Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.6% → 82.8%

Time: 29.0s

Alternatives: 19

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.6% 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: 82.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := \frac{t_0}{\sqrt{-\ell}}\\ \mathbf{if}\;\ell \leq -1.1 \cdot 10^{-64}:\\ \;\;\;\;\frac{t_0}{\sqrt{-h}} \cdot \left(t_1 \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t_1 \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left({\left(\frac{M}{2 \cdot \frac{d}{D}}\right)}^{2} \cdot -0.5\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\frac{\sqrt{h}}{\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D \cdot -0.5}{d}\right)}^{2}\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (- d))) (t_1 (/ t_0 (sqrt (- l)))))
   (if (<= l -1.1e-64)
     (*
      (/ t_0 (sqrt (- h)))
      (* t_1 (- 1.0 (* 0.5 (* (pow (* (/ M 2.0) (/ D d)) 2.0) (/ h l))))))
     (if (<= l -2e-310)
       (*
        t_1
        (*
         (sqrt (/ d h))
         (+ 1.0 (/ (* h (* (pow (/ M (* 2.0 (/ d D))) 2.0) -0.5)) l))))
       (/
        (sqrt d)
        (/
         (sqrt h)
         (*
          (/ (sqrt d) (sqrt l))
          (+ 1.0 (* -0.5 (* (/ h l) (pow (* M (/ (* D -0.5) d)) 2.0)))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(-d);
	double t_1 = t_0 / sqrt(-l);
	double tmp;
	if (l <= -1.1e-64) {
		tmp = (t_0 / sqrt(-h)) * (t_1 * (1.0 - (0.5 * (pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))));
	} else if (l <= -2e-310) {
		tmp = t_1 * (sqrt((d / h)) * (1.0 + ((h * (pow((M / (2.0 * (d / D))), 2.0) * -0.5)) / l)));
	} else {
		tmp = sqrt(d) / (sqrt(h) / ((sqrt(d) / sqrt(l)) * (1.0 + (-0.5 * ((h / l) * pow((M * ((D * -0.5) / d)), 2.0))))));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(-d)
    t_1 = t_0 / sqrt(-l)
    if (l <= (-1.1d-64)) then
        tmp = (t_0 / sqrt(-h)) * (t_1 * (1.0d0 - (0.5d0 * ((((m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (h / l)))))
    else if (l <= (-2d-310)) then
        tmp = t_1 * (sqrt((d / h)) * (1.0d0 + ((h * (((m / (2.0d0 * (d / d_1))) ** 2.0d0) * (-0.5d0))) / l)))
    else
        tmp = sqrt(d) / (sqrt(h) / ((sqrt(d) / sqrt(l)) * (1.0d0 + ((-0.5d0) * ((h / l) * ((m * ((d_1 * (-0.5d0)) / d)) ** 2.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt(-d);
	double t_1 = t_0 / Math.sqrt(-l);
	double tmp;
	if (l <= -1.1e-64) {
		tmp = (t_0 / Math.sqrt(-h)) * (t_1 * (1.0 - (0.5 * (Math.pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))));
	} else if (l <= -2e-310) {
		tmp = t_1 * (Math.sqrt((d / h)) * (1.0 + ((h * (Math.pow((M / (2.0 * (d / D))), 2.0) * -0.5)) / l)));
	} else {
		tmp = Math.sqrt(d) / (Math.sqrt(h) / ((Math.sqrt(d) / Math.sqrt(l)) * (1.0 + (-0.5 * ((h / l) * Math.pow((M * ((D * -0.5) / d)), 2.0))))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt(-d)
	t_1 = t_0 / math.sqrt(-l)
	tmp = 0
	if l <= -1.1e-64:
		tmp = (t_0 / math.sqrt(-h)) * (t_1 * (1.0 - (0.5 * (math.pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))))
	elif l <= -2e-310:
		tmp = t_1 * (math.sqrt((d / h)) * (1.0 + ((h * (math.pow((M / (2.0 * (d / D))), 2.0) * -0.5)) / l)))
	else:
		tmp = math.sqrt(d) / (math.sqrt(h) / ((math.sqrt(d) / math.sqrt(l)) * (1.0 + (-0.5 * ((h / l) * math.pow((M * ((D * -0.5) / d)), 2.0))))))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(-d))
	t_1 = Float64(t_0 / sqrt(Float64(-l)))
	tmp = 0.0
	if (l <= -1.1e-64)
		tmp = Float64(Float64(t_0 / sqrt(Float64(-h))) * Float64(t_1 * Float64(1.0 - Float64(0.5 * Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * Float64(h / l))))));
	elseif (l <= -2e-310)
		tmp = Float64(t_1 * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(h * Float64((Float64(M / Float64(2.0 * Float64(d / D))) ^ 2.0) * -0.5)) / l))));
	else
		tmp = Float64(sqrt(d) / Float64(sqrt(h) / Float64(Float64(sqrt(d) / sqrt(l)) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(M * Float64(Float64(D * -0.5) / d)) ^ 2.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(-d);
	t_1 = t_0 / sqrt(-l);
	tmp = 0.0;
	if (l <= -1.1e-64)
		tmp = (t_0 / sqrt(-h)) * (t_1 * (1.0 - (0.5 * ((((M / 2.0) * (D / d)) ^ 2.0) * (h / l)))));
	elseif (l <= -2e-310)
		tmp = t_1 * (sqrt((d / h)) * (1.0 + ((h * (((M / (2.0 * (d / D))) ^ 2.0) * -0.5)) / l)));
	else
		tmp = sqrt(d) / (sqrt(h) / ((sqrt(d) / sqrt(l)) * (1.0 + (-0.5 * ((h / l) * ((M * ((D * -0.5) / d)) ^ 2.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.1e-64], N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(1.0 - N[(0.5 * N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-310], N[(t$95$1 * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h * N[(N[Power[N[(M / N[(2.0 * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[d], $MachinePrecision] / N[(N[Sqrt[h], $MachinePrecision] / N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(M * N[(N[(D * -0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.1e-64

   1. Initial program 70.5%

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

   3. Simplified 71.7%

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

      1. frac-2neg 71.7%

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

      2. sqrt-div 86.6%

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

   5. Applied egg-rr 86.6%

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

      1. frac-2neg 86.6%

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

      2. sqrt-div 90.0%

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

   7. Applied egg-rr 90.0%

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

   if -1.1e-64 < l < -1.999999999999994e-310

   1. Initial program 70.7%

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

   3. Simplified 70.7%

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

   5. Applied egg-rr 79.2%

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

      1. add-sqr-sqrt 51.6%

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

      2. sqrt-prod 79.2%

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

      3. unpow2 79.2%

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

      4. unpow-prod-down 66.3%

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

      5. unpow-prod-down 66.3%

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

      6. metadata-eval 66.3%

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

      7. metadata-eval 66.3%

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

      8. unpow-prod-down 66.3%

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

      9. metadata-eval 66.3%

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

      10. div-inv 66.3%

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

      11. unpow-prod-down 79.2%

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

      12. unpow2 79.2%

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

      13. sqrt-prod 44.9%

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

      14. add-sqr-sqrt 79.2%

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

      15. *-commutative 79.2%

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

      16. clear-num 79.2%

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

      17. frac-times 79.2%

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

      18. *-un-lft-identity 79.2%

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

   7. Applied egg-rr 79.2%

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

      1. frac-2neg 75.0%

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

      2. sqrt-div 76.9%

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

   9. Applied egg-rr 88.9%

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

   if -1.999999999999994e-310 < l

   1. Initial program 69.2%

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

   3. Simplified 68.6%

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

      1. pow1/2 68.6%

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

      2. metadata-eval 68.6%

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

      3. add-cbrt-cube 55.7%

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

      4. pow1/3 53.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{{\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)}^{0.3333333333333333}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. sqr-pow 53.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{1}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)}^{0.3333333333333333} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. pow-prod-up 53.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(1 + \frac{1}{2}\right)}\right)}}^{0.3333333333333333} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. metadata-eval 53.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left({\left(\frac{d}{\ell}\right)}^{\left(1 + \color{blue}{0.5}\right)}\right)}^{0.3333333333333333} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. metadata-eval 53.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left({\left(\frac{d}{\ell}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   5. Applied egg-rr 53.9%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{{\left({\left(\frac{d}{\ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
   6. Step-by-step derivation

      1. unpow1/3 55.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt[3]{{\left(\frac{d}{\ell}\right)}^{1.5}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   7. Simplified 55.7%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt[3]{{\left(\frac{d}{\ell}\right)}^{1.5}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   9. Applied egg-rr 76.0%

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

       1. associate-/l* 75.9%

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

       2. associate-*r/ 75.9%

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

   11. Simplified 75.9%

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

       1. sqrt-div 85.4%

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

       2. div-inv 85.4%

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

   13. Applied egg-rr 85.4%

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

       1. associate-*r/ 85.4%

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

       2. *-rgt-identity 85.4%

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

   15. Simplified 85.4%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.1 \cdot 10^{-64}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left({\left(\frac{M}{2 \cdot \frac{d}{D}}\right)}^{2} \cdot -0.5\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\frac{\sqrt{h}}{\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D \cdot -0.5}{d}\right)}^{2}\right)\right)}}\\ \end{array} \]

Alternative 2: 80.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -6.2 \cdot 10^{-241}:\\ \;\;\;\;\frac{t_0}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \left(M \cdot -0.5\right)\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{t_0}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left({\left(\frac{M}{2 \cdot \frac{d}{D}}\right)}^{2} \cdot -0.5\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\frac{\sqrt{h}}{\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D \cdot -0.5}{d}\right)}^{2}\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (- d))))
   (if (<= l -6.2e-241)
     (*
      (/ t_0 (sqrt (- h)))
      (*
       (sqrt (/ d l))
       (- 1.0 (* 0.5 (/ (* h (pow (* (/ D d) (* M -0.5)) 2.0)) l)))))
     (if (<= l -2e-310)
       (*
        (/ t_0 (sqrt (- l)))
        (*
         (sqrt (/ d h))
         (+ 1.0 (/ (* h (* (pow (/ M (* 2.0 (/ d D))) 2.0) -0.5)) l))))
       (/
        (sqrt d)
        (/
         (sqrt h)
         (*
          (/ (sqrt d) (sqrt l))
          (+ 1.0 (* -0.5 (* (/ h l) (pow (* M (/ (* D -0.5) d)) 2.0)))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(-d);
	double tmp;
	if (l <= -6.2e-241) {
		tmp = (t_0 / sqrt(-h)) * (sqrt((d / l)) * (1.0 - (0.5 * ((h * pow(((D / d) * (M * -0.5)), 2.0)) / l))));
	} else if (l <= -2e-310) {
		tmp = (t_0 / sqrt(-l)) * (sqrt((d / h)) * (1.0 + ((h * (pow((M / (2.0 * (d / D))), 2.0) * -0.5)) / l)));
	} else {
		tmp = sqrt(d) / (sqrt(h) / ((sqrt(d) / sqrt(l)) * (1.0 + (-0.5 * ((h / l) * pow((M * ((D * -0.5) / d)), 2.0))))));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(-d)
    if (l <= (-6.2d-241)) then
        tmp = (t_0 / sqrt(-h)) * (sqrt((d / l)) * (1.0d0 - (0.5d0 * ((h * (((d_1 / d) * (m * (-0.5d0))) ** 2.0d0)) / l))))
    else if (l <= (-2d-310)) then
        tmp = (t_0 / sqrt(-l)) * (sqrt((d / h)) * (1.0d0 + ((h * (((m / (2.0d0 * (d / d_1))) ** 2.0d0) * (-0.5d0))) / l)))
    else
        tmp = sqrt(d) / (sqrt(h) / ((sqrt(d) / sqrt(l)) * (1.0d0 + ((-0.5d0) * ((h / l) * ((m * ((d_1 * (-0.5d0)) / d)) ** 2.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt(-d);
	double tmp;
	if (l <= -6.2e-241) {
		tmp = (t_0 / Math.sqrt(-h)) * (Math.sqrt((d / l)) * (1.0 - (0.5 * ((h * Math.pow(((D / d) * (M * -0.5)), 2.0)) / l))));
	} else if (l <= -2e-310) {
		tmp = (t_0 / Math.sqrt(-l)) * (Math.sqrt((d / h)) * (1.0 + ((h * (Math.pow((M / (2.0 * (d / D))), 2.0) * -0.5)) / l)));
	} else {
		tmp = Math.sqrt(d) / (Math.sqrt(h) / ((Math.sqrt(d) / Math.sqrt(l)) * (1.0 + (-0.5 * ((h / l) * Math.pow((M * ((D * -0.5) / d)), 2.0))))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt(-d)
	tmp = 0
	if l <= -6.2e-241:
		tmp = (t_0 / math.sqrt(-h)) * (math.sqrt((d / l)) * (1.0 - (0.5 * ((h * math.pow(((D / d) * (M * -0.5)), 2.0)) / l))))
	elif l <= -2e-310:
		tmp = (t_0 / math.sqrt(-l)) * (math.sqrt((d / h)) * (1.0 + ((h * (math.pow((M / (2.0 * (d / D))), 2.0) * -0.5)) / l)))
	else:
		tmp = math.sqrt(d) / (math.sqrt(h) / ((math.sqrt(d) / math.sqrt(l)) * (1.0 + (-0.5 * ((h / l) * math.pow((M * ((D * -0.5) / d)), 2.0))))))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(-d))
	tmp = 0.0
	if (l <= -6.2e-241)
		tmp = Float64(Float64(t_0 / sqrt(Float64(-h))) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(Float64(D / d) * Float64(M * -0.5)) ^ 2.0)) / l)))));
	elseif (l <= -2e-310)
		tmp = Float64(Float64(t_0 / sqrt(Float64(-l))) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(h * Float64((Float64(M / Float64(2.0 * Float64(d / D))) ^ 2.0) * -0.5)) / l))));
	else
		tmp = Float64(sqrt(d) / Float64(sqrt(h) / Float64(Float64(sqrt(d) / sqrt(l)) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(M * Float64(Float64(D * -0.5) / d)) ^ 2.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(-d);
	tmp = 0.0;
	if (l <= -6.2e-241)
		tmp = (t_0 / sqrt(-h)) * (sqrt((d / l)) * (1.0 - (0.5 * ((h * (((D / d) * (M * -0.5)) ^ 2.0)) / l))));
	elseif (l <= -2e-310)
		tmp = (t_0 / sqrt(-l)) * (sqrt((d / h)) * (1.0 + ((h * (((M / (2.0 * (d / D))) ^ 2.0) * -0.5)) / l)));
	else
		tmp = sqrt(d) / (sqrt(h) / ((sqrt(d) / sqrt(l)) * (1.0 + (-0.5 * ((h / l) * ((M * ((D * -0.5) / d)) ^ 2.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[l, -6.2e-241], N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M * -0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-310], N[(N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h * N[(N[Power[N[(M / N[(2.0 * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[d], $MachinePrecision] / N[(N[Sqrt[h], $MachinePrecision] / N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(M * N[(N[(D * -0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.1999999999999998e-241

