Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.4% → 73.4%

Time: 27.1s

Alternatives: 18

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: 66.4% 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: 73.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 - -0.5 \cdot \left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{+135}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1e-311)
   (*
    (* d (pow (* l h) -0.5))
    (- -1.0 (* -0.5 (* (pow (* 0.5 (* M (/ D d))) 2.0) (/ h l)))))
   (if (<= l 1.55e+135)
     (*
      (* (/ (sqrt d) (sqrt h)) (sqrt (/ d l)))
      (- 1.0 (* 0.5 (/ (* h (pow (* (/ D d) (* 0.5 M)) 2.0)) l))))
     (* d (* (pow h -0.5) (pow l -0.5))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1e-311) {
		tmp = (d * pow((l * h), -0.5)) * (-1.0 - (-0.5 * (pow((0.5 * (M * (D / d))), 2.0) * (h / l))));
	} else if (l <= 1.55e+135) {
		tmp = ((sqrt(d) / sqrt(h)) * sqrt((d / l))) * (1.0 - (0.5 * ((h * pow(((D / d) * (0.5 * M)), 2.0)) / l)));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-1d-311)) then
        tmp = (d * ((l * h) ** (-0.5d0))) * ((-1.0d0) - ((-0.5d0) * (((0.5d0 * (m * (d_1 / d))) ** 2.0d0) * (h / l))))
    else if (l <= 1.55d+135) then
        tmp = ((sqrt(d) / sqrt(h)) * sqrt((d / l))) * (1.0d0 - (0.5d0 * ((h * (((d_1 / d) * (0.5d0 * m)) ** 2.0d0)) / l)))
    else
        tmp = d * ((h ** (-0.5d0)) * (l ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1e-311) {
		tmp = (d * Math.pow((l * h), -0.5)) * (-1.0 - (-0.5 * (Math.pow((0.5 * (M * (D / d))), 2.0) * (h / l))));
	} else if (l <= 1.55e+135) {
		tmp = ((Math.sqrt(d) / Math.sqrt(h)) * Math.sqrt((d / l))) * (1.0 - (0.5 * ((h * Math.pow(((D / d) * (0.5 * M)), 2.0)) / l)));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -1e-311:
		tmp = (d * math.pow((l * h), -0.5)) * (-1.0 - (-0.5 * (math.pow((0.5 * (M * (D / d))), 2.0) * (h / l))))
	elif l <= 1.55e+135:
		tmp = ((math.sqrt(d) / math.sqrt(h)) * math.sqrt((d / l))) * (1.0 - (0.5 * ((h * math.pow(((D / d) * (0.5 * M)), 2.0)) / l)))
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1e-311)
		tmp = Float64(Float64(d * (Float64(l * h) ^ -0.5)) * Float64(-1.0 - Float64(-0.5 * Float64((Float64(0.5 * Float64(M * Float64(D / d))) ^ 2.0) * Float64(h / l)))));
	elseif (l <= 1.55e+135)
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * sqrt(Float64(d / l))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(Float64(D / d) * Float64(0.5 * M)) ^ 2.0)) / l))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -1e-311)
		tmp = (d * ((l * h) ^ -0.5)) * (-1.0 - (-0.5 * (((0.5 * (M * (D / d))) ^ 2.0) * (h / l))));
	elseif (l <= 1.55e+135)
		tmp = ((sqrt(d) / sqrt(h)) * sqrt((d / l))) * (1.0 - (0.5 * ((h * (((D / d) * (0.5 * M)) ^ 2.0)) / l)));
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1e-311], N[(N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(-0.5 * N[(N[Power[N[(0.5 * N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.55e+135], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(N[(D / d), $MachinePrecision] * N[(0.5 * M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -9.99999999999948e-312

