

   9. Applied egg-rr 57.6%

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

   if -2.90000000000000017e72 < l < -1.999999999999994e-310

   1. Initial program 79.4%

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

   3. Simplified 78.0%

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

      1. add-sqr-sqrt 77.9%

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

      2. pow2 77.9%

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

      3. sqrt-prod 78.0%

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

      4. sqrt-pow1 78.0%

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

      5. metadata-eval 78.0%

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

      6. pow1 78.0%

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

      7. frac-times 79.4%

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

      8. associate-/l* 78.0%

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

      9. *-un-lft-identity 78.0%

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

      10. times-frac 78.0%

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

      11. metadata-eval 78.0%

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

   6. Applied egg-rr 78.0%

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

   7. Taylor expanded in h around inf 52.2%

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

      1. sub-neg 52.2%

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

      2. distribute-rgt-in 52.2%

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

      3. lft-mult-inverse 52.2%

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

      4. distribute-lft-neg-in 52.2%

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

      5. metadata-eval 52.2%

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

      6. associate-/r* 53.8%

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

      7. associate-/l* 52.3%

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

      8. unpow2 52.3%

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

      9. unpow2 52.3%

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

      10. unpow2 52.3%

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

      11. times-frac 67.0%

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

      12. swap-sqr 85.4%

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

      13. unpow2 85.4%

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

      14. associate-*r/ 85.4%

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

   9. Simplified 85.4%

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

       1. *-commutative 85.4%

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

       2. sqrt-unprod 72.9%

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

   11. Applied egg-rr 72.9%

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

   if -1.999999999999994e-310 < l

   1. Initial program 62.9%

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

   3. Simplified 61.7%

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

      1. add-sqr-sqrt 61.7%

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

      2. pow2 61.7%

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

      3. sqrt-prod 61.6%

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

      4. sqrt-pow1 62.8%

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

      5. metadata-eval 62.8%

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

      6. pow1 62.8%

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

      7. frac-times 64.1%

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

      8. associate-/l* 62.8%

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

      9. *-un-lft-identity 62.8%

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

      10. times-frac 62.8%

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

      11. metadata-eval 62.8%

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

   6. Applied egg-rr 62.8%

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

   7. Taylor expanded in h around inf 44.4%

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

      1. sub-neg 44.4%

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

      2. distribute-rgt-in 44.4%

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

      3. lft-mult-inverse 44.4%

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

      4. distribute-lft-neg-in 44.4%

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

      5. metadata-eval 44.4%

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

      6. associate-/r* 45.8%

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

      7. associate-/l* 46.5%

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

      8. unpow2 46.5%

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

      9. unpow2 46.5%

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

      10. unpow2 46.5%

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

      11. times-frac 54.9%

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

      12. swap-sqr 65.9%

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

      13. unpow2 65.9%

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

      14. associate-*r/ 65.9%

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

   9. Simplified 65.9%

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

       1. *-commutative 65.9%

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

       2. sqrt-div 74.5%

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

       3. sqrt-div 87.7%

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

       4. frac-times 87.9%

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

       5. add-sqr-sqrt 88.0%

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

   11. Applied egg-rr 88.0%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 + \left(-0.125 \cdot \frac{{\left(\frac{D \cdot M}{d}\right)}^{2}}{\ell}\right) \cdot h\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.9 \cdot 10^{+72}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(1 + h \cdot \left(-0.125 \cdot \frac{{\left(\frac{M \cdot D}{d}\right)}^{2}}{\ell}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + h \cdot \left(-0.125 \cdot \frac{{\left(\frac{M \cdot D}{d}\right)}^{2}}{\ell}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -8.3 \cdot 10^{+67}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;\ell \leq 5.6 \cdot 10^{+28}:\\ \;\;\;\;\left(1 + h \cdot \left(-0.125 \cdot \frac{{\left(\frac{M \cdot D}{d}\right)}^{2}}{\ell}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -8.3e+67)
   (* (sqrt (/ d h)) (/ (sqrt (- d)) (sqrt (- l))))
   (if (<= l 5.6e+28)
     (*
      (+ 1.0 (* h (* -0.125 (/ (pow (/ (* M D) d) 2.0) l))))
      (sqrt (* (/ d h) (/ d l))))
     (* d (/ (pow h -0.5) (sqrt l))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -8.3e+67) {
		tmp = sqrt((d / h)) * (sqrt(-d) / sqrt(-l));
	} else if (l <= 5.6e+28) {
		tmp = (1.0 + (h * (-0.125 * (pow(((M * D) / d), 2.0) / l)))) * sqrt(((d / h) * (d / l)));
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-8.3d+67)) then
        tmp = sqrt((d / h)) * (sqrt(-d) / sqrt(-l))
    else if (l <= 5.6d+28) then
        tmp = (1.0d0 + (h * ((-0.125d0) * ((((m * d_1) / d) ** 2.0d0) / l)))) * sqrt(((d / h) * (d / l)))
    else
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    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 <= -8.3e+67) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt(-d) / Math.sqrt(-l));
	} else if (l <= 5.6e+28) {
		tmp = (1.0 + (h * (-0.125 * (Math.pow(((M * D) / d), 2.0) / l)))) * Math.sqrt(((d / h) * (d / l)));
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -8.3e+67:
		tmp = math.sqrt((d / h)) * (math.sqrt(-d) / math.sqrt(-l))
	elif l <= 5.6e+28:
		tmp = (1.0 + (h * (-0.125 * (math.pow(((M * D) / d), 2.0) / l)))) * math.sqrt(((d / h) * (d / l)))
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -8.3e+67)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))));
	elseif (l <= 5.6e+28)
		tmp = Float64(Float64(1.0 + Float64(h * Float64(-0.125 * Float64((Float64(Float64(M * D) / d) ^ 2.0) / l)))) * sqrt(Float64(Float64(d / h) * Float64(d / l))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -8.3e+67)
		tmp = sqrt((d / h)) * (sqrt(-d) / sqrt(-l));
	elseif (l <= 5.6e+28)
		tmp = (1.0 + (h * (-0.125 * ((((M * D) / d) ^ 2.0) / l)))) * sqrt(((d / h) * (d / l)));
	else
		tmp = d * ((h ^ -0.5) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -8.3e+67], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.6e+28], N[(N[(1.0 + N[(h * N[(-0.125 * N[(N[Power[N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -8.299999999999999e67