   1. Initial program 70.7%

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

   3. Simplified 71.5%

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

      1. frac-2neg 71.5%

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

      2. sqrt-div 84.1%

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

   5. Applied egg-rr 84.1%

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

      1. associate-*r/ 85.9%

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

   7. Applied egg-rr 85.9%

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

   if -6.1999999999999998e-241 < l < -1.999999999999994e-310

   1. Initial program 70.0%

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

   3. Simplified 70.0%

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

   5. Applied egg-rr 82.6%

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

      1. add-sqr-sqrt 44.7%

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

      2. sqrt-prod 82.6%

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

      3. unpow2 82.6%

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

      4. unpow-prod-down 63.8%

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

      5. unpow-prod-down 63.8%

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

      6. metadata-eval 63.8%

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

      7. metadata-eval 63.8%

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

      8. unpow-prod-down 63.8%

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

      9. metadata-eval 63.8%

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

      10. div-inv 63.8%

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

      11. unpow-prod-down 82.6%

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

      12. unpow2 82.6%

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

      13. sqrt-prod 38.9%

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

      14. add-sqr-sqrt 82.6%

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

      15. *-commutative 82.6%

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

      16. clear-num 82.6%

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

      17. frac-times 82.6%

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

      18. *-un-lft-identity 82.6%

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

   7. Applied egg-rr 82.6%

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

      1. frac-2neg 70.0%

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

      2. sqrt-div 75.7%

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

   9. Applied egg-rr 99.8%

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

   if -1.999999999999994e-310 < l

   1. Initial program 69.2%

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

   3. Simplified 68.6%

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

      1. pow1/2 68.6%

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

      2. metadata-eval 68.6%

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

      3. add-cbrt-cube 55.7%

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

      4. pow1/3 53.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{{\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)}^{0.3333333333333333}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. sqr-pow 53.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{1}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)}^{0.3333333333333333} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. pow-prod-up 53.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(1 + \frac{1}{2}\right)}\right)}}^{0.3333333333333333} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. metadata-eval 53.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left({\left(\frac{d}{\ell}\right)}^{\left(1 + \color{blue}{0.5}\right)}\right)}^{0.3333333333333333} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. metadata-eval 53.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left({\left(\frac{d}{\ell}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   5. Applied egg-rr 53.9%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{{\left({\left(\frac{d}{\ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
   6. Step-by-step derivation

      1. unpow1/3 55.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt[3]{{\left(\frac{d}{\ell}\right)}^{1.5}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   7. Simplified 55.7%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt[3]{{\left(\frac{d}{\ell}\right)}^{1.5}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   9. Applied egg-rr 76.0%

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

       1. associate-/l* 75.9%

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

       2. associate-*r/ 75.9%

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

   11. Simplified 75.9%

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

       1. sqrt-div 85.4%

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

       2. div-inv 85.4%

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

   13. Applied egg-rr 85.4%

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

       1. associate-*r/ 85.4%

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

       2. *-rgt-identity 85.4%

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

   15. Simplified 85.4%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.2 \cdot 10^{-241}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \left(M \cdot -0.5\right)\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left({\left(\frac{M}{2 \cdot \frac{d}{D}}\right)}^{2} \cdot -0.5\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\frac{\sqrt{h}}{\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D \cdot -0.5}{d}\right)}^{2}\right)\right)}}\\ \end{array} \]

Alternative 3: 79.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{D}{d} \cdot \left(M \cdot -0.5\right)\right)}^{2}\\ t_1 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -2.7 \cdot 10^{-240}:\\ \;\;\;\;\frac{t_1}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{h \cdot t_0}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{t_1}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot t_0\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D \cdot -0.5}{d}\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (pow (* (/ D d) (* M -0.5)) 2.0)) (t_1 (sqrt (- d))))
   (if (<= l -2.7e-240)
     (*
      (/ t_1 (sqrt (- h)))
      (* (sqrt (/ d l)) (- 1.0 (* 0.5 (/ (* h t_0) l)))))
     (if (<= l -2e-310)
       (*
        (/ t_1 (sqrt (- l)))
        (* (sqrt (/ d h)) (+ 1.0 (/ (* h (* -0.5 t_0)) l))))
       (*
        (+ 1.0 (* -0.5 (* (/ h l) (pow (* M (/ (* D -0.5) d)) 2.0))))
        (/ d (* (sqrt h) (sqrt l))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = pow(((D / d) * (M * -0.5)), 2.0);
	double t_1 = sqrt(-d);
	double tmp;
	if (l <= -2.7e-240) {
		tmp = (t_1 / sqrt(-h)) * (sqrt((d / l)) * (1.0 - (0.5 * ((h * t_0) / l))));
	} else if (l <= -2e-310) {
		tmp = (t_1 / sqrt(-l)) * (sqrt((d / h)) * (1.0 + ((h * (-0.5 * t_0)) / l)));
	} else {
		tmp = (1.0 + (-0.5 * ((h / l) * pow((M * ((D * -0.5) / d)), 2.0)))) * (d / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((d_1 / d) * (m * (-0.5d0))) ** 2.0d0
    t_1 = sqrt(-d)
    if (l <= (-2.7d-240)) then
        tmp = (t_1 / sqrt(-h)) * (sqrt((d / l)) * (1.0d0 - (0.5d0 * ((h * t_0) / l))))
    else if (l <= (-2d-310)) then
        tmp = (t_1 / sqrt(-l)) * (sqrt((d / h)) * (1.0d0 + ((h * ((-0.5d0) * t_0)) / l)))
    else
        tmp = (1.0d0 + ((-0.5d0) * ((h / l) * ((m * ((d_1 * (-0.5d0)) / d)) ** 2.0d0)))) * (d / (sqrt(h) * sqrt(l)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.pow(((D / d) * (M * -0.5)), 2.0);
	double t_1 = Math.sqrt(-d);
	double tmp;
	if (l <= -2.7e-240) {
		tmp = (t_1 / Math.sqrt(-h)) * (Math.sqrt((d / l)) * (1.0 - (0.5 * ((h * t_0) / l))));
	} else if (l <= -2e-310) {
		tmp = (t_1 / Math.sqrt(-l)) * (Math.sqrt((d / h)) * (1.0 + ((h * (-0.5 * t_0)) / l)));
	} else {
		tmp = (1.0 + (-0.5 * ((h / l) * Math.pow((M * ((D * -0.5) / d)), 2.0)))) * (d / (Math.sqrt(h) * Math.sqrt(l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.pow(((D / d) * (M * -0.5)), 2.0)
	t_1 = math.sqrt(-d)
	tmp = 0
	if l <= -2.7e-240:
		tmp = (t_1 / math.sqrt(-h)) * (math.sqrt((d / l)) * (1.0 - (0.5 * ((h * t_0) / l))))
	elif l <= -2e-310:
		tmp = (t_1 / math.sqrt(-l)) * (math.sqrt((d / h)) * (1.0 + ((h * (-0.5 * t_0)) / l)))
	else:
		tmp = (1.0 + (-0.5 * ((h / l) * math.pow((M * ((D * -0.5) / d)), 2.0)))) * (d / (math.sqrt(h) * math.sqrt(l)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(D / d) * Float64(M * -0.5)) ^ 2.0
	t_1 = sqrt(Float64(-d))
	tmp = 0.0
	if (l <= -2.7e-240)
		tmp = Float64(Float64(t_1 / sqrt(Float64(-h))) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * t_0) / l)))));
	elseif (l <= -2e-310)
		tmp = Float64(Float64(t_1 / sqrt(Float64(-l))) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(h * Float64(-0.5 * t_0)) / l))));
	else
		tmp = Float64(Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(M * Float64(Float64(D * -0.5) / d)) ^ 2.0)))) * Float64(d / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = ((D / d) * (M * -0.5)) ^ 2.0;
	t_1 = sqrt(-d);
	tmp = 0.0;
	if (l <= -2.7e-240)
		tmp = (t_1 / sqrt(-h)) * (sqrt((d / l)) * (1.0 - (0.5 * ((h * t_0) / l))));
	elseif (l <= -2e-310)
		tmp = (t_1 / sqrt(-l)) * (sqrt((d / h)) * (1.0 + ((h * (-0.5 * t_0)) / l)));
	else
		tmp = (1.0 + (-0.5 * ((h / l) * ((M * ((D * -0.5) / d)) ^ 2.0)))) * (d / (sqrt(h) * sqrt(l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Power[N[(N[(D / d), $MachinePrecision] * N[(M * -0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[l, -2.7e-240], N[(N[(t$95$1 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * t$95$0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-310], N[(N[(t$95$1 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h * N[(-0.5 * t$95$0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(M * N[(N[(D * -0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.70000000000000018e-240

   1. Initial program 70.7%

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

   3. Simplified 71.5%

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

      1. frac-2neg 71.5%

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

      2. sqrt-div 84.1%

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

   5. Applied egg-rr 84.1%

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

      1. associate-*r/ 85.9%

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

   7. Applied egg-rr 85.9%

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

   if -2.70000000000000018e-240 < l < -1.999999999999994e-310

   1. Initial program 70.0%

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

   3. Simplified 70.0%

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

   5. Applied egg-rr 82.6%

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

      1. frac-2neg 70.0%

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

      2. sqrt-div 75.7%

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

   7. Applied egg-rr 99.8%

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

   if -1.999999999999994e-310 < l

   1. Initial program 69.2%

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

   3. Simplified 68.6%

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

      1. pow1/2 68.6%

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

      2. metadata-eval 68.6%

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

      3. add-cbrt-cube 55.7%

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

      4. pow1/3 53.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{{\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)}^{0.3333333333333333}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. sqr-pow 53.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{1}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)}^{0.3333333333333333} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. pow-prod-up 53.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(1 + \frac{1}{2}\right)}\right)}}^{0.3333333333333333} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. metadata-eval 53.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left({\left(\frac{d}{\ell}\right)}^{\left(1 + \color{blue}{0.5}\right)}\right)}^{0.3333333333333333} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. metadata-eval 53.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left({\left(\frac{d}{\ell}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   5. Applied egg-rr 53.9%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{{\left({\left(\frac{d}{\ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
   6. Step-by-step derivation

      1. unpow1/3 55.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt[3]{{\left(\frac{d}{\ell}\right)}^{1.5}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   7. Simplified 55.7%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt[3]{{\left(\frac{d}{\ell}\right)}^{1.5}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   9. Applied egg-rr 34.0%