   1. Initial program 61.9%

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

   3. Simplified 62.3%

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

      1. associate-*r/ 62.5%

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

      2. div-inv 62.5%

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

      3. metadata-eval 62.5%

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

   5. Applied egg-rr 62.5%

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

      1. sqrt-div 0.0%

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

      2. metadata-eval 0.0%

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

      3. div-inv 0.0%

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

      4. associate-*r/ 0.0%

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

      5. pow1 0.0%

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

   7. Applied egg-rr 54.4%

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

      1. unpow1 54.4%

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

      2. *-commutative 54.4%

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

      3. *-commutative 54.4%

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

      4. associate-*r* 54.4%

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

   9. Simplified 54.4%

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

   10. Taylor expanded in d around -inf 72.7%

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

       1. mul-1-neg 72.7%

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

       2. distribute-rgt-neg-in 72.7%

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

       3. unpow-1 72.7%

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

       4. metadata-eval 72.7%

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

       5. pow-sqr 72.8%

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

       6. rem-sqrt-square 72.8%

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

       7. rem-square-sqrt 72.6%

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

       8. fabs-sqr 72.6%

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

       9. rem-square-sqrt 72.8%

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

   12. Simplified 72.8%

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

   if -9.99999999999948e-312 < l < 1.55000000000000011e135

   1. Initial program 70.2%

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

   3. Simplified 70.2%

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

      1. associate-*r/ 74.8%

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

      2. div-inv 74.8%

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

      3. metadata-eval 74.8%

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

   5. Applied egg-rr 74.8%

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

      1. sqrt-div 89.5%

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

   7. Applied egg-rr 89.5%

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

   if 1.55000000000000011e135 < l

   1. Initial program 44.4%

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

   3. Simplified 44.3%

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

      1. associate-*r/ 44.1%

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

      2. div-inv 44.1%

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

      3. metadata-eval 44.1%

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

   5. Applied egg-rr 44.1%

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

   6. Taylor expanded in d around inf 51.6%

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

      1. unpow-1 51.6%

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

      2. sqr-pow 51.7%

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

      3. rem-sqrt-square 51.7%

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

      4. metadata-eval 51.7%

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

      5. sqr-pow 51.4%

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

      6. fabs-sqr 51.4%

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

      7. sqr-pow 51.7%

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

   8. Simplified 51.7%

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

      1. unpow-prod-down 64.6%

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

   10. Applied egg-rr 64.6%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 - -0.5 \cdot \left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{+135}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 2: 71.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;h \leq -4.8 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(t_0 \cdot \mathsf{fma}\left({\left(0.5 \cdot \frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 - -0.5 \cdot \left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq 3.85 \cdot 10^{+198}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2}}{\ell}\right) \cdot \left(t_0 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot \left(M \cdot \left(M \cdot \left(D \cdot \frac{D}{d}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h))))
   (if (<= h -4.8e+153)
     (*
      (sqrt (/ d l))
      (* t_0 (fma (pow (* 0.5 (/ M (/ d D))) 2.0) (* -0.5 (/ h l)) 1.0)))
     (if (<= h -1e-310)
       (*
        (* d (pow (* l h) -0.5))
        (- -1.0 (* -0.5 (* (pow (* 0.5 (* M (/ D d))) 2.0) (/ h l)))))
       (if (<= h 3.85e+198)
         (*
          (- 1.0 (* 0.5 (/ (* h (pow (* (/ D d) (* 0.5 M)) 2.0)) l)))
          (* t_0 (/ (sqrt d) (sqrt l))))
         (* (/ (sqrt h) (pow l 1.5)) (* -0.125 (* M (* M (* D (/ D d)))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h));
	double tmp;
	if (h <= -4.8e+153) {
		tmp = sqrt((d / l)) * (t_0 * fma(pow((0.5 * (M / (d / D))), 2.0), (-0.5 * (h / l)), 1.0));
	} else if (h <= -1e-310) {
		tmp = (d * pow((l * h), -0.5)) * (-1.0 - (-0.5 * (pow((0.5 * (M * (D / d))), 2.0) * (h / l))));
	} else if (h <= 3.85e+198) {
		tmp = (1.0 - (0.5 * ((h * pow(((D / d) * (0.5 * M)), 2.0)) / l))) * (t_0 * (sqrt(d) / sqrt(l)));
	} else {
		tmp = (sqrt(h) / pow(l, 1.5)) * (-0.125 * (M * (M * (D * (D / d)))));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / h))
	tmp = 0.0
	if (h <= -4.8e+153)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(t_0 * fma((Float64(0.5 * Float64(M / Float64(d / D))) ^ 2.0), Float64(-0.5 * Float64(h / l)), 1.0)));
	elseif (h <= -1e-310)
		tmp = Float64(Float64(d * (Float64(l * h) ^ -0.5)) * Float64(-1.0 - Float64(-0.5 * Float64((Float64(0.5 * Float64(M * Float64(D / d))) ^ 2.0) * Float64(h / l)))));
	elseif (h <= 3.85e+198)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(Float64(D / d) * Float64(0.5 * M)) ^ 2.0)) / l))) * Float64(t_0 * Float64(sqrt(d) / sqrt(l))));
	else
		tmp = Float64(Float64(sqrt(h) / (l ^ 1.5)) * Float64(-0.125 * Float64(M * Float64(M * Float64(D * Float64(D / d))))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[h, -4.8e+153], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(N[Power[N[(0.5 * N[(M / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -1e-310], N[(N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(-0.5 * N[(N[Power[N[(0.5 * N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 3.85e+198], N[(N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(N[(D / d), $MachinePrecision] * N[(0.5 * M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision] * N[(-0.125 * N[(M * N[(M * N[(D * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if h < -4.79999999999999985e153