   1. Initial program 53.8%

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

   3. Simplified 51.5%

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

   5. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 2.9%

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

      4. mul-1-neg 2.9%

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

   7. Simplified 2.9%

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

      1. add-sqr-sqrt 1.3%

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

      2. sqrt-unprod 46.3%

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

      3. sqr-neg 46.3%

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

      4. add-sqr-sqrt 46.3%

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

      5. frac-2neg 46.3%

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

      6. sqrt-div 57.6%

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

   9. Applied egg-rr 57.6%

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

   if -8.299999999999999e67 < l < 5.6000000000000003e28

   1. Initial program 77.3%

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

   3. Simplified 75.9%

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

      1. add-sqr-sqrt 75.9%

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

      2. pow2 75.9%

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

      3. sqrt-prod 75.9%

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

      4. sqrt-pow1 76.6%

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

      5. metadata-eval 76.6%

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

      6. pow1 76.6%

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

      7. frac-times 77.9%

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

      8. associate-/l* 76.6%

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

      9. *-un-lft-identity 76.6%

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

      10. times-frac 76.6%

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

      11. metadata-eval 76.6%

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

   6. Applied egg-rr 76.6%

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

   7. Taylor expanded in h around inf 52.2%

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

      1. sub-neg 52.2%

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

      2. distribute-rgt-in 52.2%

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

      3. lft-mult-inverse 52.2%

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

      4. distribute-lft-neg-in 52.2%

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

      5. metadata-eval 52.2%

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

      6. associate-/r* 54.4%

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

      7. associate-/l* 53.7%

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

      8. unpow2 53.7%

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

      9. unpow2 53.7%

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

      10. unpow2 53.7%

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

      11. times-frac 67.5%

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

      12. swap-sqr 82.2%

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

      13. unpow2 82.2%

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

      14. associate-*r/ 82.3%

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

   9. Simplified 82.3%

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

       1. *-commutative 82.3%

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

       2. sqrt-unprod 69.9%

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

   11. Applied egg-rr 69.9%

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

   if 5.6000000000000003e28 < l

   1. Initial program 49.0%

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

   3. Simplified 47.7%

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

      1. add-sqr-sqrt 47.7%

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

      2. pow2 47.7%

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

      3. sqrt-prod 47.7%

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

      4. sqrt-pow1 48.8%

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

      5. metadata-eval 48.8%

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

      6. pow1 48.8%

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

      7. frac-times 50.2%

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

      8. associate-/l* 48.8%

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

      9. *-un-lft-identity 48.8%

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

      10. times-frac 48.8%

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

      11. metadata-eval 48.8%

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

   6. Applied egg-rr 48.8%

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

   7. Taylor expanded in d around inf 58.2%

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

      1. associate-/r* 59.2%

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

      2. associate-/r* 58.2%

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

      3. unpow-1 58.2%

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

      4. metadata-eval 58.2%

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

      5. pow-sqr 58.2%

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

      6. rem-sqrt-square 58.2%

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

      7. rem-square-sqrt 57.9%

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

      8. fabs-sqr 57.9%

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

      9. rem-square-sqrt 58.2%

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

   9. Simplified 58.2%

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

   10. Taylor expanded in h around -inf 0.0%

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

       1. associate-*l* 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 6.5%

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

       4. neg-mul-1 6.5%

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

       5. *-commutative 6.5%

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

       6. unpow-1 6.5%

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

       7. metadata-eval 6.5%

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

       8. pow-sqr 6.5%

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

       9. rem-sqrt-square 6.5%

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

       10. +-rgt-identity 6.5%

           \[\leadsto d \cdot \left(-\left|{\color{blue}{\left(\ell \cdot h + 0\right)}}^{-0.5}\right|\right) \]

       11. metadata-eval 6.5%

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

       12. associate-+l+ 12.0%

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

       13. fma-undefine 12.0%

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

       14. +-commutative 12.0%

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

       15. rem-square-sqrt 12.0%

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

       16. fabs-sqr 12.0%

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

       17. rem-square-sqrt 12.0%

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

   12. Simplified 6.5%

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

       1. add-sqr-sqrt 3.5%

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

       2. sqrt-unprod 58.2%

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

       3. add-sqr-sqrt 58.0%

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

       4. sqrt-unprod 58.2%

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

       5. pow-prod-up 58.2%

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

       6. metadata-eval 58.2%

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

       7. inv-pow 58.2%

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

       8. associate-/l/ 58.1%

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

       9. add-cbrt-cube 34.7%

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

       10. unpow3 34.7%

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

       11. add-sqr-sqrt 34.7%

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

       12. sqrt-unprod 34.7%

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

       13. pow-prod-up 34.7%

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

       14. metadata-eval 34.7%

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

       15. inv-pow 34.7%

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

       16. associate-/l/ 34.7%

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

       17. add-cbrt-cube 34.7%

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

       18. unpow3 34.7%

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

   14. Applied egg-rr 66.3%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8.3 \cdot 10^{+67}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;\ell \leq 5.6 \cdot 10^{+28}:\\ \;\;\;\;\left(1 + h \cdot \left(-0.125 \cdot \frac{{\left(\frac{M \cdot D}{d}\right)}^{2}}{\ell}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + h \cdot \left(-0.125 \cdot \frac{{\left(\frac{M \cdot D}{d}\right)}^{2}}{\ell}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{if}\;d \leq 2 \cdot 10^{-285}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 2.55 \cdot 10^{-133}:\\ \;\;\;\;d \cdot \frac{h \cdot \left(-0.125 \cdot \frac{{\left(\frac{\frac{d}{M}}{D}\right)}^{-2}}{\ell}\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+90}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (+ 1.0 (* h (* -0.125 (/ (pow (/ (* M D) d) 2.0) l))))
          (sqrt (* (/ d h) (/ d l))))))
   (if (<= d 2e-285)
     t_0
     (if (<= d 2.55e-133)
       (* d (/ (* h (* -0.125 (/ (pow (/ (/ d M) D) -2.0) l))) (sqrt (* l h))))
       (if (<= d 1.7e+90) t_0 (* d (/ (pow h -0.5) (sqrt l))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (1.0 + (h * (-0.125 * (pow(((M * D) / d), 2.0) / l)))) * sqrt(((d / h) * (d / l)));
	double tmp;
	if (d <= 2e-285) {
		tmp = t_0;
	} else if (d <= 2.55e-133) {
		tmp = d * ((h * (-0.125 * (pow(((d / M) / D), -2.0) / l))) / sqrt((l * h)));
	} else if (d <= 1.7e+90) {
		tmp = t_0;
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + (h * ((-0.125d0) * ((((m * d_1) / d) ** 2.0d0) / l)))) * sqrt(((d / h) * (d / l)))
    if (d <= 2d-285) then
        tmp = t_0
    else if (d <= 2.55d-133) then
        tmp = d * ((h * ((-0.125d0) * ((((d / m) / d_1) ** (-2.0d0)) / l))) / sqrt((l * h)))
    else if (d <= 1.7d+90) then
        tmp = t_0
    else
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (1.0 + (h * (-0.125 * (Math.pow(((M * D) / d), 2.0) / l)))) * Math.sqrt(((d / h) * (d / l)));
	double tmp;
	if (d <= 2e-285) {
		tmp = t_0;
	} else if (d <= 2.55e-133) {
		tmp = d * ((h * (-0.125 * (Math.pow(((d / M) / D), -2.0) / l))) / Math.sqrt((l * h)));
	} else if (d <= 1.7e+90) {
		tmp = t_0;
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (1.0 + (h * (-0.125 * (math.pow(((M * D) / d), 2.0) / l)))) * math.sqrt(((d / h) * (d / l)))
	tmp = 0
	if d <= 2e-285:
		tmp = t_0
	elif d <= 2.55e-133:
		tmp = d * ((h * (-0.125 * (math.pow(((d / M) / D), -2.0) / l))) / math.sqrt((l * h)))
	elif d <= 1.7e+90:
		tmp = t_0
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(1.0 + Float64(h * Float64(-0.125 * Float64((Float64(Float64(M * D) / d) ^ 2.0) / l)))) * sqrt(Float64(Float64(d / h) * Float64(d / l))))
	tmp = 0.0
	if (d <= 2e-285)
		tmp = t_0;
	elseif (d <= 2.55e-133)
		tmp = Float64(d * Float64(Float64(h * Float64(-0.125 * Float64((Float64(Float64(d / M) / D) ^ -2.0) / l))) / sqrt(Float64(l * h))));
	elseif (d <= 1.7e+90)
		tmp = t_0;
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (1.0 + (h * (-0.125 * ((((M * D) / d) ^ 2.0) / l)))) * sqrt(((d / h) * (d / l)));
	tmp = 0.0;
	if (d <= 2e-285)
		tmp = t_0;
	elseif (d <= 2.55e-133)
		tmp = d * ((h * (-0.125 * ((((d / M) / D) ^ -2.0) / l))) / sqrt((l * h)));
	elseif (d <= 1.7e+90)
		tmp = t_0;
	else
		tmp = d * ((h ^ -0.5) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(1.0 + N[(h * N[(-0.125 * N[(N[Power[N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, 2e-285], t$95$0, If[LessEqual[d, 2.55e-133], N[(d * N[(N[(h * N[(-0.125 * N[(N[Power[N[(N[(d / M), $MachinePrecision] / D), $MachinePrecision], -2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.7e+90], t$95$0, N[(d * N[(N[Power[h, -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < 2.00000000000000015e-285 or 2.54999999999999995e-133 < d < 1.70000000000000009e90