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

       1. expm1-def 51.9%

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

       2. expm1-log1p 81.6%

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

       3. associate-*r/ 81.6%

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

   11. Simplified 81.6%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.7 \cdot 10^{-240}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \left(M \cdot -0.5\right)\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \left(M \cdot -0.5\right)\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D \cdot -0.5}{d}\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 4: 79.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -3.8 \cdot 10^{-241}:\\ \;\;\;\;\frac{t_0}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \left(M \cdot -0.5\right)\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{t_0}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left({\left(\frac{M}{2 \cdot \frac{d}{D}}\right)}^{2} \cdot -0.5\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D \cdot -0.5}{d}\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (- d))))
   (if (<= l -3.8e-241)
     (*
      (/ t_0 (sqrt (- h)))
      (*
       (sqrt (/ d l))
       (- 1.0 (* 0.5 (/ (* h (pow (* (/ D d) (* M -0.5)) 2.0)) l)))))
     (if (<= l -2e-310)
       (*
        (/ t_0 (sqrt (- l)))
        (*
         (sqrt (/ d h))
         (+ 1.0 (/ (* h (* (pow (/ M (* 2.0 (/ d D))) 2.0) -0.5)) l))))
       (*
        (+ 1.0 (* -0.5 (* (/ h l) (pow (* M (/ (* D -0.5) d)) 2.0))))
        (/ d (* (sqrt h) (sqrt l))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(-d);
	double tmp;
	if (l <= -3.8e-241) {
		tmp = (t_0 / sqrt(-h)) * (sqrt((d / l)) * (1.0 - (0.5 * ((h * pow(((D / d) * (M * -0.5)), 2.0)) / l))));
	} else if (l <= -2e-310) {
		tmp = (t_0 / sqrt(-l)) * (sqrt((d / h)) * (1.0 + ((h * (pow((M / (2.0 * (d / D))), 2.0) * -0.5)) / l)));
	} else {
		tmp = (1.0 + (-0.5 * ((h / l) * pow((M * ((D * -0.5) / d)), 2.0)))) * (d / (sqrt(h) * 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 = sqrt(-d)
    if (l <= (-3.8d-241)) then
        tmp = (t_0 / sqrt(-h)) * (sqrt((d / l)) * (1.0d0 - (0.5d0 * ((h * (((d_1 / d) * (m * (-0.5d0))) ** 2.0d0)) / l))))
    else if (l <= (-2d-310)) then
        tmp = (t_0 / sqrt(-l)) * (sqrt((d / h)) * (1.0d0 + ((h * (((m / (2.0d0 * (d / d_1))) ** 2.0d0) * (-0.5d0))) / l)))
    else
        tmp = (1.0d0 + ((-0.5d0) * ((h / l) * ((m * ((d_1 * (-0.5d0)) / d)) ** 2.0d0)))) * (d / (sqrt(h) * sqrt(l)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt(-d);
	double tmp;
	if (l <= -3.8e-241) {
		tmp = (t_0 / Math.sqrt(-h)) * (Math.sqrt((d / l)) * (1.0 - (0.5 * ((h * Math.pow(((D / d) * (M * -0.5)), 2.0)) / l))));
	} else if (l <= -2e-310) {
		tmp = (t_0 / Math.sqrt(-l)) * (Math.sqrt((d / h)) * (1.0 + ((h * (Math.pow((M / (2.0 * (d / D))), 2.0) * -0.5)) / l)));
	} else {
		tmp = (1.0 + (-0.5 * ((h / l) * Math.pow((M * ((D * -0.5) / d)), 2.0)))) * (d / (Math.sqrt(h) * Math.sqrt(l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt(-d)
	tmp = 0
	if l <= -3.8e-241:
		tmp = (t_0 / math.sqrt(-h)) * (math.sqrt((d / l)) * (1.0 - (0.5 * ((h * math.pow(((D / d) * (M * -0.5)), 2.0)) / l))))
	elif l <= -2e-310:
		tmp = (t_0 / math.sqrt(-l)) * (math.sqrt((d / h)) * (1.0 + ((h * (math.pow((M / (2.0 * (d / D))), 2.0) * -0.5)) / l)))
	else:
		tmp = (1.0 + (-0.5 * ((h / l) * math.pow((M * ((D * -0.5) / d)), 2.0)))) * (d / (math.sqrt(h) * math.sqrt(l)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(-d))
	tmp = 0.0
	if (l <= -3.8e-241)
		tmp = Float64(Float64(t_0 / sqrt(Float64(-h))) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(Float64(D / d) * Float64(M * -0.5)) ^ 2.0)) / l)))));
	elseif (l <= -2e-310)
		tmp = Float64(Float64(t_0 / sqrt(Float64(-l))) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(h * Float64((Float64(M / Float64(2.0 * Float64(d / D))) ^ 2.0) * -0.5)) / l))));
	else
		tmp = Float64(Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(M * Float64(Float64(D * -0.5) / d)) ^ 2.0)))) * Float64(d / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(-d);
	tmp = 0.0;
	if (l <= -3.8e-241)
		tmp = (t_0 / sqrt(-h)) * (sqrt((d / l)) * (1.0 - (0.5 * ((h * (((D / d) * (M * -0.5)) ^ 2.0)) / l))));
	elseif (l <= -2e-310)
		tmp = (t_0 / sqrt(-l)) * (sqrt((d / h)) * (1.0 + ((h * (((M / (2.0 * (d / D))) ^ 2.0) * -0.5)) / l)));
	else
		tmp = (1.0 + (-0.5 * ((h / l) * ((M * ((D * -0.5) / d)) ^ 2.0)))) * (d / (sqrt(h) * sqrt(l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[l, -3.8e-241], N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M * -0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-310], N[(N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h * N[(N[Power[N[(M / N[(2.0 * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(M * N[(N[(D * -0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.7999999999999999e-241

   1. Initial program 70.7%

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

   3. Simplified 71.5%

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

      1. frac-2neg 71.5%

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

      2. sqrt-div 84.1%

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

   5. Applied egg-rr 84.1%

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

      1. associate-*r/ 85.9%

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

   7. Applied egg-rr 85.9%

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

   if -3.7999999999999999e-241 < l < -1.999999999999994e-310

   1. Initial program 70.0%

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

   3. Simplified 70.0%

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

   5. Applied egg-rr 82.6%

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

      1. add-sqr-sqrt 44.7%

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

      2. sqrt-prod 82.6%

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

      3. unpow2 82.6%

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

      4. unpow-prod-down 63.8%

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

      5. unpow-prod-down 63.8%

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

      6. metadata-eval 63.8%

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

      7. metadata-eval 63.8%

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

      8. unpow-prod-down 63.8%

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

      9. metadata-eval 63.8%

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

      10. div-inv 63.8%

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

      11. unpow-prod-down 82.6%

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

      12. unpow2 82.6%

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

      13. sqrt-prod 38.9%

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

      14. add-sqr-sqrt 82.6%

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

      15. *-commutative 82.6%

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

      16. clear-num 82.6%

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

      17. frac-times 82.6%

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

      18. *-un-lft-identity 82.6%

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

   7. Applied egg-rr 82.6%

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

      1. frac-2neg 70.0%

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

      2. sqrt-div 75.7%

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

   9. Applied egg-rr 99.8%

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

   if -1.999999999999994e-310 < l

   1. Initial program 69.2%

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

   3. Simplified 68.6%

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

      1. pow1/2 68.6%

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

      2. metadata-eval 68.6%

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

      3. add-cbrt-cube 55.7%

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

      4. pow1/3 53.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{{\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)}^{0.3333333333333333}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. sqr-pow 53.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{1}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)}^{0.3333333333333333} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. pow-prod-up 53.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(1 + \frac{1}{2}\right)}\right)}}^{0.3333333333333333} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. metadata-eval 53.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left({\left(\frac{d}{\ell}\right)}^{\left(1 + \color{blue}{0.5}\right)}\right)}^{0.3333333333333333} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. metadata-eval 53.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left({\left(\frac{d}{\ell}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   5. Applied egg-rr 53.9%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{{\left({\left(\frac{d}{\ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
   6. Step-by-step derivation

      1. unpow1/3 55.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt[3]{{\left(\frac{d}{\ell}\right)}^{1.5}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   7. Simplified 55.7%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt[3]{{\left(\frac{d}{\ell}\right)}^{1.5}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   9. Applied egg-rr 34.0%

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

       1. expm1-def 51.9%

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

       2. expm1-log1p 81.6%

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

       3. associate-*r/ 81.6%

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

   11. Simplified 81.6%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.8 \cdot 10^{-241}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \left(M \cdot -0.5\right)\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left({\left(\frac{M}{2 \cdot \frac{d}{D}}\right)}^{2} \cdot -0.5\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D \cdot -0.5}{d}\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 5: 77.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(M \cdot D\right) \cdot \frac{0.5}{d}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -1.4 \cdot 10^{-231}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \cdot t_1\right)\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{-277}:\\ \;\;\;\;t_1 \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \frac{-0.5}{\frac{1}{t_0} \cdot \frac{\ell}{t_0}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D \cdot -0.5}{d}\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (* M D) (/ 0.5 d))) (t_1 (sqrt (/ d l))))
   (if (<= l -1.4e-231)
     (*
      (/ (sqrt (- d)) (sqrt (- h)))
      (* (- 1.0 (* 0.5 (* (pow (* (/ M 2.0) (/ D d)) 2.0) (/ h l)))) t_1))
     (if (<= l 7e-277)
       (*
        t_1
        (* (sqrt (/ d h)) (+ 1.0 (* h (/ -0.5 (* (/ 1.0 t_0) (/ l t_0)))))))
       (*
        (+ 1.0 (* -0.5 (* (/ h l) (pow (* M (/ (* D -0.5) d)) 2.0))))
        (/ d (* (sqrt h) (sqrt l))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (M * D) * (0.5 / d);
	double t_1 = sqrt((d / l));
	double tmp;
	if (l <= -1.4e-231) {
		tmp = (sqrt(-d) / sqrt(-h)) * ((1.0 - (0.5 * (pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))) * t_1);
	} else if (l <= 7e-277) {
		tmp = t_1 * (sqrt((d / h)) * (1.0 + (h * (-0.5 / ((1.0 / t_0) * (l / t_0))))));
	} else {
		tmp = (1.0 + (-0.5 * ((h / l) * pow((M * ((D * -0.5) / d)), 2.0)))) * (d / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (m * d_1) * (0.5d0 / d)
    t_1 = sqrt((d / l))
    if (l <= (-1.4d-231)) then
        tmp = (sqrt(-d) / sqrt(-h)) * ((1.0d0 - (0.5d0 * ((((m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (h / l)))) * t_1)
    else if (l <= 7d-277) then
        tmp = t_1 * (sqrt((d / h)) * (1.0d0 + (h * ((-0.5d0) / ((1.0d0 / t_0) * (l / t_0))))))
    else
        tmp = (1.0d0 + ((-0.5d0) * ((h / l) * ((m * ((d_1 * (-0.5d0)) / d)) ** 2.0d0)))) * (d / (sqrt(h) * 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 = (M * D) * (0.5 / d);
	double t_1 = Math.sqrt((d / l));
	double tmp;
	if (l <= -1.4e-231) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-h)) * ((1.0 - (0.5 * (Math.pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))) * t_1);
	} else if (l <= 7e-277) {
		tmp = t_1 * (Math.sqrt((d / h)) * (1.0 + (h * (-0.5 / ((1.0 / t_0) * (l / t_0))))));
	} else {
		tmp = (1.0 + (-0.5 * ((h / l) * Math.pow((M * ((D * -0.5) / d)), 2.0)))) * (d / (Math.sqrt(h) * Math.sqrt(l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (M * D) * (0.5 / d)
	t_1 = math.sqrt((d / l))
	tmp = 0
	if l <= -1.4e-231:
		tmp = (math.sqrt(-d) / math.sqrt(-h)) * ((1.0 - (0.5 * (math.pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))) * t_1)
	elif l <= 7e-277:
		tmp = t_1 * (math.sqrt((d / h)) * (1.0 + (h * (-0.5 / ((1.0 / t_0) * (l / t_0))))))
	else:
		tmp = (1.0 + (-0.5 * ((h / l) * math.pow((M * ((D * -0.5) / d)), 2.0)))) * (d / (math.sqrt(h) * math.sqrt(l)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(M * D) * Float64(0.5 / d))
	t_1 = sqrt(Float64(d / l))
	tmp = 0.0
	if (l <= -1.4e-231)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(Float64(1.0 - Float64(0.5 * Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * Float64(h / l)))) * t_1));
	elseif (l <= 7e-277)
		tmp = Float64(t_1 * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(h * Float64(-0.5 / Float64(Float64(1.0 / t_0) * Float64(l / t_0)))))));
	else
		tmp = Float64(Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(M * Float64(Float64(D * -0.5) / d)) ^ 2.0)))) * Float64(d / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (M * D) * (0.5 / d);
	t_1 = sqrt((d / l));
	tmp = 0.0;
	if (l <= -1.4e-231)
		tmp = (sqrt(-d) / sqrt(-h)) * ((1.0 - (0.5 * ((((M / 2.0) * (D / d)) ^ 2.0) * (h / l)))) * t_1);
	elseif (l <= 7e-277)
		tmp = t_1 * (sqrt((d / h)) * (1.0 + (h * (-0.5 / ((1.0 / t_0) * (l / t_0))))));
	else
		tmp = (1.0 + (-0.5 * ((h / l) * ((M * ((D * -0.5) / d)) ^ 2.0)))) * (d / (sqrt(h) * sqrt(l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(M * D), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1.4e-231], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[(0.5 * N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 7e-277], N[(t$95$1 * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(h * N[(-0.5 / N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(l / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(M * N[(N[(D * -0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.3999999999999999e-231