   1. Initial program 56.5%

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

   3. Simplified 53.2%

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

   if -4.79999999999999985e153 < h < -9.999999999999969e-311

   1. Initial program 64.1%

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

   3. Simplified 64.1%

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

      1. associate-*r/ 65.4%

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

      2. div-inv 65.4%

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

      3. metadata-eval 65.4%

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

   5. Applied egg-rr 65.4%

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

      1. sqrt-div 0.0%

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

      2. metadata-eval 0.0%

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

      3. div-inv 0.0%

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

      4. associate-*r/ 0.0%

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

      5. pow1 0.0%

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

   7. Applied egg-rr 57.9%

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

      1. unpow1 57.9%

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

      2. *-commutative 57.9%

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

      3. *-commutative 57.9%

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

      4. associate-*r* 57.9%

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

   9. Simplified 57.9%

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

   10. Taylor expanded in d around -inf 84.5%

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

       1. mul-1-neg 84.5%

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

       2. distribute-rgt-neg-in 84.5%

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

       3. unpow-1 84.5%

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

       4. metadata-eval 84.5%

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

       5. pow-sqr 84.5%

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

       6. rem-sqrt-square 84.5%

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

       7. rem-square-sqrt 84.4%

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

       8. fabs-sqr 84.4%

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

       9. rem-square-sqrt 84.5%

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

   12. Simplified 84.5%

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

   if -9.999999999999969e-311 < h < 3.85000000000000019e198

   1. Initial program 68.2%

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

   3. Simplified 68.1%

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

      1. associate-*r/ 71.4%

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

      2. div-inv 71.4%

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

      3. metadata-eval 71.4%

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

   5. Applied egg-rr 71.4%

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

      1. sqrt-div 78.8%

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

   7. Applied egg-rr 78.8%

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

   if 3.85000000000000019e198 < h

   1. Initial program 39.6%

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

   3. Simplified 39.6%

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

   4. Taylor expanded in d around 0 48.6%

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

      1. associate-*r* 48.6%

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

      2. *-commutative 48.6%

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

      3. associate-/l* 48.6%

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

      4. associate-/r/ 53.0%

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

      5. unpow2 53.0%

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

      6. unpow2 53.0%

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

   6. Simplified 53.0%

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

   7. Taylor expanded in D around 0 48.6%

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

      1. unpow2 48.6%

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

      2. unpow2 48.6%

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

      3. unpow2 48.6%

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

      4. associate-*l/ 53.0%

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

      5. *-commutative 53.0%

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

      6. unpow2 53.0%

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

      7. associate-*l* 57.3%

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

      8. associate-/l* 57.3%

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

      9. associate-/r/ 57.3%

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

   9. Simplified 57.3%

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

       1. sqrt-div 61.4%

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

   11. Applied egg-rr 61.4%

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

       1. sqr-pow 61.4%

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

       2. rem-sqrt-square 65.7%

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

       3. sqr-pow 65.6%

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

       4. fabs-sqr 65.6%

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

       5. sqr-pow 65.7%

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

       6. metadata-eval 65.7%

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

   13. Simplified 65.7%

       \[\leadsto \color{blue}{\frac{\sqrt{h}}{{\ell}^{1.5}}} \cdot \left(-0.125 \cdot \left(M \cdot \left(M \cdot \left(\frac{D}{d} \cdot D\right)\right)\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -4.8 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 - -0.5 \cdot \left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq 3.85 \cdot 10^{+198}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2}}{\ell}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot \left(M \cdot \left(M \cdot \left(D \cdot \frac{D}{d}\right)\right)\right)\right)\\ \end{array} \]