   1. Initial program 75.5%

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

   3. Simplified 73.7%

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

      1. add-sqr-sqrt 73.7%

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

      2. pow2 73.7%

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

      3. sqrt-prod 73.7%

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

      4. sqrt-pow1 74.7%

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

      5. metadata-eval 74.7%

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

      6. pow1 74.7%

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

      7. frac-times 76.5%

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

      8. associate-/l* 74.7%

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

      9. *-un-lft-identity 74.7%

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

      10. times-frac 74.7%

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

      11. metadata-eval 74.7%

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

   6. Applied egg-rr 74.7%

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

   7. Taylor expanded in h around inf 53.9%

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

      1. sub-neg 53.9%

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

      2. distribute-rgt-in 53.9%

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

      3. lft-mult-inverse 53.9%

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

      4. distribute-lft-neg-in 53.9%

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

      5. metadata-eval 53.9%

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

      6. associate-/r* 55.2%

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

      7. associate-/l* 53.4%

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

      8. unpow2 53.4%

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

      9. unpow2 53.4%

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

      10. unpow2 53.4%

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

      11. times-frac 61.8%

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

      12. swap-sqr 79.1%

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

      13. unpow2 79.1%

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

      14. associate-*r/ 79.1%

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

   9. Simplified 79.1%

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

       1. *-commutative 79.1%

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

       2. sqrt-unprod 66.0%

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

   11. Applied egg-rr 66.0%

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

   if 2.00000000000000015e-285 < d < 2.54999999999999995e-133

   1. Initial program 36.9%

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

   3. Simplified 34.3%

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

   5. Taylor expanded in M around inf 21.9%

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

      1. associate-*r/ 21.9%

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

      2. associate-*r* 22.0%

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

      3. associate-*r* 22.0%

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

      4. associate-*l/ 22.1%

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

      5. associate-*r/ 22.1%

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

      6. *-commutative 22.1%

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

      7. associate-*l* 22.1%

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

   7. Simplified 32.8%

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

      1. pow1 32.8%

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

      2. associate-*r* 32.9%

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

      3. sqrt-unprod 17.1%

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

      4. associate-*l/ 19.5%

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

      5. *-commutative 19.5%

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

   9. Applied egg-rr 19.5%

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

       1. unpow1 19.5%

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

       2. associate-*r/ 16.9%

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

       3. associate-*l/ 11.7%

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

       4. unpow2 11.7%

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

       5. associate-/l* 8.9%

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

       6. associate-*r/ 8.9%

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

       7. associate-/l* 8.9%

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

   11. Simplified 8.9%

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

       1. pow1 8.9%

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

       2. associate-/l/ 8.9%

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

       3. *-commutative 8.9%

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

       4. sqrt-div 14.4%

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

       5. sqrt-pow1 45.8%

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

       6. metadata-eval 45.8%

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

       7. pow1 45.8%

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

       8. *-commutative 45.8%

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

       9. *-commutative 45.8%

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

       10. associate-*r/ 45.8%

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

       11. associate-*r/ 45.9%

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

   13. Applied egg-rr 45.9%

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

       1. unpow1 45.9%

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

       2. *-commutative 45.9%

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

       3. associate-*r/ 56.3%

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

       4. associate-*l/ 53.9%

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

       5. associate-*l/ 61.7%

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

       6. associate-/l* 61.7%

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

   15. Simplified 61.7%

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

   if 1.70000000000000009e90 < d

   1. Initial program 55.6%

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

   3. Simplified 55.7%

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

      1. add-sqr-sqrt 55.7%

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

      2. pow2 55.7%

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

      3. sqrt-prod 55.7%

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

      4. sqrt-pow1 57.7%

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

      5. metadata-eval 57.7%

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

      6. pow1 57.7%

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

      7. frac-times 57.5%

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

      8. associate-/l* 57.7%

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

      9. *-un-lft-identity 57.7%

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

      10. times-frac 57.7%

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

      11. metadata-eval 57.7%

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

   6. Applied egg-rr 57.7%

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

   7. Taylor expanded in d around inf 74.8%

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

      1. associate-/r* 75.3%

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

      2. associate-/r* 74.8%

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

      3. unpow-1 74.8%

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

      4. metadata-eval 74.8%

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

      5. pow-sqr 74.8%

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

      6. rem-sqrt-square 74.8%

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

      7. rem-square-sqrt 74.4%

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

      8. fabs-sqr 74.4%

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

      9. rem-square-sqrt 74.8%

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

   9. Simplified 74.8%

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

   10. Taylor expanded in h around -inf 0.0%

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

       1. associate-*l* 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 5.4%

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

       4. neg-mul-1 5.4%

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

       5. *-commutative 5.4%

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

       6. unpow-1 5.4%

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

       7. metadata-eval 5.4%

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

       8. pow-sqr 5.4%

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

       9. rem-sqrt-square 5.4%

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

       10. +-rgt-identity 5.4%

           \[\leadsto d \cdot \left(-\left|{\color{blue}{\left(\ell \cdot h + 0\right)}}^{-0.5}\right|\right) \]

       11. metadata-eval 5.4%

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

       12. associate-+l+ 12.5%

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

       13. fma-undefine 12.5%

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

       14. +-commutative 12.5%

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

       15. rem-square-sqrt 12.5%

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

       16. fabs-sqr 12.5%

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

       17. rem-square-sqrt 12.5%

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

   12. Simplified 5.4%

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

       1. add-sqr-sqrt 0.2%

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

       2. sqrt-unprod 74.8%

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

       3. add-sqr-sqrt 74.6%

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

       4. sqrt-unprod 74.8%

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

       5. pow-prod-up 74.8%

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

       6. metadata-eval 74.8%

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

       7. inv-pow 74.8%

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

       8. associate-/l/ 74.7%

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

       9. add-cbrt-cube 34.5%

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

       10. unpow3 34.4%

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

       11. add-sqr-sqrt 34.4%

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

       12. sqrt-unprod 34.4%

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

       13. pow-prod-up 34.4%

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

       14. metadata-eval 34.4%

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

       15. inv-pow 34.4%

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

       16. associate-/l/ 34.4%

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

       17. add-cbrt-cube 34.4%

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

       18. unpow3 34.4%

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

   14. Applied egg-rr 80.3%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 2 \cdot 10^{-285}:\\ \;\;\;\;\left(1 + h \cdot \left(-0.125 \cdot \frac{{\left(\frac{M \cdot D}{d}\right)}^{2}}{\ell}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 2.55 \cdot 10^{-133}:\\ \;\;\;\;d \cdot \frac{h \cdot \left(-0.125 \cdot \frac{{\left(\frac{\frac{d}{M}}{D}\right)}^{-2}}{\ell}\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+90}:\\ \;\;\;\;\left(1 + h \cdot \left(-0.125 \cdot \frac{{\left(\frac{M \cdot D}{d}\right)}^{2}}{\ell}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 2.7 \cdot 10^{-282}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 5.2 \cdot 10^{-29}:\\ \;\;\;\;d \cdot \frac{h \cdot \left(-0.125 \cdot \frac{{\left(\frac{\frac{d}{M}}{D}\right)}^{-2}}{\ell}\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d 2.7e-282)
   (* (sqrt (/ d h)) (sqrt (/ d l)))
   (if (<= d 5.2e-29)
     (* d (/ (* h (* -0.125 (/ (pow (/ (/ d M) D) -2.0) l))) (sqrt (* l h))))
     (* d (/ (pow h -0.5) (sqrt l))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 2.7e-282) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (d <= 5.2e-29) {
		tmp = d * ((h * (-0.125 * (pow(((d / M) / D), -2.0) / l))) / sqrt((l * h)));
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= 2.7d-282) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else if (d <= 5.2d-29) then
        tmp = d * ((h * ((-0.125d0) * ((((d / m) / d_1) ** (-2.0d0)) / l))) / sqrt((l * h)))
    else
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 2.7e-282) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (d <= 5.2e-29) {
		tmp = d * ((h * (-0.125 * (Math.pow(((d / M) / D), -2.0) / l))) / Math.sqrt((l * h)));
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= 2.7e-282:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	elif d <= 5.2e-29:
		tmp = d * ((h * (-0.125 * (math.pow(((d / M) / D), -2.0) / l))) / math.sqrt((l * h)))
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= 2.7e-282)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (d <= 5.2e-29)
		tmp = Float64(d * Float64(Float64(h * Float64(-0.125 * Float64((Float64(Float64(d / M) / D) ^ -2.0) / l))) / sqrt(Float64(l * h))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= 2.7e-282)
		tmp = sqrt((d / h)) * sqrt((d / l));
	elseif (d <= 5.2e-29)
		tmp = d * ((h * (-0.125 * ((((d / M) / D) ^ -2.0) / l))) / sqrt((l * h)));
	else
		tmp = d * ((h ^ -0.5) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, 2.7e-282], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5.2e-29], N[(d * N[(N[(h * N[(-0.125 * N[(N[Power[N[(N[(d / M), $MachinePrecision] / D), $MachinePrecision], -2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < 2.69999999999999982e-282