   1. Initial program 71.1%

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

   3. Simplified 71.9%

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

      1. frac-2neg 71.9%

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

      2. sqrt-div 84.7%

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

   5. Applied egg-rr 84.7%

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

   if -1.3999999999999999e-231 < l < 6.99999999999999966e-277

   1. Initial program 64.8%

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

   3. Simplified 64.8%

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

   5. Applied egg-rr 76.7%

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

      1. add-sqr-sqrt 45.0%

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

      2. sqrt-prod 76.7%

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

      3. unpow2 76.7%

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

      4. unpow-prod-down 52.6%

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

      5. unpow-prod-down 52.6%

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

      6. metadata-eval 52.6%

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

      7. metadata-eval 52.6%

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

      8. unpow-prod-down 52.6%

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

      9. metadata-eval 52.6%

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

      10. div-inv 52.6%

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

      11. unpow-prod-down 76.7%

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

      12. unpow2 76.7%

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

      13. sqrt-prod 32.7%

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

      14. add-sqr-sqrt 76.7%

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

      15. *-commutative 76.7%

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

      16. clear-num 76.7%

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

      17. frac-times 76.7%

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

      18. *-un-lft-identity 76.7%

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

   7. Applied egg-rr 76.7%

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

      1. expm1-log1p-u 4.7%

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

      2. expm1-udef 4.7%

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

      3. log1p-udef 4.7%

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

      4. +-commutative 4.7%

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

      5. add-exp-log 76.7%

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

      6. associate-/l* 64.8%

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

   9. Applied egg-rr 64.8%

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

       1. associate--l+ 64.8%

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

       2. metadata-eval 64.8%

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

       3. +-rgt-identity 64.8%

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

       4. associate-/r/ 76.7%

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

       5. *-commutative 76.7%

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

       6. *-commutative 76.7%

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

       7. associate-/l* 76.7%

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

       8. associate-*l/ 76.7%

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

       9. associate-/r/ 72.6%

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

       10. *-commutative 72.6%

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

   11. Simplified 72.6%

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

       1. *-un-lft-identity 72.6%

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

       2. unpow2 72.6%

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

       3. times-frac 77.2%

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

       4. *-commutative 77.2%

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

       5. div-inv 77.3%

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

       6. associate-*r* 77.3%

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

       7. associate-/r* 77.3%

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

       8. metadata-eval 77.3%

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

       9. *-commutative 77.3%

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

       10. div-inv 77.2%

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

       11. associate-*r* 81.3%

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

       12. associate-/r* 81.3%

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

       13. metadata-eval 81.3%

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

   13. Applied egg-rr 81.3%

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

   if 6.99999999999999966e-277 < l

   1. Initial program 69.9%

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

   3. Simplified 69.3%

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

      1. pow1/2 69.3%

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

      2. metadata-eval 69.3%

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

      3. add-cbrt-cube 55.6%

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

      4. pow1/3 53.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{{\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)}^{0.3333333333333333}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. sqr-pow 53.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{1}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)}^{0.3333333333333333} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. pow-prod-up 53.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(1 + \frac{1}{2}\right)}\right)}}^{0.3333333333333333} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. metadata-eval 53.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left({\left(\frac{d}{\ell}\right)}^{\left(1 + \color{blue}{0.5}\right)}\right)}^{0.3333333333333333} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. metadata-eval 53.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left({\left(\frac{d}{\ell}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   5. Applied egg-rr 53.7%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{{\left({\left(\frac{d}{\ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
   6. Step-by-step derivation

      1. unpow1/3 55.6%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt[3]{{\left(\frac{d}{\ell}\right)}^{1.5}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   7. Simplified 55.6%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt[3]{{\left(\frac{d}{\ell}\right)}^{1.5}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   9. Applied egg-rr 36.0%

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

       1. expm1-def 54.9%

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

       2. expm1-log1p 83.0%

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

       3. associate-*r/ 83.0%

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

   11. Simplified 83.0%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.4 \cdot 10^{-231}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{-277}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \frac{-0.5}{\frac{1}{\left(M \cdot D\right) \cdot \frac{0.5}{d}} \cdot \frac{\ell}{\left(M \cdot D\right) \cdot \frac{0.5}{d}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D \cdot -0.5}{d}\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 6: 79.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -2e-310)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(Float64(D / d) * Float64(M * -0.5)) ^ 2.0)) / l)))));
	else
		tmp = Float64(Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(M * Float64(Float64(D * -0.5) / d)) ^ 2.0)))) * Float64(d / Float64(sqrt(h) * sqrt(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) / sqrt(-h)) * (sqrt((d / l)) * (1.0 - (0.5 * ((h * (((D / d) * (M * -0.5)) ^ 2.0)) / l))));
	else
		tmp = (1.0 + (-0.5 * ((h / l) * ((M * ((D * -0.5) / d)) ^ 2.0)))) * (d / (sqrt(h) * sqrt(l)));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if l < -1.999999999999994e-310

   1. Initial program 70.6%

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

   3. Simplified 71.3%

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

      1. frac-2neg 71.3%

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

      2. sqrt-div 82.4%

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

   5. Applied egg-rr 82.4%

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

      1. associate-*r/ 84.1%

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

   7. Applied egg-rr 84.1%

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

   if -1.999999999999994e-310 < l

   1. Initial program 69.2%

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

   3. Simplified 68.6%

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

      1. pow1/2 68.6%

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

      2. metadata-eval 68.6%

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

      3. add-cbrt-cube 55.7%

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

      4. pow1/3 53.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{{\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)}^{0.3333333333333333}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. sqr-pow 53.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{1}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)}^{0.3333333333333333} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. pow-prod-up 53.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(1 + \frac{1}{2}\right)}\right)}}^{0.3333333333333333} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. metadata-eval 53.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left({\left(\frac{d}{\ell}\right)}^{\left(1 + \color{blue}{0.5}\right)}\right)}^{0.3333333333333333} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. metadata-eval 53.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left({\left(\frac{d}{\ell}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   5. Applied egg-rr 53.9%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{{\left({\left(\frac{d}{\ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
   6. Step-by-step derivation

      1. unpow1/3 55.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt[3]{{\left(\frac{d}{\ell}\right)}^{1.5}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   7. Simplified 55.7%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt[3]{{\left(\frac{d}{\ell}\right)}^{1.5}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   9. Applied egg-rr 34.0%

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

       1. expm1-def 51.9%

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

       2. expm1-log1p 81.6%

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

       3. associate-*r/ 81.6%

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

   11. Simplified 81.6%

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

4. Final simplification 82.8%

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

Alternative 7: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(M \cdot D\right) \cdot \frac{0.5}{d}\\ \mathbf{if}\;\ell \leq -8 \cdot 10^{+134}:\\ \;\;\;\;d \cdot \left({\left(-\ell\right)}^{-0.5} \cdot \left(-{\left(\frac{-1}{h}\right)}^{0.5}\right)\right)\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{-275}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \frac{-0.5}{\frac{1}{t_0} \cdot \frac{\ell}{t_0}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D \cdot -0.5}{d}\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (* M D) (/ 0.5 d))))
   (if (<= l -8e+134)
     (* d (* (pow (- l) -0.5) (- (pow (/ -1.0 h) 0.5))))
     (if (<= l 2.7e-275)
       (*
        (sqrt (/ d l))
        (* (sqrt (/ d h)) (+ 1.0 (* h (/ -0.5 (* (/ 1.0 t_0) (/ l t_0)))))))
       (*
        (+ 1.0 (* -0.5 (* (/ h l) (pow (* M (/ (* D -0.5) d)) 2.0))))
        (/ d (* (sqrt h) (sqrt l))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (M * D) * (0.5 / d);
	double tmp;
	if (l <= -8e+134) {
		tmp = d * (pow(-l, -0.5) * -pow((-1.0 / h), 0.5));
	} else if (l <= 2.7e-275) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + (h * (-0.5 / ((1.0 / t_0) * (l / t_0))))));
	} else {
		tmp = (1.0 + (-0.5 * ((h / l) * pow((M * ((D * -0.5) / d)), 2.0)))) * (d / (sqrt(h) * 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 = (m * d_1) * (0.5d0 / d)
    if (l <= (-8d+134)) then
        tmp = d * ((-l ** (-0.5d0)) * -(((-1.0d0) / h) ** 0.5d0))
    else if (l <= 2.7d-275) then
        tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0d0 + (h * ((-0.5d0) / ((1.0d0 / t_0) * (l / t_0))))))
    else
        tmp = (1.0d0 + ((-0.5d0) * ((h / l) * ((m * ((d_1 * (-0.5d0)) / d)) ** 2.0d0)))) * (d / (sqrt(h) * 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 = (M * D) * (0.5 / d);
	double tmp;
	if (l <= -8e+134) {
		tmp = d * (Math.pow(-l, -0.5) * -Math.pow((-1.0 / h), 0.5));
	} else if (l <= 2.7e-275) {
		tmp = Math.sqrt((d / l)) * (Math.sqrt((d / h)) * (1.0 + (h * (-0.5 / ((1.0 / t_0) * (l / t_0))))));
	} else {
		tmp = (1.0 + (-0.5 * ((h / l) * Math.pow((M * ((D * -0.5) / d)), 2.0)))) * (d / (Math.sqrt(h) * Math.sqrt(l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (M * D) * (0.5 / d)
	tmp = 0
	if l <= -8e+134:
		tmp = d * (math.pow(-l, -0.5) * -math.pow((-1.0 / h), 0.5))
	elif l <= 2.7e-275:
		tmp = math.sqrt((d / l)) * (math.sqrt((d / h)) * (1.0 + (h * (-0.5 / ((1.0 / t_0) * (l / t_0))))))
	else:
		tmp = (1.0 + (-0.5 * ((h / l) * math.pow((M * ((D * -0.5) / d)), 2.0)))) * (d / (math.sqrt(h) * math.sqrt(l)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(M * D) * Float64(0.5 / d))
	tmp = 0.0
	if (l <= -8e+134)
		tmp = Float64(d * Float64((Float64(-l) ^ -0.5) * Float64(-(Float64(-1.0 / h) ^ 0.5))));
	elseif (l <= 2.7e-275)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(h * Float64(-0.5 / Float64(Float64(1.0 / t_0) * Float64(l / t_0)))))));
	else
		tmp = Float64(Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(M * Float64(Float64(D * -0.5) / d)) ^ 2.0)))) * Float64(d / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (M * D) * (0.5 / d);
	tmp = 0.0;
	if (l <= -8e+134)
		tmp = d * ((-l ^ -0.5) * -((-1.0 / h) ^ 0.5));
	elseif (l <= 2.7e-275)
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + (h * (-0.5 / ((1.0 / t_0) * (l / t_0))))));
	else
		tmp = (1.0 + (-0.5 * ((h / l) * ((M * ((D * -0.5) / d)) ^ 2.0)))) * (d / (sqrt(h) * sqrt(l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(M * D), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -8e+134], N[(d * N[(N[Power[(-l), -0.5], $MachinePrecision] * (-N[Power[N[(-1.0 / h), $MachinePrecision], 0.5], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.7e-275], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(h * N[(-0.5 / N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(l / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(M * N[(N[(D * -0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -7.99999999999999937e134

   1. Initial program 56.0%

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

   3. Simplified 58.5%

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

      1. frac-2neg 58.5%

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

      2. sqrt-div 79.7%

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

   5. Applied egg-rr 79.7%

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

   6. Taylor expanded in d around -inf 61.3%

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

      1. mul-1-neg 61.3%

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

      2. distribute-rgt-neg-in 61.3%

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

   8. Simplified 61.3%

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

   9. Taylor expanded in d around 0 61.3%

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

       1. mul-1-neg 61.3%

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

       2. *-commutative 61.3%

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

       3. unpow1/2 61.3%

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

       4. rem-exp-log 57.3%

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

       5. exp-neg 57.3%

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

       6. exp-prod 57.3%

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

       7. distribute-lft-neg-out 57.3%

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

       8. distribute-rgt-neg-in 57.3%

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

       9. metadata-eval 57.3%

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

       10. exp-to-pow 61.3%

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

       11. distribute-rgt-neg-in 61.3%

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

   11. Simplified 61.3%

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

   12. Taylor expanded in h around -inf 71.0%

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

       1. distribute-lft-in 71.0%

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

       2. exp-sum 71.1%

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

       3. *-commutative 71.1%

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

       4. neg-mul-1 71.1%

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

       5. exp-to-pow 72.6%

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

       6. *-commutative 72.6%

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

       7. *-commutative 72.6%

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

       8. associate-*l* 72.6%

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

       9. metadata-eval 72.6%

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

       10. exp-to-pow 77.7%

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

   14. Simplified 77.7%

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

   if -7.99999999999999937e134 < l < 2.69999999999999993e-275

   1. Initial program 75.6%

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

   3. Simplified 75.5%

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

   5. Applied egg-rr 80.8%

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

      1. add-sqr-sqrt 54.1%

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

      2. sqrt-prod 80.8%

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

      3. unpow2 80.8%

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

      4. unpow-prod-down 66.3%

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

      5. unpow-prod-down 66.3%

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

      6. metadata-eval 66.3%

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

      7. metadata-eval 66.3%

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

      8. unpow-prod-down 66.3%

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

      9. metadata-eval 66.3%

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

      10. div-inv 66.3%

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

      11. unpow-prod-down 80.8%

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

      12. unpow2 80.8%

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

      13. sqrt-prod 41.6%

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

      14. add-sqr-sqrt 80.8%

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

      15. *-commutative 80.8%

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

      16. clear-num 80.8%

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

      17. frac-times 80.8%

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

      18. *-un-lft-identity 80.8%

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

   7. Applied egg-rr 80.8%

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

      1. expm1-log1p-u 32.3%

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

      2. expm1-udef 32.3%

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

      3. log1p-udef 32.3%

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

      4. +-commutative 32.3%

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

      5. add-exp-log 80.8%

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

      6. associate-/l* 75.5%

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

   9. Applied egg-rr 75.5%

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

       1. associate--l+ 75.5%

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

       2. metadata-eval 75.5%

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

       3. +-rgt-identity 75.5%

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

       4. associate-/r/ 79.8%

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

       5. *-commutative 79.8%

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

       6. *-commutative 79.8%

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

       7. associate-/l* 79.8%

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

       8. associate-*l/ 79.8%

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

       9. associate-/r/ 77.2%

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

       10. *-commutative 77.2%

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

   11. Simplified 77.2%

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

       1. *-un-lft-identity 77.2%

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

       2. unpow2 77.2%

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

       3. times-frac 79.4%

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

       4. *-commutative 79.4%

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

       5. div-inv 79.4%

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

       6. associate-*r* 79.4%

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

       7. associate-/r* 79.4%

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

       8. metadata-eval 79.4%

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

       9. *-commutative 79.4%

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

       10. div-inv 79.4%

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

       11. associate-*r* 82.0%

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

       12. associate-/r* 82.0%

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

       13. metadata-eval 82.0%

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

   13. Applied egg-rr 82.0%

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

   if 2.69999999999999993e-275 < l

   1. Initial program 69.9%

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

   3. Simplified 69.3%

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

      1. pow1/2 69.3%

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

      2. metadata-eval 69.3%

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

      3. add-cbrt-cube 55.6%

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

      4. pow1/3 53.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{{\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)}^{0.3333333333333333}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. sqr-pow 53.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{1}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)}^{0.3333333333333333} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. pow-prod-up 53.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(1 + \frac{1}{2}\right)}\right)}}^{0.3333333333333333} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. metadata-eval 53.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left({\left(\frac{d}{\ell}\right)}^{\left(1 + \color{blue}{0.5}\right)}\right)}^{0.3333333333333333} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. metadata-eval 53.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left({\left(\frac{d}{\ell}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   5. Applied egg-rr 53.7%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{{\left({\left(\frac{d}{\ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
   6. Step-by-step derivation