Alternative 3: 72.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 - -0.5 \cdot \left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{+135}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1e-311)
   (*
    (* d (pow (* l h) -0.5))
    (- -1.0 (* -0.5 (* (pow (* 0.5 (* M (/ D d))) 2.0) (/ h l)))))
   (if (<= l 1.3e+135)
     (*
      (* (/ (sqrt d) (sqrt h)) (sqrt (/ d l)))
      (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D d) (/ M 2.0)) 2.0)))))
     (* d (* (pow h -0.5) (pow l -0.5))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1e-311) {
		tmp = (d * pow((l * h), -0.5)) * (-1.0 - (-0.5 * (pow((0.5 * (M * (D / d))), 2.0) * (h / l))));
	} else if (l <= 1.3e+135) {
		tmp = ((sqrt(d) / sqrt(h)) * sqrt((d / l))) * (1.0 - (0.5 * ((h / l) * pow(((D / d) * (M / 2.0)), 2.0))));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-1d-311)) then
        tmp = (d * ((l * h) ** (-0.5d0))) * ((-1.0d0) - ((-0.5d0) * (((0.5d0 * (m * (d_1 / d))) ** 2.0d0) * (h / l))))
    else if (l <= 1.3d+135) then
        tmp = ((sqrt(d) / sqrt(h)) * sqrt((d / l))) * (1.0d0 - (0.5d0 * ((h / l) * (((d_1 / d) * (m / 2.0d0)) ** 2.0d0))))
    else
        tmp = d * ((h ** (-0.5d0)) * (l ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1e-311) {
		tmp = (d * Math.pow((l * h), -0.5)) * (-1.0 - (-0.5 * (Math.pow((0.5 * (M * (D / d))), 2.0) * (h / l))));
	} else if (l <= 1.3e+135) {
		tmp = ((Math.sqrt(d) / Math.sqrt(h)) * Math.sqrt((d / l))) * (1.0 - (0.5 * ((h / l) * Math.pow(((D / d) * (M / 2.0)), 2.0))));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -1e-311:
		tmp = (d * math.pow((l * h), -0.5)) * (-1.0 - (-0.5 * (math.pow((0.5 * (M * (D / d))), 2.0) * (h / l))))
	elif l <= 1.3e+135:
		tmp = ((math.sqrt(d) / math.sqrt(h)) * math.sqrt((d / l))) * (1.0 - (0.5 * ((h / l) * math.pow(((D / d) * (M / 2.0)), 2.0))))
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1e-311)
		tmp = Float64(Float64(d * (Float64(l * h) ^ -0.5)) * Float64(-1.0 - Float64(-0.5 * Float64((Float64(0.5 * Float64(M * Float64(D / d))) ^ 2.0) * Float64(h / l)))));
	elseif (l <= 1.3e+135)
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * sqrt(Float64(d / l))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D / d) * Float64(M / 2.0)) ^ 2.0)))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -1e-311)
		tmp = (d * ((l * h) ^ -0.5)) * (-1.0 - (-0.5 * (((0.5 * (M * (D / d))) ^ 2.0) * (h / l))));
	elseif (l <= 1.3e+135)
		tmp = ((sqrt(d) / sqrt(h)) * sqrt((d / l))) * (1.0 - (0.5 * ((h / l) * (((D / d) * (M / 2.0)) ^ 2.0))));
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1e-311], N[(N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(-0.5 * N[(N[Power[N[(0.5 * N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.3e+135], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -9.99999999999948e-312