   1. Initial program 69.0%

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

   3. Simplified 66.4%

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

   5. Taylor expanded in d around inf 44.5%

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

   if 2.69999999999999982e-282 < d < 5.2000000000000004e-29

   1. Initial program 56.1%

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

   3. Simplified 54.5%

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

   5. Taylor expanded in M around inf 33.4%

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

      1. associate-*r/ 33.4%

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

      2. associate-*r* 33.4%

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

      3. associate-*r* 33.4%

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

      4. associate-*l/ 33.5%

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

      5. associate-*r/ 33.5%

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

      6. *-commutative 33.5%

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

      7. associate-*l* 33.5%

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

   7. Simplified 43.9%

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

      1. pow1 43.9%

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

      2. associate-*r* 43.9%

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

      3. sqrt-unprod 33.6%

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

      4. associate-*l/ 35.2%

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

      5. *-commutative 35.2%

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

   9. Applied egg-rr 35.2%

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

       1. unpow1 35.2%

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

       2. associate-*r/ 31.8%

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

       3. associate-*l/ 28.5%

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

       4. unpow2 28.5%

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

       5. associate-/l* 26.6%

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

       6. associate-*r/ 26.6%

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

       7. associate-/l* 26.6%

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

   11. Simplified 26.6%

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

       1. pow1 26.6%

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

       2. associate-/l/ 28.3%

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

       3. *-commutative 28.3%

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

       4. sqrt-div 31.7%

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

       5. sqrt-pow1 52.0%

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

       6. metadata-eval 52.0%

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

       7. pow1 52.0%

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

       8. *-commutative 52.0%

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

       9. *-commutative 52.0%

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

       10. associate-*r/ 52.0%

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

       11. associate-*r/ 52.0%

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

   13. Applied egg-rr 52.0%

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

       1. unpow1 52.0%

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

       2. *-commutative 52.0%

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

       3. associate-*r/ 58.8%

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

       4. associate-*l/ 57.3%

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

       5. associate-*l/ 62.3%

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

       6. associate-/l* 60.8%

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

   15. Simplified 60.9%

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

   if 5.2000000000000004e-29 < d

   1. Initial program 68.0%

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

   3. Simplified 68.1%

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

      1. add-sqr-sqrt 68.1%

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

      2. pow2 68.1%

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

      3. sqrt-prod 68.1%

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

      4. sqrt-pow1 70.2%

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

      5. metadata-eval 70.2%

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

      6. pow1 70.2%

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

      7. frac-times 70.1%

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

      8. associate-/l* 70.2%

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

      9. *-un-lft-identity 70.2%

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

      10. times-frac 70.2%

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

      11. metadata-eval 70.2%

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

   6. Applied egg-rr 70.2%

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

   7. Taylor expanded in d around inf 62.0%

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

      1. associate-/r* 62.9%

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

      2. associate-/r* 62.0%

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

      3. unpow-1 62.0%

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

      4. metadata-eval 62.0%

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

      5. pow-sqr 62.0%

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

      6. rem-sqrt-square 62.0%

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

      7. rem-square-sqrt 61.6%

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

      8. fabs-sqr 61.6%

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

      9. rem-square-sqrt 62.0%

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

   9. Simplified 62.0%

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

   10. Taylor expanded in h around -inf 0.0%

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

       1. associate-*l* 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 6.7%

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

       4. neg-mul-1 6.7%

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

       5. *-commutative 6.7%

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

       6. unpow-1 6.7%

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

       7. metadata-eval 6.7%

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

       8. pow-sqr 6.7%

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

       9. rem-sqrt-square 6.7%

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

       10. +-rgt-identity 6.7%

           \[\leadsto d \cdot \left(-\left|{\color{blue}{\left(\ell \cdot h + 0\right)}}^{-0.5}\right|\right) \]

       11. metadata-eval 6.7%

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

       12. associate-+l+ 19.4%

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

       13. fma-undefine 19.4%

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

       14. +-commutative 19.4%

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

       15. rem-square-sqrt 19.4%

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

       16. fabs-sqr 19.4%

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

       17. rem-square-sqrt 19.4%

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

   12. Simplified 6.7%

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

       1. add-sqr-sqrt 0.2%

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

       2. sqrt-unprod 62.0%

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

       3. add-sqr-sqrt 61.8%

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

       4. sqrt-unprod 62.0%

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

       5. pow-prod-up 62.0%

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

       6. metadata-eval 62.0%

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

       7. inv-pow 62.0%

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

       8. associate-/l/ 61.9%

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

       9. add-cbrt-cube 29.4%

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

       10. unpow3 29.4%

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

       11. add-sqr-sqrt 29.4%

           \[\leadsto d \cdot \sqrt{\left(-\sqrt{\sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}}}\right) \cdot \left(-\color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-0.5}} \cdot \sqrt{{\left(\ell \cdot h\right)}^{-0.5}}}\right)} \]

       12. sqrt-unprod 29.4%

           \[\leadsto d \cdot \sqrt{\left(-\sqrt{\sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}}}\right) \cdot \left(-\color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-0.5} \cdot {\left(\ell \cdot h\right)}^{-0.5}}}\right)} \]

       13. pow-prod-up 29.4%

           \[\leadsto d \cdot \sqrt{\left(-\sqrt{\sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}}}\right) \cdot \left(-\sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{\left(-0.5 + -0.5\right)}}}\right)} \]

       14. metadata-eval 29.4%

           \[\leadsto d \cdot \sqrt{\left(-\sqrt{\sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}}}\right) \cdot \left(-\sqrt{{\left(\ell \cdot h\right)}^{\color{blue}{-1}}}\right)} \]

       15. inv-pow 29.4%

           \[\leadsto d \cdot \sqrt{\left(-\sqrt{\sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}}}\right) \cdot \left(-\sqrt{\color{blue}{\frac{1}{\ell \cdot h}}}\right)} \]

       16. associate-/l/ 29.4%

           \[\leadsto d \cdot \sqrt{\left(-\sqrt{\sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}}}\right) \cdot \left(-\sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}}\right)} \]

       17. add-cbrt-cube 29.4%

           \[\leadsto d \cdot \sqrt{\left(-\sqrt{\sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}}}\right) \cdot \left(-\sqrt{\color{blue}{\sqrt[3]{\left(\frac{\frac{1}{h}}{\ell} \cdot \frac{\frac{1}{h}}{\ell}\right) \cdot \frac{\frac{1}{h}}{\ell}}}}\right)} \]

       18. unpow3 29.4%

           \[\leadsto d \cdot \sqrt{\left(-\sqrt{\sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}}}\right) \cdot \left(-\sqrt{\sqrt[3]{\color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}}}}\right)} \]

   14. Applied egg-rr 70.0%

       \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 46.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 4.9 \cdot 10^{-265}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d 4.9e-265)
   (* (sqrt (/ d h)) (sqrt (/ d l)))
   (* d (/ (pow h -0.5) (sqrt l)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 4.9e-265) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= 4.9d-265) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 4.9e-265) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= 4.9e-265:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= 4.9e-265)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= 4.9e-265)
		tmp = sqrt((d / h)) * sqrt((d / l));
	else
		tmp = d * ((h ^ -0.5) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, 4.9e-265], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 4.89999999999999999e-265

   1. Initial program 68.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 65.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 43.4%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

   if 4.89999999999999999e-265 < d

   1. Initial program 63.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 63.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 63.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 63.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 63.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 64.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. metadata-eval 64.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. pow1 64.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. frac-times 64.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. associate-/l* 64.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. *-un-lft-identity 64.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(M \cdot \frac{\color{blue}{1 \cdot D}}{2 \cdot d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. times-frac 64.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(M \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{D}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. metadata-eval 64.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(M \cdot \left(\color{blue}{0.5} \cdot \frac{D}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 64.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in d around inf 47.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. associate-/r* 47.9%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

      2. associate-/r* 47.4%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      3. unpow-1 47.4%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

      4. metadata-eval 47.4%

         \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      5. pow-sqr 47.4%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

      6. rem-sqrt-square 47.4%

         \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

      7. rem-square-sqrt 47.2%

         \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

      8. fabs-sqr 47.2%

         \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

      9. rem-square-sqrt 47.4%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 47.4%

      \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Taylor expanded in h around -inf 0.0%

       \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. associate-*l* 0.0%

          \[\leadsto \color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

       2. unpow2 0.0%

          \[\leadsto d \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

       3. rem-square-sqrt 9.4%

          \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

       4. neg-mul-1 9.4%

          \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

       5. *-commutative 9.4%

          \[\leadsto d \cdot \left(-\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right) \]