      1. unpow1/3 55.6%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt[3]{{\left(\frac{d}{\ell}\right)}^{1.5}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   7. Simplified 55.6%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt[3]{{\left(\frac{d}{\ell}\right)}^{1.5}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   9. Applied egg-rr 36.0%

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

       1. expm1-def 54.9%

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

       2. expm1-log1p 83.0%

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

       3. associate-*r/ 83.0%

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

   11. Simplified 83.0%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8 \cdot 10^{+134}:\\ \;\;\;\;d \cdot \left({\left(-\ell\right)}^{-0.5} \cdot \left(-{\left(\frac{-1}{h}\right)}^{0.5}\right)\right)\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{-275}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \frac{-0.5}{\frac{1}{\left(M \cdot D\right) \cdot \frac{0.5}{d}} \cdot \frac{\ell}{\left(M \cdot D\right) \cdot \frac{0.5}{d}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D \cdot -0.5}{d}\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 8: 71.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(M \cdot D\right) \cdot \frac{0.5}{d}\\ \mathbf{if}\;\ell \leq -1.3 \cdot 10^{+134}:\\ \;\;\;\;d \cdot \left({\left(-\ell\right)}^{-0.5} \cdot \left(-{\left(\frac{-1}{h}\right)}^{0.5}\right)\right)\\ \mathbf{elif}\;\ell \leq 8.5 \cdot 10^{+121}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \frac{-0.5}{\frac{1}{t_0} \cdot \frac{\ell}{t_0}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (* M D) (/ 0.5 d))))
   (if (<= l -1.3e+134)
     (* d (* (pow (- l) -0.5) (- (pow (/ -1.0 h) 0.5))))
     (if (<= l 8.5e+121)
       (*
        (sqrt (/ d l))
        (* (sqrt (/ d h)) (+ 1.0 (* h (/ -0.5 (* (/ 1.0 t_0) (/ l t_0)))))))
       (/ d (* (sqrt h) (sqrt l)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (M * D) * (0.5 / d);
	double tmp;
	if (l <= -1.3e+134) {
		tmp = d * (pow(-l, -0.5) * -pow((-1.0 / h), 0.5));
	} else if (l <= 8.5e+121) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + (h * (-0.5 / ((1.0 / t_0) * (l / t_0))))));
	} else {
		tmp = d / (sqrt(h) * 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 = (m * d_1) * (0.5d0 / d)
    if (l <= (-1.3d+134)) then
        tmp = d * ((-l ** (-0.5d0)) * -(((-1.0d0) / h) ** 0.5d0))
    else if (l <= 8.5d+121) then
        tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0d0 + (h * ((-0.5d0) / ((1.0d0 / t_0) * (l / t_0))))))
    else
        tmp = d / (sqrt(h) * 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 = (M * D) * (0.5 / d);
	double tmp;
	if (l <= -1.3e+134) {
		tmp = d * (Math.pow(-l, -0.5) * -Math.pow((-1.0 / h), 0.5));
	} else if (l <= 8.5e+121) {
		tmp = Math.sqrt((d / l)) * (Math.sqrt((d / h)) * (1.0 + (h * (-0.5 / ((1.0 / t_0) * (l / t_0))))));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (M * D) * (0.5 / d)
	tmp = 0
	if l <= -1.3e+134:
		tmp = d * (math.pow(-l, -0.5) * -math.pow((-1.0 / h), 0.5))
	elif l <= 8.5e+121:
		tmp = math.sqrt((d / l)) * (math.sqrt((d / h)) * (1.0 + (h * (-0.5 / ((1.0 / t_0) * (l / t_0))))))
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(M * D) * Float64(0.5 / d))
	tmp = 0.0
	if (l <= -1.3e+134)
		tmp = Float64(d * Float64((Float64(-l) ^ -0.5) * Float64(-(Float64(-1.0 / h) ^ 0.5))));
	elseif (l <= 8.5e+121)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(h * Float64(-0.5 / Float64(Float64(1.0 / t_0) * Float64(l / t_0)))))));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (M * D) * (0.5 / d);
	tmp = 0.0;
	if (l <= -1.3e+134)
		tmp = d * ((-l ^ -0.5) * -((-1.0 / h) ^ 0.5));
	elseif (l <= 8.5e+121)
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + (h * (-0.5 / ((1.0 / t_0) * (l / t_0))))));
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(M * D), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.3e+134], N[(d * N[(N[Power[(-l), -0.5], $MachinePrecision] * (-N[Power[N[(-1.0 / h), $MachinePrecision], 0.5], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 8.5e+121], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(h * N[(-0.5 / N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(l / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.3000000000000001e134

   1. Initial program 56.0%

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

   3. Simplified 58.5%

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

      1. frac-2neg 58.5%

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

      2. sqrt-div 79.7%

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

   5. Applied egg-rr 79.7%

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

   6. Taylor expanded in d around -inf 61.3%

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

      1. mul-1-neg 61.3%

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

      2. distribute-rgt-neg-in 61.3%

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

   8. Simplified 61.3%

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

   9. Taylor expanded in d around 0 61.3%

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

       1. mul-1-neg 61.3%

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

       2. *-commutative 61.3%

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

       3. unpow1/2 61.3%

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

       4. rem-exp-log 57.3%

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

       5. exp-neg 57.3%

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

       6. exp-prod 57.3%

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

       7. distribute-lft-neg-out 57.3%

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

       8. distribute-rgt-neg-in 57.3%

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

       9. metadata-eval 57.3%

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

       10. exp-to-pow 61.3%

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

       11. distribute-rgt-neg-in 61.3%

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

   11. Simplified 61.3%

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

   12. Taylor expanded in h around -inf 71.0%

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

       1. distribute-lft-in 71.0%

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

       2. exp-sum 71.1%

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

       3. *-commutative 71.1%

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

       4. neg-mul-1 71.1%

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

       5. exp-to-pow 72.6%

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

       6. *-commutative 72.6%

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

       7. *-commutative 72.6%

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

       8. associate-*l* 72.6%

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

       9. metadata-eval 72.6%

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

       10. exp-to-pow 77.7%

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

   14. Simplified 77.7%

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

   if -1.3000000000000001e134 < l < 8.5e121

   1. Initial program 74.8%

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

   3. Simplified 74.3%

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

   5. Applied egg-rr 78.2%

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

      1. add-sqr-sqrt 48.7%

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

      2. sqrt-prod 78.2%

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

      3. unpow2 78.2%

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

      4. unpow-prod-down 62.9%

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

      5. unpow-prod-down 62.9%

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

      6. metadata-eval 62.9%

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

      7. metadata-eval 62.9%

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

      8. unpow-prod-down 62.9%

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

      9. metadata-eval 62.9%

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

      10. div-inv 62.9%

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

      11. unpow-prod-down 78.2%

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

      12. unpow2 78.2%

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

      13. sqrt-prod 47.3%

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

      14. add-sqr-sqrt 78.2%

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

      15. *-commutative 78.2%

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

      16. clear-num 78.2%

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

      17. frac-times 78.2%

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

      18. *-un-lft-identity 78.2%

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

   7. Applied egg-rr 78.2%

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

      1. expm1-log1p-u 36.6%

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

      2. expm1-udef 36.6%

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

      3. log1p-udef 36.6%

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

      4. +-commutative 36.6%

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

      5. add-exp-log 78.2%

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

      6. associate-/l* 74.4%

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

   9. Applied egg-rr 74.4%

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

       1. associate--l+ 74.4%

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

       2. metadata-eval 74.4%

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

       3. +-rgt-identity 74.4%

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

       4. associate-/r/ 77.7%

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

       5. *-commutative 77.7%

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

       6. *-commutative 77.7%

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

       7. associate-/l* 77.7%

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

       8. associate-*l/ 77.7%

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

       9. associate-/r/ 76.3%

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

       10. *-commutative 76.3%

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

   11. Simplified 76.3%

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

       1. *-un-lft-identity 76.3%

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

       2. unpow2 76.3%

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

       3. times-frac 78.0%

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

       4. *-commutative 78.0%

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

       5. div-inv 78.0%

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

       6. associate-*r* 78.0%

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

       7. associate-/r* 78.0%

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

       8. metadata-eval 78.0%

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

       9. *-commutative 78.0%

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

       10. div-inv 78.0%

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

       11. associate-*r* 79.8%

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

       12. associate-/r* 79.8%

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

       13. metadata-eval 79.8%

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

   13. Applied egg-rr 79.8%

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

   if 8.5e121 < l

   1. Initial program 59.5%

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

   3. Simplified 59.5%

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

   4. Taylor expanded in M around 0 65.8%

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

      1. *-commutative 65.8%

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

      2. sqrt-div 73.2%

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

      3. sqrt-div 78.8%

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

      4. frac-times 78.6%

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

      5. add-sqr-sqrt 78.8%

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

   6. Applied egg-rr 78.8%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.3 \cdot 10^{+134}:\\ \;\;\;\;d \cdot \left({\left(-\ell\right)}^{-0.5} \cdot \left(-{\left(\frac{-1}{h}\right)}^{0.5}\right)\right)\\ \mathbf{elif}\;\ell \leq 8.5 \cdot 10^{+121}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \frac{-0.5}{\frac{1}{\left(M \cdot D\right) \cdot \frac{0.5}{d}} \cdot \frac{\ell}{\left(M \cdot D\right) \cdot \frac{0.5}{d}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 9: 47.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.3 \cdot 10^{-242}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-309}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{-253}:\\ \;\;\;\;d \cdot \left(-e^{0.5 \cdot \left(-\mathsf{log1p}\left(-1 + \ell \cdot h\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -4.3e-242)
   (* d (- (sqrt (/ (/ 1.0 l) h))))
   (if (<= l -5e-309)
     (/ d (sqrt (* l h)))
     (if (<= l 1.55e-253)
       (* d (- (exp (* 0.5 (- (log1p (+ -1.0 (* l h))))))))
       (/ d (* (sqrt h) (sqrt l)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -4.3e-242) {
		tmp = d * -sqrt(((1.0 / l) / h));
	} else if (l <= -5e-309) {
		tmp = d / sqrt((l * h));
	} else if (l <= 1.55e-253) {
		tmp = d * -exp((0.5 * -log1p((-1.0 + (l * h)))));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -4.3e-242) {
		tmp = d * -Math.sqrt(((1.0 / l) / h));
	} else if (l <= -5e-309) {
		tmp = d / Math.sqrt((l * h));
	} else if (l <= 1.55e-253) {
		tmp = d * -Math.exp((0.5 * -Math.log1p((-1.0 + (l * h)))));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -4.3e-242:
		tmp = d * -math.sqrt(((1.0 / l) / h))
	elif l <= -5e-309:
		tmp = d / math.sqrt((l * h))
	elif l <= 1.55e-253:
		tmp = d * -math.exp((0.5 * -math.log1p((-1.0 + (l * h)))))
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -4.3e-242)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / l) / h))));
	elseif (l <= -5e-309)
		tmp = Float64(d / sqrt(Float64(l * h)));
	elseif (l <= 1.55e-253)
		tmp = Float64(d * Float64(-exp(Float64(0.5 * Float64(-log1p(Float64(-1.0 + Float64(l * h))))))));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -4.3e-242], N[(d * (-N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, -5e-309], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.55e-253], N[(d * (-N[Exp[N[(0.5 * (-N[Log[1 + N[(-1.0 + N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -4.3000000000000002e-242