   1. Initial program 61.9%

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

   3. Simplified 62.3%

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

      1. associate-*r/ 62.5%

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

      2. div-inv 62.5%

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

      3. metadata-eval 62.5%

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

   5. Applied egg-rr 62.5%

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

      1. sqrt-div 0.0%

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

      2. metadata-eval 0.0%

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

      3. div-inv 0.0%

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

      4. associate-*r/ 0.0%

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

      5. pow1 0.0%

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

   7. Applied egg-rr 54.4%

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

      1. unpow1 54.4%

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

      2. *-commutative 54.4%

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

      3. *-commutative 54.4%

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

      4. associate-*r* 54.4%

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

   9. Simplified 54.4%

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

   10. Taylor expanded in d around -inf 72.7%

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

       1. mul-1-neg 72.7%

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

       2. distribute-rgt-neg-in 72.7%

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

       3. unpow-1 72.7%

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

       4. metadata-eval 72.7%

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

       5. pow-sqr 72.8%

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

       6. rem-sqrt-square 72.8%

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

       7. rem-square-sqrt 72.6%

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

       8. fabs-sqr 72.6%

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

       9. rem-square-sqrt 72.8%

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

   12. Simplified 72.8%

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

   if -9.99999999999948e-312 < l < 1.3e135

   1. Initial program 70.2%

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

   3. Simplified 70.2%

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

      1. sqrt-div 89.5%

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

   5. Applied egg-rr 84.1%

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

   if 1.3e135 < l

   1. Initial program 44.4%

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

   3. Simplified 44.3%

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

      1. associate-*r/ 44.1%

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

      2. div-inv 44.1%

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

      3. metadata-eval 44.1%

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

   5. Applied egg-rr 44.1%

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

   6. Taylor expanded in d around inf 51.6%

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

      1. unpow-1 51.6%

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

      2. sqr-pow 51.7%

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

      3. rem-sqrt-square 51.7%

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

      4. metadata-eval 51.7%

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

      5. sqr-pow 51.4%

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

      6. fabs-sqr 51.4%

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

      7. sqr-pow 51.7%

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

   8. Simplified 51.7%

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

      1. unpow-prod-down 64.6%

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

   10. Applied egg-rr 64.6%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 - -0.5 \cdot \left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{+135}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 4: 68.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := M \cdot \frac{D}{d}\\ \mathbf{if}\;\ell \leq -6.2 \cdot 10^{-272}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 - -0.5 \cdot \left({\left(0.5 \cdot t_0\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 3.75 \cdot 10^{+133}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(-0.5 \cdot \frac{h}{\ell}\right) \cdot \left(0.25 \cdot {t_0}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* M (/ D d))))
   (if (<= l -6.2e-272)
     (*
      (* d (pow (* l h) -0.5))
      (- -1.0 (* -0.5 (* (pow (* 0.5 t_0) 2.0) (/ h l)))))
     (if (<= l 3.75e+133)
       (*
        (sqrt (/ d l))
        (* (sqrt (/ d h)) (+ 1.0 (* (* -0.5 (/ h l)) (* 0.25 (pow t_0 2.0))))))
       (* d (* (pow h -0.5) (pow l -0.5)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = M * (D / d);
	double tmp;
	if (l <= -6.2e-272) {
		tmp = (d * pow((l * h), -0.5)) * (-1.0 - (-0.5 * (pow((0.5 * t_0), 2.0) * (h / l))));
	} else if (l <= 3.75e+133) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((-0.5 * (h / l)) * (0.25 * pow(t_0, 2.0)))));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = m * (d_1 / d)
    if (l <= (-6.2d-272)) then
        tmp = (d * ((l * h) ** (-0.5d0))) * ((-1.0d0) - ((-0.5d0) * (((0.5d0 * t_0) ** 2.0d0) * (h / l))))
    else if (l <= 3.75d+133) then
        tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0d0 + (((-0.5d0) * (h / l)) * (0.25d0 * (t_0 ** 2.0d0)))))
    else
        tmp = d * ((h ** (-0.5d0)) * (l ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = M * (D / d);
	double tmp;
	if (l <= -6.2e-272) {
		tmp = (d * Math.pow((l * h), -0.5)) * (-1.0 - (-0.5 * (Math.pow((0.5 * t_0), 2.0) * (h / l))));
	} else if (l <= 3.75e+133) {
		tmp = Math.sqrt((d / l)) * (Math.sqrt((d / h)) * (1.0 + ((-0.5 * (h / l)) * (0.25 * Math.pow(t_0, 2.0)))));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = M * (D / d)
	tmp = 0
	if l <= -6.2e-272:
		tmp = (d * math.pow((l * h), -0.5)) * (-1.0 - (-0.5 * (math.pow((0.5 * t_0), 2.0) * (h / l))))
	elif l <= 3.75e+133:
		tmp = math.sqrt((d / l)) * (math.sqrt((d / h)) * (1.0 + ((-0.5 * (h / l)) * (0.25 * math.pow(t_0, 2.0)))))
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(M * Float64(D / d))
	tmp = 0.0
	if (l <= -6.2e-272)
		tmp = Float64(Float64(d * (Float64(l * h) ^ -0.5)) * Float64(-1.0 - Float64(-0.5 * Float64((Float64(0.5 * t_0) ^ 2.0) * Float64(h / l)))));
	elseif (l <= 3.75e+133)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(-0.5 * Float64(h / l)) * Float64(0.25 * (t_0 ^ 2.0))))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = M * (D / d);
	tmp = 0.0;
	if (l <= -6.2e-272)
		tmp = (d * ((l * h) ^ -0.5)) * (-1.0 - (-0.5 * (((0.5 * t_0) ^ 2.0) * (h / l))));
	elseif (l <= 3.75e+133)
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((-0.5 * (h / l)) * (0.25 * (t_0 ^ 2.0)))));
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -6.2e-272], N[(N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(-0.5 * N[(N[Power[N[(0.5 * t$95$0), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.75e+133], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[(0.25 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.20000000000000059e-272