       6. unpow-1 9.4%

          \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}}\right) \]

       7. metadata-eval 9.4%

          \[\leadsto d \cdot \left(-\sqrt{{\left(\ell \cdot h\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \]

       8. pow-sqr 9.4%

          \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-0.5} \cdot {\left(\ell \cdot h\right)}^{-0.5}}}\right) \]

       9. rem-sqrt-square 9.4%

          \[\leadsto d \cdot \left(-\color{blue}{\left|{\left(\ell \cdot h\right)}^{-0.5}\right|}\right) \]

       10. +-rgt-identity 9.4%

           \[\leadsto d \cdot \left(-\left|{\color{blue}{\left(\ell \cdot h + 0\right)}}^{-0.5}\right|\right) \]

       11. metadata-eval 9.4%

           \[\leadsto d \cdot \left(-\left|{\left(\ell \cdot h + \color{blue}{\left(1 + -1\right)}\right)}^{-0.5}\right|\right) \]

       12. associate-+l+ 24.7%

           \[\leadsto d \cdot \left(-\left|{\color{blue}{\left(\left(\ell \cdot h + 1\right) + -1\right)}}^{-0.5}\right|\right) \]

       13. fma-undefine 24.7%

           \[\leadsto d \cdot \left(-\left|{\left(\color{blue}{\mathsf{fma}\left(\ell, h, 1\right)} + -1\right)}^{-0.5}\right|\right) \]

       14. +-commutative 24.7%

           \[\leadsto d \cdot \left(-\left|{\color{blue}{\left(-1 + \mathsf{fma}\left(\ell, h, 1\right)\right)}}^{-0.5}\right|\right) \]

       15. rem-square-sqrt 24.7%

           \[\leadsto d \cdot \left(-\left|\color{blue}{\sqrt{{\left(-1 + \mathsf{fma}\left(\ell, h, 1\right)\right)}^{-0.5}} \cdot \sqrt{{\left(-1 + \mathsf{fma}\left(\ell, h, 1\right)\right)}^{-0.5}}}\right|\right) \]

       16. fabs-sqr 24.7%

           \[\leadsto d \cdot \left(-\color{blue}{\sqrt{{\left(-1 + \mathsf{fma}\left(\ell, h, 1\right)\right)}^{-0.5}} \cdot \sqrt{{\left(-1 + \mathsf{fma}\left(\ell, h, 1\right)\right)}^{-0.5}}}\right) \]

       17. rem-square-sqrt 24.7%

           \[\leadsto d \cdot \left(-\color{blue}{{\left(-1 + \mathsf{fma}\left(\ell, h, 1\right)\right)}^{-0.5}}\right) \]

   12. Simplified 9.4%

       \[\leadsto \color{blue}{d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)} \]
   13. Step-by-step derivation

       1. add-sqr-sqrt 1.8%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{-{\left(\ell \cdot h\right)}^{-0.5}} \cdot \sqrt{-{\left(\ell \cdot h\right)}^{-0.5}}\right)} \]

       2. sqrt-unprod 47.4%

          \[\leadsto d \cdot \color{blue}{\sqrt{\left(-{\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)}} \]

       3. add-sqr-sqrt 47.3%

          \[\leadsto d \cdot \sqrt{\left(-\color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-0.5}} \cdot \sqrt{{\left(\ell \cdot h\right)}^{-0.5}}}\right) \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)} \]

       4. sqrt-unprod 47.4%

          \[\leadsto d \cdot \sqrt{\left(-\color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-0.5} \cdot {\left(\ell \cdot h\right)}^{-0.5}}}\right) \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)} \]

       5. pow-prod-up 47.4%

          \[\leadsto d \cdot \sqrt{\left(-\sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{\left(-0.5 + -0.5\right)}}}\right) \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)} \]

       6. metadata-eval 47.4%

          \[\leadsto d \cdot \sqrt{\left(-\sqrt{{\left(\ell \cdot h\right)}^{\color{blue}{-1}}}\right) \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)} \]

       7. inv-pow 47.4%

          \[\leadsto d \cdot \sqrt{\left(-\sqrt{\color{blue}{\frac{1}{\ell \cdot h}}}\right) \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)} \]

       8. associate-/l/ 47.4%

          \[\leadsto d \cdot \sqrt{\left(-\sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}}\right) \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)} \]

       9. add-cbrt-cube 24.6%

          \[\leadsto d \cdot \sqrt{\left(-\sqrt{\color{blue}{\sqrt[3]{\left(\frac{\frac{1}{h}}{\ell} \cdot \frac{\frac{1}{h}}{\ell}\right) \cdot \frac{\frac{1}{h}}{\ell}}}}\right) \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)} \]

       10. unpow3 24.6%

           \[\leadsto d \cdot \sqrt{\left(-\sqrt{\sqrt[3]{\color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}}}}\right) \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)} \]

       11. add-sqr-sqrt 24.6%

           \[\leadsto d \cdot \sqrt{\left(-\sqrt{\sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}}}\right) \cdot \left(-\color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-0.5}} \cdot \sqrt{{\left(\ell \cdot h\right)}^{-0.5}}}\right)} \]

       12. sqrt-unprod 24.6%

           \[\leadsto d \cdot \sqrt{\left(-\sqrt{\sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}}}\right) \cdot \left(-\color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-0.5} \cdot {\left(\ell \cdot h\right)}^{-0.5}}}\right)} \]

       13. pow-prod-up 24.6%

           \[\leadsto d \cdot \sqrt{\left(-\sqrt{\sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}}}\right) \cdot \left(-\sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{\left(-0.5 + -0.5\right)}}}\right)} \]

       14. metadata-eval 24.6%

           \[\leadsto d \cdot \sqrt{\left(-\sqrt{\sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}}}\right) \cdot \left(-\sqrt{{\left(\ell \cdot h\right)}^{\color{blue}{-1}}}\right)} \]

       15. inv-pow 24.6%

           \[\leadsto d \cdot \sqrt{\left(-\sqrt{\sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}}}\right) \cdot \left(-\sqrt{\color{blue}{\frac{1}{\ell \cdot h}}}\right)} \]

       16. associate-/l/ 24.6%

           \[\leadsto d \cdot \sqrt{\left(-\sqrt{\sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}}}\right) \cdot \left(-\sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}}\right)} \]

       17. add-cbrt-cube 24.6%

           \[\leadsto d \cdot \sqrt{\left(-\sqrt{\sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}}}\right) \cdot \left(-\sqrt{\color{blue}{\sqrt[3]{\left(\frac{\frac{1}{h}}{\ell} \cdot \frac{\frac{1}{h}}{\ell}\right) \cdot \frac{\frac{1}{h}}{\ell}}}}\right)} \]

       18. unpow3 24.6%

           \[\leadsto d \cdot \sqrt{\left(-\sqrt{\sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}}}\right) \cdot \left(-\sqrt{\sqrt[3]{\color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}}}}\right)} \]

   14. Applied egg-rr 53.5%

       \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 47.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 6 \cdot 10^{-218}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l 6e-218)
   (* (- d) (pow (* l h) -0.5))
   (* d (/ (pow h -0.5) (sqrt l)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 6e-218) {
		tmp = -d * pow((l * h), -0.5);
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= 6d-218) then
        tmp = -d * ((l * h) ** (-0.5d0))
    else
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    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 <= 6e-218) {
		tmp = -d * Math.pow((l * h), -0.5);
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= 6e-218:
		tmp = -d * math.pow((l * h), -0.5)
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 6e-218)
		tmp = Float64(Float64(-d) * (Float64(l * h) ^ -0.5));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= 6e-218)
		tmp = -d * ((l * h) ^ -0.5);
	else
		tmp = d * ((h ^ -0.5) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, 6e-218], N[((-d) * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.9999999999999997e-218

   1. Initial program 68.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 66.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 9.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   6. Taylor expanded in h around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. associate-*l* 0.0%

         \[\leadsto \color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. unpow2 0.0%

         \[\leadsto d \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. rem-square-sqrt 41.2%

         \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. neg-mul-1 41.2%

         \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      5. unpow-1 41.2%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \]

      6. metadata-eval 41.2%

         \[\leadsto d \cdot \left(-\sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \]

      7. pow-sqr 41.2%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      8. rem-sqrt-square 41.4%

         \[\leadsto d \cdot \left(-\color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \]