   1. Initial program 70.1%

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

   3. Simplified 70.9%

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

      1. frac-2neg 70.9%

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

      2. sqrt-div 83.4%

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

   5. Applied egg-rr 83.4%

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

   6. Taylor expanded in d around -inf 50.2%

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

      1. mul-1-neg 50.2%

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

      2. distribute-rgt-neg-in 50.2%

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

   8. Simplified 50.2%

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

      1. expm1-log1p-u 48.9%

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

      2. expm1-udef 25.1%

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

   10. Applied egg-rr 25.1%

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

       1. expm1-def 48.9%

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

       2. expm1-log1p 50.2%

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

       3. *-commutative 50.2%

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

       4. associate-/r* 51.7%

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

   12. Simplified 51.7%

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

   if -4.3000000000000002e-242 < l < -4.9999999999999995e-309

   1. Initial program 74.6%

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

   3. Simplified 68.0%

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

   4. Taylor expanded in M around 0 1.2%

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

      1. sqrt-div 0.0%

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

      2. div-inv 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. associate-*r/ 0.0%

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

      2. *-rgt-identity 0.0%

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

   8. Simplified 0.0%

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

      1. expm1-log1p-u 0.0%

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

      2. expm1-udef 0.0%

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

      3. *-commutative 0.0%

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

      4. sqrt-div 0.0%

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

      5. frac-times 0.0%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-prod 0.3%

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

   10. Applied egg-rr 0.3%

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

       1. expm1-def 0.6%

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

       2. expm1-log1p 42.2%

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

   12. Simplified 42.2%

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

   if -4.9999999999999995e-309 < l < 1.54999999999999998e-253

   1. Initial program 69.1%

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

   3. Simplified 69.9%

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

      1. frac-2neg 69.9%

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

      2. sqrt-div 0.0%

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

   5. Applied egg-rr 0.0%

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

   6. Taylor expanded in d around -inf 18.5%

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

      1. mul-1-neg 18.5%

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

      2. distribute-rgt-neg-in 18.5%

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

   8. Simplified 18.5%

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

      1. pow1/2 18.5%

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

      2. pow-to-exp 18.5%

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

      3. log-rec 18.5%

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

   10. Applied egg-rr 18.5%

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

       1. log1p-expm1-u 69.5%

          \[\leadsto d \cdot \left(-e^{\left(-\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(h \cdot \ell\right)\right)\right)}\right) \cdot 0.5}\right) \]

       2. expm1-udef 69.5%

          \[\leadsto d \cdot \left(-e^{\left(-\mathsf{log1p}\left(\color{blue}{e^{\log \left(h \cdot \ell\right)} - 1}\right)\right) \cdot 0.5}\right) \]

       3. add-exp-log 69.5%

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

       4. *-commutative 69.5%

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

   12. Applied egg-rr 69.5%

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

   if 1.54999999999999998e-253 < l

   1. Initial program 69.2%

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

   3. Simplified 68.5%

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

   4. Taylor expanded in M around 0 51.1%

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

      1. *-commutative 51.1%

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

      2. sqrt-div 56.3%

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

      3. sqrt-div 61.1%

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

      4. frac-times 61.0%

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

      5. add-sqr-sqrt 61.1%

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

   6. Applied egg-rr 61.1%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot 1 \]
3. Recombined 4 regimes into one program.

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.3 \cdot 10^{-242}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-309}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{-253}:\\ \;\;\;\;d \cdot \left(-e^{0.5 \cdot \left(-\mathsf{log1p}\left(-1 + \ell \cdot h\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 10: 51.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.3 \cdot 10^{-244}:\\ \;\;\;\;d \cdot \left({\left(-\ell\right)}^{-0.5} \cdot \left(-{\left(\frac{-1}{h}\right)}^{0.5}\right)\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{-254}:\\ \;\;\;\;d \cdot \left(-e^{0.5 \cdot \left(-\mathsf{log1p}\left(-1 + \ell \cdot h\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -2.3e-244)
   (* d (* (pow (- l) -0.5) (- (pow (/ -1.0 h) 0.5))))
   (if (<= l -2e-310)
     (/ d (sqrt (* l h)))
     (if (<= l 3.2e-254)
       (* d (- (exp (* 0.5 (- (log1p (+ -1.0 (* l h))))))))
       (/ d (* (sqrt h) (sqrt l)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -2.3e-244) {
		tmp = d * (pow(-l, -0.5) * -pow((-1.0 / h), 0.5));
	} else if (l <= -2e-310) {
		tmp = d / sqrt((l * h));
	} else if (l <= 3.2e-254) {
		tmp = d * -exp((0.5 * -log1p((-1.0 + (l * h)))));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -2.3e-244) {
		tmp = d * (Math.pow(-l, -0.5) * -Math.pow((-1.0 / h), 0.5));
	} else if (l <= -2e-310) {
		tmp = d / Math.sqrt((l * h));
	} else if (l <= 3.2e-254) {
		tmp = d * -Math.exp((0.5 * -Math.log1p((-1.0 + (l * h)))));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -2.3e-244:
		tmp = d * (math.pow(-l, -0.5) * -math.pow((-1.0 / h), 0.5))
	elif l <= -2e-310:
		tmp = d / math.sqrt((l * h))
	elif l <= 3.2e-254:
		tmp = d * -math.exp((0.5 * -math.log1p((-1.0 + (l * h)))))
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -2.3e-244)
		tmp = Float64(d * Float64((Float64(-l) ^ -0.5) * Float64(-(Float64(-1.0 / h) ^ 0.5))));
	elseif (l <= -2e-310)
		tmp = Float64(d / sqrt(Float64(l * h)));
	elseif (l <= 3.2e-254)
		tmp = Float64(d * Float64(-exp(Float64(0.5 * Float64(-log1p(Float64(-1.0 + Float64(l * h))))))));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -2.3e-244], N[(d * N[(N[Power[(-l), -0.5], $MachinePrecision] * (-N[Power[N[(-1.0 / h), $MachinePrecision], 0.5], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-310], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.2e-254], N[(d * (-N[Exp[N[(0.5 * (-N[Log[1 + N[(-1.0 + N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.3e-244

   1. Initial program 70.1%

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

   3. Simplified 70.9%

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

      1. frac-2neg 70.9%

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

      2. sqrt-div 83.4%

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

   5. Applied egg-rr 83.4%

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

   6. Taylor expanded in d around -inf 50.2%

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

      1. mul-1-neg 50.2%

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

      2. distribute-rgt-neg-in 50.2%

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

   8. Simplified 50.2%

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

   9. Taylor expanded in d around 0 50.2%

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

       1. mul-1-neg 50.2%

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

       2. *-commutative 50.2%

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

       3. unpow1/2 50.2%

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

       4. rem-exp-log 47.2%

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

       5. exp-neg 47.2%

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

       6. exp-prod 47.6%

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

       7. distribute-lft-neg-out 47.6%

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

       8. distribute-rgt-neg-in 47.6%

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

       9. metadata-eval 47.6%

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

       10. exp-to-pow 50.6%

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

       11. distribute-rgt-neg-in 50.6%

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

   11. Simplified 50.6%

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

   12. Taylor expanded in h around -inf 53.9%

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

       1. distribute-lft-in 53.9%

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

       2. exp-sum 54.2%

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

       3. *-commutative 54.2%

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

       4. neg-mul-1 54.2%

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

       5. exp-to-pow 54.8%

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

       6. *-commutative 54.8%

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

       7. *-commutative 54.8%

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

       8. associate-*l* 54.8%

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

       9. metadata-eval 54.8%

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

       10. exp-to-pow 58.3%

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

   14. Simplified 58.3%

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

   if -2.3e-244 < l < -1.999999999999994e-310

   1. Initial program 74.6%

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

   3. Simplified 68.0%

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

   4. Taylor expanded in M around 0 1.2%

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

      1. sqrt-div 0.0%

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

      2. div-inv 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. associate-*r/ 0.0%

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

      2. *-rgt-identity 0.0%

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

   8. Simplified 0.0%

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

      1. expm1-log1p-u 0.0%

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

      2. expm1-udef 0.0%

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

      3. *-commutative 0.0%

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

      4. sqrt-div 0.0%

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

      5. frac-times 0.0%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-prod 0.3%

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

   10. Applied egg-rr 0.3%

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

       1. expm1-def 0.6%

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

       2. expm1-log1p 42.2%

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

   12. Simplified 42.2%

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

   if -1.999999999999994e-310 < l < 3.2e-254

   1. Initial program 69.1%

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

   3. Simplified 69.9%

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

      1. frac-2neg 69.9%

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

      2. sqrt-div 0.0%

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

   5. Applied egg-rr 0.0%

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

   6. Taylor expanded in d around -inf 18.5%

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

      1. mul-1-neg 18.5%

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

      2. distribute-rgt-neg-in 18.5%

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

   8. Simplified 18.5%

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

      1. pow1/2 18.5%

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

      2. pow-to-exp 18.5%

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

      3. log-rec 18.5%

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

   10. Applied egg-rr 18.5%

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

       1. log1p-expm1-u 69.5%

          \[\leadsto d \cdot \left(-e^{\left(-\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(h \cdot \ell\right)\right)\right)}\right) \cdot 0.5}\right) \]

       2. expm1-udef 69.5%

          \[\leadsto d \cdot \left(-e^{\left(-\mathsf{log1p}\left(\color{blue}{e^{\log \left(h \cdot \ell\right)} - 1}\right)\right) \cdot 0.5}\right) \]

       3. add-exp-log 69.5%

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

       4. *-commutative 69.5%

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

   12. Applied egg-rr 69.5%

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

   if 3.2e-254 < l

   1. Initial program 69.2%

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

   3. Simplified 68.5%

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

   4. Taylor expanded in M around 0 51.1%

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

      1. *-commutative 51.1%

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

      2. sqrt-div 56.3%

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

      3. sqrt-div 61.1%

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

      4. frac-times 61.0%

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

      5. add-sqr-sqrt 61.1%

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

   6. Applied egg-rr 61.1%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot 1 \]
3. Recombined 4 regimes into one program.

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.3 \cdot 10^{-244}:\\ \;\;\;\;d \cdot \left({\left(-\ell\right)}^{-0.5} \cdot \left(-{\left(\frac{-1}{h}\right)}^{0.5}\right)\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{-254}:\\ \;\;\;\;d \cdot \left(-e^{0.5 \cdot \left(-\mathsf{log1p}\left(-1 + \ell \cdot h\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 11: 41.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.8 \cdot 10^{-244}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{+86}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -4.8e-244)
   (* d (- (sqrt (/ (/ 1.0 l) h))))
   (if (<= l 2e+86)
     (* d (sqrt (/ 1.0 (* l h))))
     (* (sqrt (/ d h)) (sqrt (/ d l))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -4.8e-244) {
		tmp = d * -sqrt(((1.0 / l) / h));
	} else if (l <= 2e+86) {
		tmp = d * sqrt((1.0 / (l * h)));
	} else {
		tmp = sqrt((d / h)) * sqrt((d / 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 <= (-4.8d-244)) then
        tmp = d * -sqrt(((1.0d0 / l) / h))
    else if (l <= 2d+86) then
        tmp = d * sqrt((1.0d0 / (l * h)))
    else
        tmp = sqrt((d / h)) * sqrt((d / 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 <= -4.8e-244) {
		tmp = d * -Math.sqrt(((1.0 / l) / h));
	} else if (l <= 2e+86) {
		tmp = d * Math.sqrt((1.0 / (l * h)));
	} else {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -4.8e-244:
		tmp = d * -math.sqrt(((1.0 / l) / h))
	elif l <= 2e+86:
		tmp = d * math.sqrt((1.0 / (l * h)))
	else:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -4.8e-244)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / l) / h))));
	elseif (l <= 2e+86)
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(l * h))));
	else
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -4.8e-244)
		tmp = d * -sqrt(((1.0 / l) / h));
	elseif (l <= 2e+86)
		tmp = d * sqrt((1.0 / (l * h)));
	else
		tmp = sqrt((d / h)) * sqrt((d / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -4.8e-244], N[(d * (-N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, 2e+86], N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.80000000000000032e-244

   1. Initial program 70.1%

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

   3. Simplified 70.9%

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

      1. frac-2neg 70.9%

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

      2. sqrt-div 83.4%

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

   5. Applied egg-rr 83.4%

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

   6. Taylor expanded in d around -inf 50.2%

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

      1. mul-1-neg 50.2%

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

      2. distribute-rgt-neg-in 50.2%

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

   8. Simplified 50.2%

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

      1. expm1-log1p-u 48.9%

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

      2. expm1-udef 25.1%

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

   10. Applied egg-rr 25.1%

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

       1. expm1-def 48.9%

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

       2. expm1-log1p 50.2%

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

       3. *-commutative 50.2%

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

       4. associate-/r* 51.7%

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

   12. Simplified 51.7%

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

   if -4.80000000000000032e-244 < l < 2e86

   1. Initial program 73.7%

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

   3. Simplified 71.9%

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

   4. Taylor expanded in M around 0 32.3%

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

   5. Taylor expanded in d around 0 45.1%

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

   if 2e86 < l

   1. Initial program 60.3%

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

   3. Simplified 60.3%

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

   5. Applied egg-rr 62.5%

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

      1. add-sqr-sqrt 42.1%

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

      2. sqrt-prod 62.5%

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

      3. unpow2 62.5%

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

      4. unpow-prod-down 57.5%

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

      5. unpow-prod-down 57.5%

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

      6. metadata-eval 57.5%

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

      7. metadata-eval 57.5%

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

      8. unpow-prod-down 57.5%

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

      9. metadata-eval 57.5%

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

      10. div-inv 57.5%

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

      11. unpow-prod-down 62.5%

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

      12. unpow2 62.5%

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

      13. sqrt-prod 45.4%

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

      14. add-sqr-sqrt 62.5%

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

      15. *-commutative 62.5%

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

      16. clear-num 62.5%

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

      17. frac-times 62.5%

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

      18. *-un-lft-identity 62.5%

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

   7. Applied egg-rr 62.5%

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

   8. Taylor expanded in M around 0 63.0%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.8 \cdot 10^{-244}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{+86}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \]