   1. Initial program 60.0%

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

   3. Simplified 60.4%

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

      1. associate-*r/ 60.7%

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

      2. div-inv 60.7%

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

      3. metadata-eval 60.7%

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

   5. Applied egg-rr 60.7%

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

      1. sqrt-div 0.0%

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

      2. metadata-eval 0.0%

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

      3. div-inv 0.0%

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

      4. associate-*r/ 0.0%

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

      5. pow1 0.0%

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

   7. Applied egg-rr 52.2%

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

      1. unpow1 52.2%

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

      2. *-commutative 52.2%

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

      3. *-commutative 52.2%

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

      4. associate-*r* 52.2%

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

   9. Simplified 52.2%

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

   10. Taylor expanded in d around -inf 71.4%

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

       1. mul-1-neg 71.4%

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

       2. distribute-rgt-neg-in 71.4%

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

       3. unpow-1 71.4%

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

       4. metadata-eval 71.4%

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

       5. pow-sqr 71.4%

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

       6. rem-sqrt-square 71.4%

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

       7. rem-square-sqrt 71.3%

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

       8. fabs-sqr 71.3%

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

       9. rem-square-sqrt 71.4%

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

   12. Simplified 71.4%

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

   if -6.20000000000000059e-272 < l < 3.74999999999999996e133

   1. Initial program 71.5%

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

   3. Simplified 71.5%

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

      1. fma-udef 71.5%

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

      2. unpow-prod-down 71.5%

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

      3. metadata-eval 71.5%

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

      4. div-inv 71.4%

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

      5. clear-num 71.4%

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

   5. Applied egg-rr 71.4%

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

   if 3.74999999999999996e133 < l

   1. Initial program 44.4%

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

   3. Simplified 44.3%

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

      1. associate-*r/ 44.1%

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

      2. div-inv 44.1%

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

      3. metadata-eval 44.1%

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

   5. Applied egg-rr 44.1%

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

   6. Taylor expanded in d around inf 51.6%

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

      1. unpow-1 51.6%

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

      2. sqr-pow 51.7%

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

      3. rem-sqrt-square 51.7%

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

      4. metadata-eval 51.7%

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

      5. sqr-pow 51.4%

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

      6. fabs-sqr 51.4%

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

      7. sqr-pow 51.7%

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

   8. Simplified 51.7%

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

      1. unpow-prod-down 64.6%

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

   10. Applied egg-rr 64.6%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.2 \cdot 10^{-272}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 - -0.5 \cdot \left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 3.75 \cdot 10^{+133}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(-0.5 \cdot \frac{h}{\ell}\right) \cdot \left(0.25 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 5: 70.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3.2 \cdot 10^{-244}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 - -0.5 \cdot \left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 2.05 \cdot 10^{+129}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{0.5}{\frac{d}{M \cdot D}}\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -3.2e-244)
   (*
    (* d (pow (* l h) -0.5))
    (- -1.0 (* -0.5 (* (pow (* 0.5 (* M (/ D d))) 2.0) (/ h l)))))
   (if (<= l 2.05e+129)
     (*
      (* (sqrt (/ d l)) (sqrt (/ d h)))
      (- 1.0 (* 0.5 (/ (* h (pow (/ 0.5 (/ d (* M D))) 2.0)) l))))
     (* d (* (pow h -0.5) (pow l -0.5))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -3.2e-244) {
		tmp = (d * pow((l * h), -0.5)) * (-1.0 - (-0.5 * (pow((0.5 * (M * (D / d))), 2.0) * (h / l))));
	} else if (l <= 2.05e+129) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.5 * ((h * pow((0.5 / (d / (M * D))), 2.0)) / l)));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-3.2d-244)) then
        tmp = (d * ((l * h) ** (-0.5d0))) * ((-1.0d0) - ((-0.5d0) * (((0.5d0 * (m * (d_1 / d))) ** 2.0d0) * (h / l))))
    else if (l <= 2.05d+129) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - (0.5d0 * ((h * ((0.5d0 / (d / (m * d_1))) ** 2.0d0)) / l)))
    else
        tmp = d * ((h ** (-0.5d0)) * (l ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -3.2e-244) {
		tmp = (d * Math.pow((l * h), -0.5)) * (-1.0 - (-0.5 * (Math.pow((0.5 * (M * (D / d))), 2.0) * (h / l))));
	} else if (l <= 2.05e+129) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (0.5 * ((h * Math.pow((0.5 / (d / (M * D))), 2.0)) / l)));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -3.2e-244:
		tmp = (d * math.pow((l * h), -0.5)) * (-1.0 - (-0.5 * (math.pow((0.5 * (M * (D / d))), 2.0) * (h / l))))
	elif l <= 2.05e+129:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (0.5 * ((h * math.pow((0.5 / (d / (M * D))), 2.0)) / l)))
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -3.2e-244)
		tmp = Float64(Float64(d * (Float64(l * h) ^ -0.5)) * Float64(-1.0 - Float64(-0.5 * Float64((Float64(0.5 * Float64(M * Float64(D / d))) ^ 2.0) * Float64(h / l)))));
	elseif (l <= 2.05e+129)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(0.5 / Float64(d / Float64(M * D))) ^ 2.0)) / l))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -3.2e-244)
		tmp = (d * ((l * h) ^ -0.5)) * (-1.0 - (-0.5 * (((0.5 * (M * (D / d))) ^ 2.0) * (h / l))));
	elseif (l <= 2.05e+129)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.5 * ((h * ((0.5 / (d / (M * D))) ^ 2.0)) / l)));
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -3.2e-244], N[(N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(-0.5 * N[(N[Power[N[(0.5 * N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.05e+129], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(0.5 / N[(d / N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.1999999999999998e-244