      9. rem-square-sqrt 41.3%

         \[\leadsto d \cdot \left(-\left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right|\right) \]

      10. fabs-sqr 41.3%

          \[\leadsto d \cdot \left(-\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      11. rem-square-sqrt 41.4%

          \[\leadsto d \cdot \left(-\color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \]

   8. Simplified 41.4%

      \[\leadsto \color{blue}{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \]

   if 5.9999999999999997e-218 < l

   1. Initial program 63.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 61.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 61.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 61.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 61.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 63.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. metadata-eval 63.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. pow1 63.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. frac-times 64.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. associate-/l* 63.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. *-un-lft-identity 63.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(M \cdot \frac{\color{blue}{1 \cdot D}}{2 \cdot d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. times-frac 63.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(M \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{D}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. metadata-eval 63.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(M \cdot \left(\color{blue}{0.5} \cdot \frac{D}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 63.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in d around inf 50.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. associate-/r* 51.3%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

      2. associate-/r* 50.8%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      3. unpow-1 50.8%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

      4. metadata-eval 50.8%

         \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      5. pow-sqr 50.8%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

      6. rem-sqrt-square 50.8%

         \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

      7. rem-square-sqrt 50.5%

         \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

      8. fabs-sqr 50.5%

         \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

      9. rem-square-sqrt 50.8%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 50.8%

      \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Taylor expanded in h around -inf 0.0%

       \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. associate-*l* 0.0%

          \[\leadsto \color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

       2. unpow2 0.0%

          \[\leadsto d \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

       3. rem-square-sqrt 5.5%

          \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

       4. neg-mul-1 5.5%

          \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

       5. *-commutative 5.5%

          \[\leadsto d \cdot \left(-\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right) \]

       6. unpow-1 5.5%

          \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}}\right) \]

       7. metadata-eval 5.5%

          \[\leadsto d \cdot \left(-\sqrt{{\left(\ell \cdot h\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \]

       8. pow-sqr 5.5%

          \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-0.5} \cdot {\left(\ell \cdot h\right)}^{-0.5}}}\right) \]

       9. rem-sqrt-square 5.5%

          \[\leadsto d \cdot \left(-\color{blue}{\left|{\left(\ell \cdot h\right)}^{-0.5}\right|}\right) \]

       10. +-rgt-identity 5.5%

           \[\leadsto d \cdot \left(-\left|{\color{blue}{\left(\ell \cdot h + 0\right)}}^{-0.5}\right|\right) \]

       11. metadata-eval 5.5%

           \[\leadsto d \cdot \left(-\left|{\left(\ell \cdot h + \color{blue}{\left(1 + -1\right)}\right)}^{-0.5}\right|\right) \]

       12. associate-+l+ 18.0%

           \[\leadsto d \cdot \left(-\left|{\color{blue}{\left(\left(\ell \cdot h + 1\right) + -1\right)}}^{-0.5}\right|\right) \]

       13. fma-undefine 18.0%

           \[\leadsto d \cdot \left(-\left|{\left(\color{blue}{\mathsf{fma}\left(\ell, h, 1\right)} + -1\right)}^{-0.5}\right|\right) \]

       14. +-commutative 18.0%

           \[\leadsto d \cdot \left(-\left|{\color{blue}{\left(-1 + \mathsf{fma}\left(\ell, h, 1\right)\right)}}^{-0.5}\right|\right) \]

       15. rem-square-sqrt 18.0%

           \[\leadsto d \cdot \left(-\left|\color{blue}{\sqrt{{\left(-1 + \mathsf{fma}\left(\ell, h, 1\right)\right)}^{-0.5}} \cdot \sqrt{{\left(-1 + \mathsf{fma}\left(\ell, h, 1\right)\right)}^{-0.5}}}\right|\right) \]

       16. fabs-sqr 18.0%

           \[\leadsto d \cdot \left(-\color{blue}{\sqrt{{\left(-1 + \mathsf{fma}\left(\ell, h, 1\right)\right)}^{-0.5}} \cdot \sqrt{{\left(-1 + \mathsf{fma}\left(\ell, h, 1\right)\right)}^{-0.5}}}\right) \]

       17. rem-square-sqrt 18.0%

           \[\leadsto d \cdot \left(-\color{blue}{{\left(-1 + \mathsf{fma}\left(\ell, h, 1\right)\right)}^{-0.5}}\right) \]

   12. Simplified 5.5%

       \[\leadsto \color{blue}{d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)} \]
   13. Step-by-step derivation

       1. add-sqr-sqrt 2.0%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{-{\left(\ell \cdot h\right)}^{-0.5}} \cdot \sqrt{-{\left(\ell \cdot h\right)}^{-0.5}}\right)} \]

       2. sqrt-unprod 50.8%

          \[\leadsto d \cdot \color{blue}{\sqrt{\left(-{\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)}} \]

       3. add-sqr-sqrt 50.7%

          \[\leadsto d \cdot \sqrt{\left(-\color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-0.5}} \cdot \sqrt{{\left(\ell \cdot h\right)}^{-0.5}}}\right) \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)} \]

       4. sqrt-unprod 50.8%

          \[\leadsto d \cdot \sqrt{\left(-\color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-0.5} \cdot {\left(\ell \cdot h\right)}^{-0.5}}}\right) \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)} \]

       5. pow-prod-up 50.8%

          \[\leadsto d \cdot \sqrt{\left(-\sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{\left(-0.5 + -0.5\right)}}}\right) \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)} \]

       6. metadata-eval 50.8%

          \[\leadsto d \cdot \sqrt{\left(-\sqrt{{\left(\ell \cdot h\right)}^{\color{blue}{-1}}}\right) \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)} \]

       7. inv-pow 50.8%

          \[\leadsto d \cdot \sqrt{\left(-\sqrt{\color{blue}{\frac{1}{\ell \cdot h}}}\right) \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)} \]

       8. associate-/l/ 50.7%

          \[\leadsto d \cdot \sqrt{\left(-\sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}}\right) \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)} \]

       9. add-cbrt-cube 27.5%

          \[\leadsto d \cdot \sqrt{\left(-\sqrt{\color{blue}{\sqrt[3]{\left(\frac{\frac{1}{h}}{\ell} \cdot \frac{\frac{1}{h}}{\ell}\right) \cdot \frac{\frac{1}{h}}{\ell}}}}\right) \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)} \]

       10. unpow3 27.5%

           \[\leadsto d \cdot \sqrt{\left(-\sqrt{\sqrt[3]{\color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}}}}\right) \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)} \]

       11. add-sqr-sqrt 27.5%

           \[\leadsto d \cdot \sqrt{\left(-\sqrt{\sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}}}\right) \cdot \left(-\color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-0.5}} \cdot \sqrt{{\left(\ell \cdot h\right)}^{-0.5}}}\right)} \]

       12. sqrt-unprod 27.5%

           \[\leadsto d \cdot \sqrt{\left(-\sqrt{\sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}}}\right) \cdot \left(-\color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-0.5} \cdot {\left(\ell \cdot h\right)}^{-0.5}}}\right)} \]

       13. pow-prod-up 27.5%

           \[\leadsto d \cdot \sqrt{\left(-\sqrt{\sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}}}\right) \cdot \left(-\sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{\left(-0.5 + -0.5\right)}}}\right)} \]

       14. metadata-eval 27.5%

           \[\leadsto d \cdot \sqrt{\left(-\sqrt{\sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}}}\right) \cdot \left(-\sqrt{{\left(\ell \cdot h\right)}^{\color{blue}{-1}}}\right)} \]

       15. inv-pow 27.5%

           \[\leadsto d \cdot \sqrt{\left(-\sqrt{\sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}}}\right) \cdot \left(-\sqrt{\color{blue}{\frac{1}{\ell \cdot h}}}\right)} \]

       16. associate-/l/ 27.4%

           \[\leadsto d \cdot \sqrt{\left(-\sqrt{\sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}}}\right) \cdot \left(-\sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}}\right)} \]

       17. add-cbrt-cube 27.5%

           \[\leadsto d \cdot \sqrt{\left(-\sqrt{\sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}}}\right) \cdot \left(-\sqrt{\color{blue}{\sqrt[3]{\left(\frac{\frac{1}{h}}{\ell} \cdot \frac{\frac{1}{h}}{\ell}\right) \cdot \frac{\frac{1}{h}}{\ell}}}}\right)} \]