Alternative 12: 46.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.75 \cdot 10^{-244}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1.75e-244)
   (* d (- (sqrt (/ (/ 1.0 l) h))))
   (if (<= l -2e-310) (/ d (sqrt (* l h))) (/ d (* (sqrt h) (sqrt l))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.75e-244) {
		tmp = d * -sqrt(((1.0 / l) / h));
	} else if (l <= -2e-310) {
		tmp = d / sqrt((l * h));
	} else {
		tmp = d / (sqrt(h) * 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 <= (-1.75d-244)) then
        tmp = d * -sqrt(((1.0d0 / l) / h))
    else if (l <= (-2d-310)) then
        tmp = d / sqrt((l * h))
    else
        tmp = d / (sqrt(h) * 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 <= -1.75e-244) {
		tmp = d * -Math.sqrt(((1.0 / l) / h));
	} else if (l <= -2e-310) {
		tmp = d / Math.sqrt((l * h));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -1.75e-244:
		tmp = d * -math.sqrt(((1.0 / l) / h))
	elif l <= -2e-310:
		tmp = d / math.sqrt((l * h))
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1.75e-244)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / l) / h))));
	elseif (l <= -2e-310)
		tmp = Float64(d / sqrt(Float64(l * h)));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -1.75e-244)
		tmp = d * -sqrt(((1.0 / l) / h));
	elseif (l <= -2e-310)
		tmp = d / sqrt((l * h));
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1.75e-244], N[(d * (-N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, -2e-310], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.74999999999999996e-244

   1. Initial program 70.1%

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

   3. Simplified 70.9%

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

      1. frac-2neg 70.9%

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

      2. sqrt-div 83.4%

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

   5. Applied egg-rr 83.4%

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

   6. Taylor expanded in d around -inf 50.2%

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

      1. mul-1-neg 50.2%

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

      2. distribute-rgt-neg-in 50.2%

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

   8. Simplified 50.2%

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

      1. expm1-log1p-u 48.9%

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

      2. expm1-udef 25.1%

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

   10. Applied egg-rr 25.1%

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

       1. expm1-def 48.9%

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

       2. expm1-log1p 50.2%

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

       3. *-commutative 50.2%

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

       4. associate-/r* 51.7%

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

   12. Simplified 51.7%

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

   if -1.74999999999999996e-244 < l < -1.999999999999994e-310

   1. Initial program 74.6%

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

   3. Simplified 68.0%

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

   4. Taylor expanded in M around 0 1.2%

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

      1. sqrt-div 0.0%

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

      2. div-inv 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. associate-*r/ 0.0%

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

      2. *-rgt-identity 0.0%

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

   8. Simplified 0.0%

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

      1. expm1-log1p-u 0.0%

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

      2. expm1-udef 0.0%

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

      3. *-commutative 0.0%

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

      4. sqrt-div 0.0%

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

      5. frac-times 0.0%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-prod 0.3%

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

   10. Applied egg-rr 0.3%

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

       1. expm1-def 0.6%

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

       2. expm1-log1p 42.2%

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

   12. Simplified 42.2%

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

   if -1.999999999999994e-310 < l

   1. Initial program 69.2%

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

   3. Simplified 68.6%

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

   4. Taylor expanded in M around 0 46.0%

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

      1. *-commutative 46.0%

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

      2. sqrt-div 52.9%

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

      3. sqrt-div 57.2%

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

      4. frac-times 57.1%

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

      5. add-sqr-sqrt 57.2%

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

   6. Applied egg-rr 57.2%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.75 \cdot 10^{-244}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 13: 42.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.95 \cdot 10^{-244}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1.95e-244) (* d (- (sqrt (/ (/ 1.0 h) l)))) (/ d (sqrt (* l h)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.95e-244) {
		tmp = d * -sqrt(((1.0 / h) / l));
	} 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 <= (-1.95d-244)) then
        tmp = d * -sqrt(((1.0d0 / h) / l))
    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 <= -1.95e-244) {
		tmp = d * -Math.sqrt(((1.0 / h) / l));
	} else {
		tmp = d / Math.sqrt((l * h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -1.95e-244:
		tmp = d * -math.sqrt(((1.0 / h) / l))
	else:
		tmp = d / math.sqrt((l * h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1.95e-244)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / h) / l))));
	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 <= -1.95e-244)
		tmp = d * -sqrt(((1.0 / h) / l));
	else
		tmp = d / sqrt((l * h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1.95e-244], N[(d * (-N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.9499999999999999e-244

   1. Initial program 70.1%

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

   3. Simplified 70.9%

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

      1. frac-2neg 70.9%

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

      2. sqrt-div 83.4%

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

   5. Applied egg-rr 83.4%

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

   6. Taylor expanded in d around -inf 50.2%

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

      1. mul-1-neg 50.2%

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

      2. distribute-rgt-neg-in 50.2%

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

   8. Simplified 50.2%

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

   9. Taylor expanded in d around 0 50.2%

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

       1. mul-1-neg 50.2%

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

       2. *-commutative 50.2%

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

       3. unpow1/2 50.2%

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

       4. rem-exp-log 47.2%

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

       5. exp-neg 47.2%

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

       6. exp-prod 47.6%

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

       7. distribute-lft-neg-out 47.6%

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

       8. distribute-rgt-neg-in 47.6%

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

       9. metadata-eval 47.6%

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

       10. exp-to-pow 50.6%

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

       11. distribute-rgt-neg-in 50.6%

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

   11. Simplified 50.6%

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

   12. Taylor expanded in d around 0 50.2%

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

       1. mul-1-neg 50.2%

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

       2. distribute-rgt-neg-in 50.2%

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

       3. associate-/r* 51.7%

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

   14. Simplified 51.7%

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

   if -1.9499999999999999e-244 < l

   1. Initial program 69.8%

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

   3. Simplified 68.5%

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

   4. Taylor expanded in M around 0 41.2%

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

      1. sqrt-div 76.3%

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

      2. div-inv 76.3%

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

   6. Applied egg-rr 47.2%

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

      1. associate-*r/ 76.3%

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

      2. *-rgt-identity 76.3%

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

   8. Simplified 47.3%

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

      1. expm1-log1p-u 45.8%

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

      2. expm1-udef 30.6%

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

      3. *-commutative 30.6%

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

      4. sqrt-div 34.1%

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

      5. frac-times 34.1%

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

      6. add-sqr-sqrt 34.1%

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

      7. sqrt-prod 28.6%

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

   10. Applied egg-rr 28.6%

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

       1. expm1-def 40.6%

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

       2. expm1-log1p 46.5%

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

   12. Simplified 46.5%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.95 \cdot 10^{-244}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]

Alternative 14: 42.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.7 \cdot 10^{-244}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1.7e-244) (* d (- (sqrt (/ (/ 1.0 l) h)))) (/ d (sqrt (* l h)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.7e-244) {
		tmp = d * -sqrt(((1.0 / l) / h));
	} 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 <= (-1.7d-244)) then
        tmp = d * -sqrt(((1.0d0 / l) / h))
    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 <= -1.7e-244) {
		tmp = d * -Math.sqrt(((1.0 / l) / h));
	} else {
		tmp = d / Math.sqrt((l * h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -1.7e-244:
		tmp = d * -math.sqrt(((1.0 / l) / h))
	else:
		tmp = d / math.sqrt((l * h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1.7e-244)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / l) / h))));
	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 <= -1.7e-244)
		tmp = d * -sqrt(((1.0 / l) / h));
	else
		tmp = d / sqrt((l * h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1.7e-244], N[(d * (-N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.70000000000000004e-244

   1. Initial program 70.1%

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

   3. Simplified 70.9%

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

      1. frac-2neg 70.9%

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

      2. sqrt-div 83.4%

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

   5. Applied egg-rr 83.4%

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

   6. Taylor expanded in d around -inf 50.2%

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

      1. mul-1-neg 50.2%

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

      2. distribute-rgt-neg-in 50.2%

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

   8. Simplified 50.2%

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

      1. expm1-log1p-u 48.9%

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

      2. expm1-udef 25.1%

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

   10. Applied egg-rr 25.1%

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

       1. expm1-def 48.9%

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

       2. expm1-log1p 50.2%

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

       3. *-commutative 50.2%

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

       4. associate-/r* 51.7%

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

   12. Simplified 51.7%

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

   if -1.70000000000000004e-244 < l

   1. Initial program 69.8%

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

   3. Simplified 68.5%

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

   4. Taylor expanded in M around 0 41.2%

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

      1. sqrt-div 76.3%

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

      2. div-inv 76.3%

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

   6. Applied egg-rr 47.2%

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

      1. associate-*r/ 76.3%

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

      2. *-rgt-identity 76.3%

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

   8. Simplified 47.3%

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

      1. expm1-log1p-u 45.8%

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

      2. expm1-udef 30.6%

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

      3. *-commutative 30.6%

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

      4. sqrt-div 34.1%

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

      5. frac-times 34.1%

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

      6. add-sqr-sqrt 34.1%

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

      7. sqrt-prod 28.6%

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

   10. Applied egg-rr 28.6%

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

       1. expm1-def 40.6%

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

       2. expm1-log1p 46.5%

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

   12. Simplified 46.5%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.7 \cdot 10^{-244}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]

Alternative 15: 42.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.2 \cdot 10^{-243}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -5.2e-243)
   (* d (- (sqrt (/ (/ 1.0 l) h))))
   (* d (pow (* l h) -0.5))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -5.2e-243) {
		tmp = d * -sqrt(((1.0 / l) / h));
	} else {
		tmp = d * pow((l * h), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-5.2d-243)) then
        tmp = d * -sqrt(((1.0d0 / l) / h))
    else
        tmp = d * ((l * h) ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -5.2e-243) {
		tmp = d * -Math.sqrt(((1.0 / l) / h));
	} else {
		tmp = d * Math.pow((l * h), -0.5);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -5.2e-243:
		tmp = d * -math.sqrt(((1.0 / l) / h))
	else:
		tmp = d * math.pow((l * h), -0.5)
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -5.2e-243)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / l) / h))));
	else
		tmp = Float64(d * (Float64(l * h) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -5.2e-243)
		tmp = d * -sqrt(((1.0 / l) / h));
	else
		tmp = d * ((l * h) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -5.2e-243], N[(d * (-N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -5.1999999999999995e-243

   1. Initial program 70.1%

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

   3. Simplified 70.9%

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

      1. frac-2neg 70.9%

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

      2. sqrt-div 83.4%

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

   5. Applied egg-rr 83.4%

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

   6. Taylor expanded in d around -inf 50.2%

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

      1. mul-1-neg 50.2%

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

      2. distribute-rgt-neg-in 50.2%

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

   8. Simplified 50.2%

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

      1. expm1-log1p-u 48.9%

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

      2. expm1-udef 25.1%

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

   10. Applied egg-rr 25.1%

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

       1. expm1-def 48.9%

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

       2. expm1-log1p 50.2%

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

       3. *-commutative 50.2%

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

       4. associate-/r* 51.7%

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

   12. Simplified 51.7%

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

   if -5.1999999999999995e-243 < l

   1. Initial program 69.8%

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

   3. Simplified 68.5%

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

   4. Taylor expanded in M around 0 41.2%

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

      1. pow1/2 41.2%

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

      2. metadata-eval 41.2%

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

      3. add-cbrt-cube 35.2%

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

      4. pow1/3 33.8%

         \[\leadsto \left(\color{blue}{{\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)}^{0.3333333333333333}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot 1 \]

      5. sqr-pow 33.8%

         \[\leadsto \left({\left(\color{blue}{{\left(\frac{d}{h}\right)}^{1}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)}^{0.3333333333333333} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot 1 \]

      6. pow-prod-up 33.8%

         \[\leadsto \left({\color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(1 + \frac{1}{2}\right)}\right)}}^{0.3333333333333333} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot 1 \]

      7. metadata-eval 33.8%

         \[\leadsto \left({\left({\left(\frac{d}{h}\right)}^{\left(1 + \color{blue}{0.5}\right)}\right)}^{0.3333333333333333} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot 1 \]

      8. metadata-eval 33.8%

         \[\leadsto \left({\left({\left(\frac{d}{h}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot 1 \]

   6. Applied egg-rr 33.8%

      \[\leadsto \left(\color{blue}{{\left({\left(\frac{d}{h}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot 1 \]
   7. Step-by-step derivation

      1. unpow1/3 35.2%

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

   8. Simplified 35.2%

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

   9. Taylor expanded in d around 0 46.6%

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

       1. unpow1/2 46.6%

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

       2. rem-exp-log 44.3%

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

       3. exp-neg 44.3%

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

       4. exp-prod 44.3%

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

       5. distribute-lft-neg-out 44.3%

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

       6. distribute-rgt-neg-in 44.3%

          \[\leadsto \left(d \cdot e^{\color{blue}{\log \left(h \cdot \ell\right) \cdot \left(-0.5\right)}}\right) \cdot 1 \]

       7. metadata-eval 44.3%

          \[\leadsto \left(d \cdot e^{\log \left(h \cdot \ell\right) \cdot \color{blue}{-0.5}}\right) \cdot 1 \]