   1. Initial program 60.2%

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

   3. Simplified 60.7%

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

      1. associate-*r/ 59.9%

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

      2. div-inv 59.9%

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

      3. metadata-eval 59.9%

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

   5. Applied egg-rr 59.9%

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

      1. sqrt-div 0.0%

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

      2. metadata-eval 0.0%

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

      3. div-inv 0.0%

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

      4. associate-*r/ 0.0%

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

      5. pow1 0.0%

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

   7. Applied egg-rr 51.6%

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

      1. unpow1 51.6%

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

      2. *-commutative 51.6%

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

      3. *-commutative 51.6%

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

      4. associate-*r* 51.6%

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

   9. Simplified 51.6%

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

   10. Taylor expanded in d around -inf 71.7%

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

       1. mul-1-neg 71.7%

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

       2. distribute-rgt-neg-in 71.7%

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

       3. unpow-1 71.7%

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

       4. metadata-eval 71.7%

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

       5. pow-sqr 71.7%

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

       6. rem-sqrt-square 71.7%

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

       7. rem-square-sqrt 71.6%

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

       8. fabs-sqr 71.6%

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

       9. rem-square-sqrt 71.7%

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

   12. Simplified 71.7%

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

   if -3.1999999999999998e-244 < l < 2.0500000000000001e129

   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 70.5%

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

      1. associate-*r/ 75.4%

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

      2. div-inv 75.4%

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

      3. metadata-eval 75.4%

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

   5. Applied egg-rr 75.4%

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

      1. pow1 75.4%

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

      2. *-commutative 75.4%

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

      3. metadata-eval 75.4%

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

      4. div-inv 75.4%

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

      5. *-commutative 75.4%

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

      6. div-inv 75.4%

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

      7. metadata-eval 75.4%

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

   7. Applied egg-rr 75.4%

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

      1. unpow1 75.4%

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

      2. associate-*l/ 75.4%

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

      3. associate-*r* 75.4%

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

      4. associate-*l/ 75.4%

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

      5. *-commutative 75.4%

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

      6. associate-*r/ 75.4%

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

      7. associate-/l* 75.3%

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

   9. Simplified 75.3%

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

   if 2.0500000000000001e129 < l

   1. Initial program 44.4%

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

   3. Simplified 44.3%

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

      1. associate-*r/ 44.1%

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

      2. div-inv 44.1%

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

      3. metadata-eval 44.1%

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

   5. Applied egg-rr 44.1%

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

   6. Taylor expanded in d around inf 51.6%

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

      1. unpow-1 51.6%

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

      2. sqr-pow 51.7%

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

      3. rem-sqrt-square 51.7%

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

      4. metadata-eval 51.7%

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

      5. sqr-pow 51.4%

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

      6. fabs-sqr 51.4%

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

      7. sqr-pow 51.7%

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

   8. Simplified 51.7%

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

      1. unpow-prod-down 64.6%

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

   10. Applied egg-rr 64.6%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.2 \cdot 10^{-244}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 - -0.5 \cdot \left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 2.05 \cdot 10^{+129}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{0.5}{\frac{d}{M \cdot D}}\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 6: 70.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3.2 \cdot 10^{-244}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 - -0.5 \cdot \left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.08 \cdot 10^{+126}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2}}{\ell}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -3.2e-244)
   (*
    (* d (pow (* l h) -0.5))
    (- -1.0 (* -0.5 (* (pow (* 0.5 (* M (/ D d))) 2.0) (/ h l)))))
   (if (<= l 1.08e+126)
     (*
      (- 1.0 (* 0.5 (/ (* h (pow (* (/ D d) (* 0.5 M)) 2.0)) l)))
      (* (sqrt (/ d l)) (sqrt (/ d h))))
     (* d (* (pow h -0.5) (pow l -0.5))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -3.2e-244) {
		tmp = (d * pow((l * h), -0.5)) * (-1.0 - (-0.5 * (pow((0.5 * (M * (D / d))), 2.0) * (h / l))));