       18. unpow3 27.5%

           \[\leadsto d \cdot \sqrt{\left(-\sqrt{\sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}}}\right) \cdot \left(-\sqrt{\sqrt[3]{\color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}}}}\right)} \]

   14. Applied egg-rr 56.9%

       \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6 \cdot 10^{-218}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 43.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4 \cdot 10^{-177}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l 4e-177) (* (- d) (pow (* l h) -0.5)) (* d (sqrt (/ (/ 1.0 l) h)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 4e-177) {
		tmp = -d * pow((l * h), -0.5);
	} else {
		tmp = d * sqrt(((1.0 / l) / h));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= 4d-177) then
        tmp = -d * ((l * h) ** (-0.5d0))
    else
        tmp = d * sqrt(((1.0d0 / l) / h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 4e-177) {
		tmp = -d * Math.pow((l * h), -0.5);
	} else {
		tmp = d * Math.sqrt(((1.0 / l) / h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= 4e-177:
		tmp = -d * math.pow((l * h), -0.5)
	else:
		tmp = d * math.sqrt(((1.0 / l) / h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 4e-177)
		tmp = Float64(Float64(-d) * (Float64(l * h) ^ -0.5));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / l) / h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= 4e-177)
		tmp = -d * ((l * h) ^ -0.5);
	else
		tmp = d * sqrt(((1.0 / l) / h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, 4e-177], N[((-d) * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.99999999999999981e-177

   1. Initial program 68.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 66.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 10.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   6. Taylor expanded in h around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. associate-*l* 0.0%

         \[\leadsto \color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. unpow2 0.0%

         \[\leadsto d \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. rem-square-sqrt 40.0%

         \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. neg-mul-1 40.0%

         \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      5. unpow-1 40.0%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \]

      6. metadata-eval 40.0%

         \[\leadsto d \cdot \left(-\sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \]

      7. pow-sqr 40.0%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      8. rem-sqrt-square 40.2%

         \[\leadsto d \cdot \left(-\color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \]

      9. rem-square-sqrt 40.1%

         \[\leadsto d \cdot \left(-\left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right|\right) \]

      10. fabs-sqr 40.1%

          \[\leadsto d \cdot \left(-\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      11. rem-square-sqrt 40.2%

          \[\leadsto d \cdot \left(-\color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \]

   8. Simplified 40.2%

      \[\leadsto \color{blue}{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \]

   if 3.99999999999999981e-177 < l

   1. Initial program 62.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 61.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 52.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. add-cbrt-cube 29.1%

         \[\leadsto d \cdot \sqrt{\color{blue}{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}} \]

      2. pow3 29.1%

         \[\leadsto d \cdot \sqrt{\sqrt[3]{\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{3}}}} \]

      3. associate-/r* 29.0%

         \[\leadsto d \cdot \sqrt{\sqrt[3]{{\color{blue}{\left(\frac{\frac{1}{h}}{\ell}\right)}}^{3}}} \]

   7. Applied egg-rr 29.0%

      \[\leadsto d \cdot \sqrt{\color{blue}{\sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}}}} \]

   8. Taylor expanded in h around 0 52.9%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. associate-/l/ 53.6%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   10. Simplified 53.6%

       \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4 \cdot 10^{-177}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 43.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.6 \cdot 10^{-177}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l 4.6e-177)
   (* (- d) (pow (* l h) -0.5))
   (* d (sqrt (/ (/ 1.0 h) l)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 4.6e-177) {
		tmp = -d * pow((l * h), -0.5);
	} else {
		tmp = d * sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= 4.6d-177) then
        tmp = -d * ((l * h) ** (-0.5d0))
    else
        tmp = d * sqrt(((1.0d0 / h) / l))
    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 <= 4.6e-177) {
		tmp = -d * Math.pow((l * h), -0.5);
	} else {
		tmp = d * Math.sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= 4.6e-177:
		tmp = -d * math.pow((l * h), -0.5)
	else:
		tmp = d * math.sqrt(((1.0 / h) / l))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 4.6e-177)
		tmp = Float64(Float64(-d) * (Float64(l * h) ^ -0.5));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / h) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= 4.6e-177)
		tmp = -d * ((l * h) ^ -0.5);
	else
		tmp = d * sqrt(((1.0 / h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, 4.6e-177], N[((-d) * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.60000000000000044e-177

   1. Initial program 68.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 66.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 10.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   6. Taylor expanded in h around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. associate-*l* 0.0%

         \[\leadsto \color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. unpow2 0.0%

         \[\leadsto d \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. rem-square-sqrt 40.0%

         \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. neg-mul-1 40.0%

         \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      5. unpow-1 40.0%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \]

      6. metadata-eval 40.0%

         \[\leadsto d \cdot \left(-\sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \]

      7. pow-sqr 40.0%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      8. rem-sqrt-square 40.2%

         \[\leadsto d \cdot \left(-\color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \]

      9. rem-square-sqrt 40.1%

         \[\leadsto d \cdot \left(-\left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right|\right) \]

      10. fabs-sqr 40.1%

          \[\leadsto d \cdot \left(-\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      11. rem-square-sqrt 40.2%

          \[\leadsto d \cdot \left(-\color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \]

   8. Simplified 40.2%

      \[\leadsto \color{blue}{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \]

   if 4.60000000000000044e-177 < l

   1. Initial program 62.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 61.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 61.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 61.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 61.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 61.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. metadata-eval 61.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. pow1 61.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. frac-times 63.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. associate-/l* 61.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. *-un-lft-identity 61.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(M \cdot \frac{\color{blue}{1 \cdot D}}{2 \cdot d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. times-frac 61.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(M \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{D}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. metadata-eval 61.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(M \cdot \left(\color{blue}{0.5} \cdot \frac{D}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 61.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in d around inf 52.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. associate-/r* 53.5%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   9. Simplified 53.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.6 \cdot 10^{-177}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 43.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5.8 \cdot 10^{-176}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l 5.8e-176) (* (- d) (pow (* l h) -0.5)) (/ d (sqrt (* l h)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 5.8e-176) {
		tmp = -d * pow((l * h), -0.5);
	} else {
		tmp = d / sqrt((l * h));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= 5.8d-176) then
        tmp = -d * ((l * h) ** (-0.5d0))
    else
        tmp = d / sqrt((l * 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 <= 5.8e-176) {
		tmp = -d * Math.pow((l * h), -0.5);
	} else {
		tmp = d / Math.sqrt((l * h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= 5.8e-176:
		tmp = -d * math.pow((l * h), -0.5)
	else:
		tmp = d / math.sqrt((l * h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 5.8e-176)
		tmp = Float64(Float64(-d) * (Float64(l * h) ^ -0.5));
	else
		tmp = Float64(d / sqrt(Float64(l * h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= 5.8e-176)
		tmp = -d * ((l * h) ^ -0.5);
	else
		tmp = d / sqrt((l * h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, 5.8e-176], N[((-d) * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.80000000000000012e-176

   1. Initial program 68.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 66.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 10.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   6. Taylor expanded in h around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. associate-*l* 0.0%

         \[\leadsto \color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. unpow2 0.0%

         \[\leadsto d \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. rem-square-sqrt 40.0%

         \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. neg-mul-1 40.0%

         \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      5. unpow-1 40.0%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \]

      6. metadata-eval 40.0%

         \[\leadsto d \cdot \left(-\sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \]

      7. pow-sqr 40.0%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      8. rem-sqrt-square 40.2%

         \[\leadsto d \cdot \left(-\color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \]

      9. rem-square-sqrt 40.1%

         \[\leadsto d \cdot \left(-\left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right|\right) \]

      10. fabs-sqr 40.1%

          \[\leadsto d \cdot \left(-\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      11. rem-square-sqrt 40.2%

          \[\leadsto d \cdot \left(-\color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \]

   8. Simplified 40.2%

      \[\leadsto \color{blue}{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \]

   if 5.80000000000000012e-176 < l

   1. Initial program 62.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 61.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in l around -inf 0.0%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{d}{\ell}}\right)} \]

      2. unpow2 0.0%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \]

      3. rem-square-sqrt 7.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{d}{\ell}}\right) \]

      4. mul-1-neg 7.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\left(-\sqrt{\frac{d}{\ell}}\right)} \]

   7. Simplified 7.2%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\left(-\sqrt{\frac{d}{\ell}}\right)} \]
   8. Step-by-step derivation