       8. exp-to-pow 46.6%

          \[\leadsto \left(d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \cdot 1 \]

   11. Simplified 46.6%

       \[\leadsto \color{blue}{\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.2 \cdot 10^{-243}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]

Alternative 16: 42.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.3 \cdot 10^{-243}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1.3e-243)
   (* d (- (sqrt (/ (/ 1.0 l) h))))
   (* d (sqrt (/ 1.0 (* l h))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.3e-243) {
		tmp = d * -sqrt(((1.0 / l) / h));
	} 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 <= (-1.3d-243)) then
        tmp = d * -sqrt(((1.0d0 / l) / h))
    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 <= -1.3e-243) {
		tmp = d * -Math.sqrt(((1.0 / l) / h));
	} else {
		tmp = d * Math.sqrt((1.0 / (l * h)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -1.3e-243:
		tmp = d * -math.sqrt(((1.0 / l) / h))
	else:
		tmp = d * math.sqrt((1.0 / (l * h)))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1.3e-243)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / l) / h))));
	else
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(l * h))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -1.3e-243)
		tmp = d * -sqrt(((1.0 / l) / h));
	else
		tmp = d * sqrt((1.0 / (l * h)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1.3e-243], N[(d * (-N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.2999999999999999e-243

   1. Initial program 70.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 70.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. frac-2neg 70.9%

         \[\leadsto \sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      2. sqrt-div 83.4%

         \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   5. Applied egg-rr 83.4%

      \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   6. Taylor expanded in d around -inf 50.2%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 50.2%

         \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. distribute-rgt-neg-in 50.2%

         \[\leadsto \color{blue}{d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   8. Simplified 50.2%

      \[\leadsto \color{blue}{d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 48.9%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{h \cdot \ell}\right)\right)}}\right) \]

      2. expm1-udef 25.1%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{e^{\mathsf{log1p}\left(\frac{1}{h \cdot \ell}\right)} - 1}}\right) \]

   10. Applied egg-rr 25.1%

       \[\leadsto d \cdot \left(-\sqrt{\color{blue}{e^{\mathsf{log1p}\left(\frac{1}{h \cdot \ell}\right)} - 1}}\right) \]
   11. Step-by-step derivation

       1. expm1-def 48.9%

          \[\leadsto d \cdot \left(-\sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{h \cdot \ell}\right)\right)}}\right) \]

       2. expm1-log1p 50.2%

          \[\leadsto d \cdot \left(-\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right) \]

       3. *-commutative 50.2%

          \[\leadsto d \cdot \left(-\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right) \]

       4. associate-/r* 51.7%

          \[\leadsto d \cdot \left(-\sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right) \]

   12. Simplified 51.7%

       \[\leadsto d \cdot \left(-\sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right) \]

   if -1.2999999999999999e-243 < l

   1. Initial program 69.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 68.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(\frac{-0.5 \cdot M}{d} \cdot D\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

   4. Taylor expanded in M around 0 41.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{1} \]

   5. Taylor expanded in d around 0 46.6%

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.3 \cdot 10^{-243}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \end{array} \]

Alternative 17: 42.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -7.5 \cdot 10^{-244}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -7.5e-244) (* 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 <= -7.5e-244) {
		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 <= (-7.5d-244)) 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 <= -7.5e-244) {
		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 <= -7.5e-244:
		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 <= -7.5e-244)
		tmp = Float64(d * Float64(-(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 <= -7.5e-244)
		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, -7.5e-244], 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 < -7.5000000000000003e-244

   1. Initial program 70.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 70.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. frac-2neg 70.9%

         \[\leadsto \sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      2. sqrt-div 83.4%

         \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   5. Applied egg-rr 83.4%

      \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   6. Taylor expanded in d around -inf 50.2%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 50.2%

         \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. distribute-rgt-neg-in 50.2%

         \[\leadsto \color{blue}{d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   8. Simplified 50.2%

      \[\leadsto \color{blue}{d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   9. Taylor expanded in d around 0 50.2%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg 50.2%

          \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

       2. *-commutative 50.2%

          \[\leadsto -\color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

       3. unpow1/2 50.2%

          \[\leadsto -\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}} \cdot d \]

       4. rem-exp-log 47.2%

          \[\leadsto -{\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \cdot d \]

       5. exp-neg 47.2%

          \[\leadsto -{\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \cdot d \]

       6. exp-prod 47.6%

          \[\leadsto -\color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \cdot d \]

       7. distribute-lft-neg-out 47.6%

          \[\leadsto -e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \cdot d \]

       8. distribute-rgt-neg-in 47.6%

          \[\leadsto -e^{\color{blue}{\log \left(h \cdot \ell\right) \cdot \left(-0.5\right)}} \cdot d \]

       9. metadata-eval 47.6%

          \[\leadsto -e^{\log \left(h \cdot \ell\right) \cdot \color{blue}{-0.5}} \cdot d \]

       10. exp-to-pow 50.6%

           \[\leadsto -\color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]

       11. distribute-rgt-neg-in 50.6%

           \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-d\right)} \]

   11. Simplified 50.6%

       \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-d\right)} \]

   if -7.5000000000000003e-244 < l

   1. Initial program 69.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 68.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(\frac{-0.5 \cdot M}{d} \cdot D\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

   4. Taylor expanded in M around 0 41.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

      1. sqrt-div 76.3%

         \[\leadsto \frac{\sqrt{d}}{\frac{\sqrt{h}}{\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \left(1 + -0.5 \cdot \left({\left(M \cdot \frac{-0.5 \cdot D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}} \]

      2. div-inv 76.3%

         \[\leadsto \frac{\sqrt{d}}{\frac{\sqrt{h}}{\color{blue}{\left(\sqrt{d} \cdot \frac{1}{\sqrt{\ell}}\right)} \cdot \left(1 + -0.5 \cdot \left({\left(M \cdot \frac{-0.5 \cdot D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}} \]

   6. Applied egg-rr 47.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\left(\sqrt{d} \cdot \frac{1}{\sqrt{\ell}}\right)}\right) \cdot 1 \]
   7. Step-by-step derivation

      1. associate-*r/ 76.3%

         \[\leadsto \frac{\sqrt{d}}{\frac{\sqrt{h}}{\color{blue}{\frac{\sqrt{d} \cdot 1}{\sqrt{\ell}}} \cdot \left(1 + -0.5 \cdot \left({\left(M \cdot \frac{-0.5 \cdot D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}} \]

      2. *-rgt-identity 76.3%

         \[\leadsto \frac{\sqrt{d}}{\frac{\sqrt{h}}{\frac{\color{blue}{\sqrt{d}}}{\sqrt{\ell}} \cdot \left(1 + -0.5 \cdot \left({\left(M \cdot \frac{-0.5 \cdot D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}} \]

   8. Simplified 47.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot 1 \]
   9. Step-by-step derivation

      1. expm1-log1p-u 45.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\right)} \cdot 1 \]

      2. expm1-udef 30.6%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)} - 1\right)} \cdot 1 \]

      3. *-commutative 30.6%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \sqrt{\frac{d}{h}}}\right)} - 1\right) \cdot 1 \]

      4. sqrt-div 34.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}}\right)} - 1\right) \cdot 1 \]

      5. frac-times 34.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \sqrt{h}}}\right)} - 1\right) \cdot 1 \]

      6. add-sqr-sqrt 34.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{d}}{\sqrt{\ell} \cdot \sqrt{h}}\right)} - 1\right) \cdot 1 \]

      7. sqrt-prod 28.6%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{d}{\color{blue}{\sqrt{\ell \cdot h}}}\right)} - 1\right) \cdot 1 \]

   10. Applied egg-rr 28.6%

       \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)} - 1\right)} \cdot 1 \]
   11. Step-by-step derivation

       1. expm1-def 40.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)\right)} \cdot 1 \]

       2. expm1-log1p 46.5%

          \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \cdot 1 \]

   12. Simplified 46.5%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7.5 \cdot 10^{-244}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]

Alternative 18: 4.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ d \cdot \sqrt{\ell \cdot h} \end{array} \]

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 69.9%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 70.0%

   \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} \]
4. Step-by-step derivation

   1. frac-2neg 70.0%

      \[\leadsto \sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   2. sqrt-div 41.8%

      \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

5. Applied egg-rr 41.8%

   \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

6. Taylor expanded in d around -inf 27.7%

   \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
7. Step-by-step derivation

   1. mul-1-neg 27.7%

      \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   2. distribute-rgt-neg-in 27.7%

      \[\leadsto \color{blue}{d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

8. Simplified 27.7%

   \[\leadsto \color{blue}{d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

9. Taylor expanded in d around 0 27.7%

   \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
10. Step-by-step derivation

    1. mul-1-neg 27.7%

       \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

    2. *-commutative 27.7%

       \[\leadsto -\color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

    3. unpow1/2 27.7%

       \[\leadsto -\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}} \cdot d \]

    4. rem-exp-log 26.3%

       \[\leadsto -{\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \cdot d \]

    5. exp-neg 26.3%

       \[\leadsto -{\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \cdot d \]

    6. exp-prod 26.5%

       \[\leadsto -\color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \cdot d \]

    7. distribute-lft-neg-out 26.5%

       \[\leadsto -e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \cdot d \]

    8. distribute-rgt-neg-in 26.5%

       \[\leadsto -e^{\color{blue}{\log \left(h \cdot \ell\right) \cdot \left(-0.5\right)}} \cdot d \]

    9. metadata-eval 26.5%

       \[\leadsto -e^{\log \left(h \cdot \ell\right) \cdot \color{blue}{-0.5}} \cdot d \]

    10. exp-to-pow 27.9%

        \[\leadsto -\color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]

    11. distribute-rgt-neg-in 27.9%

        \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-d\right)} \]

11. Simplified 27.9%

    \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-d\right)} \]

13. Applied egg-rr 2.5%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{\ell \cdot h}\right)} - 1} \]
14. Step-by-step derivation

    1. expm1-def 2.1%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{\ell \cdot h}\right)\right)} \]

    2. expm1-log1p 3.9%

       \[\leadsto \color{blue}{d \cdot \sqrt{\ell \cdot h}} \]

15. Simplified 3.9%

    \[\leadsto \color{blue}{d \cdot \sqrt{\ell \cdot h}} \]

16. Final simplification 3.9%

    \[\leadsto d \cdot \sqrt{\ell \cdot h} \]

Alternative 19: 25.5% 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 69.9%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 68.6%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(\frac{-0.5 \cdot M}{d} \cdot D\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

4. Taylor expanded in M around 0 43.1%

   \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{1} \]
5. Step-by-step derivation

   1. sqrt-div 42.0%

      \[\leadsto \frac{\sqrt{d}}{\frac{\sqrt{h}}{\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \left(1 + -0.5 \cdot \left({\left(M \cdot \frac{-0.5 \cdot D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}} \]

   2. div-inv 42.0%

      \[\leadsto \frac{\sqrt{d}}{\frac{\sqrt{h}}{\color{blue}{\left(\sqrt{d} \cdot \frac{1}{\sqrt{\ell}}\right)} \cdot \left(1 + -0.5 \cdot \left({\left(M \cdot \frac{-0.5 \cdot D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}} \]

6. Applied egg-rr 26.0%

   \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\left(\sqrt{d} \cdot \frac{1}{\sqrt{\ell}}\right)}\right) \cdot 1 \]
7. Step-by-step derivation

   1. associate-*r/ 42.0%

      \[\leadsto \frac{\sqrt{d}}{\frac{\sqrt{h}}{\color{blue}{\frac{\sqrt{d} \cdot 1}{\sqrt{\ell}}} \cdot \left(1 + -0.5 \cdot \left({\left(M \cdot \frac{-0.5 \cdot D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}} \]

   2. *-rgt-identity 42.0%

      \[\leadsto \frac{\sqrt{d}}{\frac{\sqrt{h}}{\frac{\color{blue}{\sqrt{d}}}{\sqrt{\ell}} \cdot \left(1 + -0.5 \cdot \left({\left(M \cdot \frac{-0.5 \cdot D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}} \]

8. Simplified 26.0%

   \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot 1 \]
9. Step-by-step derivation

   1. expm1-log1p-u 25.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\right)} \cdot 1 \]

   2. expm1-udef 16.9%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)} - 1\right)} \cdot 1 \]

   3. *-commutative 16.9%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \sqrt{\frac{d}{h}}}\right)} - 1\right) \cdot 1 \]

   4. sqrt-div 18.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}}\right)} - 1\right) \cdot 1 \]

   5. frac-times 18.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \sqrt{h}}}\right)} - 1\right) \cdot 1 \]

   6. add-sqr-sqrt 18.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{d}}{\sqrt{\ell} \cdot \sqrt{h}}\right)} - 1\right) \cdot 1 \]

   7. sqrt-prod 16.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{d}{\color{blue}{\sqrt{\ell \cdot h}}}\right)} - 1\right) \cdot 1 \]

10. Applied egg-rr 16.8%

    \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)} - 1\right)} \cdot 1 \]
11. Step-by-step derivation

    1. expm1-def 23.4%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)\right)} \cdot 1 \]

    2. expm1-log1p 29.0%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \cdot 1 \]

12. Simplified 29.0%

    \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \cdot 1 \]

13. Final simplification 29.0%

    \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

Reproduce

herbie shell --seed 2023299 
(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)))))