      1. sqrt-div 5.7%

         \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \left(-\sqrt{\frac{d}{\ell}}\right) \]

      2. add-sqr-sqrt 3.0%

         \[\leadsto \frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\left(\sqrt{-\sqrt{\frac{d}{\ell}}} \cdot \sqrt{-\sqrt{\frac{d}{\ell}}}\right)} \]

      3. sqrt-unprod 52.5%

         \[\leadsto \frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\sqrt{\left(-\sqrt{\frac{d}{\ell}}\right) \cdot \left(-\sqrt{\frac{d}{\ell}}\right)}} \]

      4. sqr-neg 52.5%

         \[\leadsto \frac{\sqrt{d}}{\sqrt{h}} \cdot \sqrt{\color{blue}{\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{\ell}}}} \]

      5. add-sqr-sqrt 52.5%

         \[\leadsto \frac{\sqrt{d}}{\sqrt{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}} \]

      6. sqrt-div 58.2%

         \[\leadsto \frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \]

      7. frac-times 58.3%

         \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      8. add-sqr-sqrt 58.5%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      9. sqrt-prod 53.0%

         \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      10. *-commutative 53.0%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\ell \cdot h}}} \]

   9. Applied egg-rr 53.0%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.8 \cdot 10^{-176}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 43.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;\ell \leq 3.8 \cdot 10^{-177}:\\ \;\;\;\;\frac{-d}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (* l h)))) (if (<= l 3.8e-177) (/ (- d) t_0) (/ d t_0))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((l * h));
	double tmp;
	if (l <= 3.8e-177) {
		tmp = -d / t_0;
	} else {
		tmp = d / t_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) :: tmp
    t_0 = sqrt((l * h))
    if (l <= 3.8d-177) then
        tmp = -d / t_0
    else
        tmp = d / t_0
    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((l * h));
	double tmp;
	if (l <= 3.8e-177) {
		tmp = -d / t_0;
	} else {
		tmp = d / t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((l * h))
	tmp = 0
	if l <= 3.8e-177:
		tmp = -d / t_0
	else:
		tmp = d / t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(l * h))
	tmp = 0.0
	if (l <= 3.8e-177)
		tmp = Float64(Float64(-d) / t_0);
	else
		tmp = Float64(d / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((l * h));
	tmp = 0.0;
	if (l <= 3.8e-177)
		tmp = -d / t_0;
	else
		tmp = d / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, 3.8e-177], N[((-d) / t$95$0), $MachinePrecision], N[(d / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 3.80000000000000004e-177

   1. Initial program 68.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 66.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in l around -inf 0.0%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{d}{\ell}}\right)} \]

      2. unpow2 0.0%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \]

      3. rem-square-sqrt 9.6%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{d}{\ell}}\right) \]

      4. mul-1-neg 9.6%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\left(-\sqrt{\frac{d}{\ell}}\right)} \]

   7. Simplified 9.6%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\left(-\sqrt{\frac{d}{\ell}}\right)} \]
   8. Step-by-step derivation

      1. distribute-rgt-neg-out 9.6%

         \[\leadsto \color{blue}{-\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

      2. sqrt-div 4.1%

         \[\leadsto -\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \sqrt{\frac{d}{\ell}} \]

      3. sqrt-div 2.1%

         \[\leadsto -\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \]

      4. frac-times 2.1%

         \[\leadsto -\color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      5. add-sqr-sqrt 2.1%

         \[\leadsto -\frac{\color{blue}{d}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      6. sqrt-prod 40.2%

         \[\leadsto -\frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      7. *-commutative 40.2%

         \[\leadsto -\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}} \]

   9. Applied egg-rr 40.2%

      \[\leadsto \color{blue}{-\frac{d}{\sqrt{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. distribute-neg-frac2 40.2%

          \[\leadsto \color{blue}{\frac{d}{-\sqrt{\ell \cdot h}}} \]

       2. *-commutative 40.2%

          \[\leadsto \frac{d}{-\sqrt{\color{blue}{h \cdot \ell}}} \]

   11. Simplified 40.2%

       \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   if 3.80000000000000004e-177 < l

   1. Initial program 62.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 61.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in l around -inf 0.0%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{d}{\ell}}\right)} \]

      2. unpow2 0.0%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \]

      3. rem-square-sqrt 7.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{d}{\ell}}\right) \]

      4. mul-1-neg 7.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\left(-\sqrt{\frac{d}{\ell}}\right)} \]

   7. Simplified 7.2%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\left(-\sqrt{\frac{d}{\ell}}\right)} \]
   8. Step-by-step derivation

      1. sqrt-div 5.7%

         \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \left(-\sqrt{\frac{d}{\ell}}\right) \]

      2. add-sqr-sqrt 3.0%

         \[\leadsto \frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\left(\sqrt{-\sqrt{\frac{d}{\ell}}} \cdot \sqrt{-\sqrt{\frac{d}{\ell}}}\right)} \]

      3. sqrt-unprod 52.5%

         \[\leadsto \frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\sqrt{\left(-\sqrt{\frac{d}{\ell}}\right) \cdot \left(-\sqrt{\frac{d}{\ell}}\right)}} \]

      4. sqr-neg 52.5%

         \[\leadsto \frac{\sqrt{d}}{\sqrt{h}} \cdot \sqrt{\color{blue}{\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{\ell}}}} \]

      5. add-sqr-sqrt 52.5%

         \[\leadsto \frac{\sqrt{d}}{\sqrt{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}} \]

      6. sqrt-div 58.2%

         \[\leadsto \frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \]

      7. frac-times 58.3%

         \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      8. add-sqr-sqrt 58.5%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      9. sqrt-prod 53.0%

         \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      10. *-commutative 53.0%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\ell \cdot h}}} \]

   9. Applied egg-rr 53.0%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.8 \cdot 10^{-177}:\\ \;\;\;\;\frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 27.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \frac{d}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (/ d (sqrt (* l h))))

C

double code(double d, double h, double l, double M, double D) {
	return d / sqrt((l * h));
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = d / sqrt((l * h))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return d / Math.sqrt((l * h));
}

Python

def code(d, h, l, M, D):
	return d / math.sqrt((l * h))

Julia

function code(d, h, l, M, D)
	return Float64(d / sqrt(Float64(l * h)))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = d / sqrt((l * h));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.7%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 64.2%

   \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in l around -inf 0.0%

   \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
6. Step-by-step derivation

   1. *-commutative 0.0%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{d}{\ell}}\right)} \]

   2. unpow2 0.0%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \]

   3. rem-square-sqrt 8.5%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{d}{\ell}}\right) \]

   4. mul-1-neg 8.5%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\left(-\sqrt{\frac{d}{\ell}}\right)} \]

7. Simplified 8.5%

   \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\left(-\sqrt{\frac{d}{\ell}}\right)} \]
8. Step-by-step derivation

   1. sqrt-div 4.8%

      \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \left(-\sqrt{\frac{d}{\ell}}\right) \]

   2. add-sqr-sqrt 1.4%

      \[\leadsto \frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\left(\sqrt{-\sqrt{\frac{d}{\ell}}} \cdot \sqrt{-\sqrt{\frac{d}{\ell}}}\right)} \]

   3. sqrt-unprod 25.5%

      \[\leadsto \frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\sqrt{\left(-\sqrt{\frac{d}{\ell}}\right) \cdot \left(-\sqrt{\frac{d}{\ell}}\right)}} \]

   4. sqr-neg 25.5%

      \[\leadsto \frac{\sqrt{d}}{\sqrt{h}} \cdot \sqrt{\color{blue}{\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{\ell}}}} \]

   5. add-sqr-sqrt 25.5%

      \[\leadsto \frac{\sqrt{d}}{\sqrt{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}} \]

   6. sqrt-div 29.1%

      \[\leadsto \frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \]

   7. frac-times 29.2%

      \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h} \cdot \sqrt{\ell}}} \]

   8. add-sqr-sqrt 29.3%

      \[\leadsto \frac{\color{blue}{d}}{\sqrt{h} \cdot \sqrt{\ell}} \]

   9. sqrt-prod 29.5%

      \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

   10. *-commutative 29.5%

       \[\leadsto \frac{d}{\sqrt{\color{blue}{\ell \cdot h}}} \]

9. Applied egg-rr 29.5%

   \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024156 
(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)))))