	} else if (l <= 1.08e+126) {
		tmp = (1.0 - (0.5 * ((h * pow(((D / d) * (0.5 * M)), 2.0)) / l))) * (sqrt((d / l)) * sqrt((d / h)));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-3.2d-244)) then
        tmp = (d * ((l * h) ** (-0.5d0))) * ((-1.0d0) - ((-0.5d0) * (((0.5d0 * (m * (d_1 / d))) ** 2.0d0) * (h / l))))
    else if (l <= 1.08d+126) then
        tmp = (1.0d0 - (0.5d0 * ((h * (((d_1 / d) * (0.5d0 * m)) ** 2.0d0)) / l))) * (sqrt((d / l)) * sqrt((d / h)))
    else
        tmp = d * ((h ** (-0.5d0)) * (l ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -3.2e-244) {
		tmp = (d * Math.pow((l * h), -0.5)) * (-1.0 - (-0.5 * (Math.pow((0.5 * (M * (D / d))), 2.0) * (h / l))));
	} else if (l <= 1.08e+126) {
		tmp = (1.0 - (0.5 * ((h * Math.pow(((D / d) * (0.5 * M)), 2.0)) / l))) * (Math.sqrt((d / l)) * Math.sqrt((d / h)));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -3.2e-244:
		tmp = (d * math.pow((l * h), -0.5)) * (-1.0 - (-0.5 * (math.pow((0.5 * (M * (D / d))), 2.0) * (h / l))))
	elif l <= 1.08e+126:
		tmp = (1.0 - (0.5 * ((h * math.pow(((D / d) * (0.5 * M)), 2.0)) / l))) * (math.sqrt((d / l)) * math.sqrt((d / h)))
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -3.2e-244)
		tmp = Float64(Float64(d * (Float64(l * h) ^ -0.5)) * Float64(-1.0 - Float64(-0.5 * Float64((Float64(0.5 * Float64(M * Float64(D / d))) ^ 2.0) * Float64(h / l)))));
	elseif (l <= 1.08e+126)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(Float64(D / d) * Float64(0.5 * M)) ^ 2.0)) / l))) * Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -3.2e-244)
		tmp = (d * ((l * h) ^ -0.5)) * (-1.0 - (-0.5 * (((0.5 * (M * (D / d))) ^ 2.0) * (h / l))));
	elseif (l <= 1.08e+126)
		tmp = (1.0 - (0.5 * ((h * (((D / d) * (0.5 * M)) ^ 2.0)) / l))) * (sqrt((d / l)) * sqrt((d / h)));
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -3.2e-244], N[(N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(-0.5 * N[(N[Power[N[(0.5 * N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.08e+126], N[(N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(N[(D / d), $MachinePrecision] * N[(0.5 * M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.1999999999999998e-244

   1. Initial program 60.2%

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

   3. Simplified 60.7%

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

      1. associate-*r/ 59.9%

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

      2. div-inv 59.9%

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

      3. metadata-eval 59.9%

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

   5. Applied egg-rr 59.9%

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

      1. sqrt-div 0.0%

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

      2. metadata-eval 0.0%

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

      3. div-inv 0.0%

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

      4. associate-*r/ 0.0%

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

      5. pow1 0.0%

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

   7. Applied egg-rr 51.6%

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

      1. unpow1 51.6%

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

      2. *-commutative 51.6%

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

      3. *-commutative 51.6%

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

      4. associate-*r* 51.6%

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

   9. Simplified 51.6%

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

   10. Taylor expanded in d around -inf 71.7%

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

       1. mul-1-neg 71.7%

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

       2. distribute-rgt-neg-in 71.7%

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

       3. unpow-1 71.7%

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

       4. metadata-eval 71.7%

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

       5. pow-sqr 71.7%

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

       6. rem-sqrt-square 71.7%

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

       7. rem-square-sqrt 71.6%

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

       8. fabs-sqr 71.6%

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

       9. rem-square-sqrt 71.7%

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

   12. Simplified 71.7%

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

   if -3.1999999999999998e-244 < l < 1.0799999999999999e126

   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 70.5%

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

      1. associate-*r/ 75.4%

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

      2. div-inv 75.4%

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

      3. metadata-eval 75.4%

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

   5. Applied egg-rr 75.4%

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

   if 1.0799999999999999e126 < l

   1. Initial program 44.4%

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

   3. Simplified 44.3%

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

      1. associate-*r/ 44.1%

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

      2. div-inv 44.1%

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

      3. metadata-eval 44.1%

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

   5. Applied egg-rr 44.1%

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

   6. Taylor expanded in d around inf 51.6%

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

      1. unpow-1 51.6%

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

      2. sqr-pow 51.7%

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

      3. rem-sqrt-square 51.7%

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

      4. metadata-eval 51.7%

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

      5. sqr-pow 51.4%

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

      6. fabs-sqr 51.4%

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

      7. sqr-pow 51.7%

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

   8. Simplified 51.7%

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

      1. unpow-prod-down 64.6%

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

   10. Applied egg-rr 64.6%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.2 \cdot 10^{-244}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 - -0.5 \cdot \left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.08 \cdot 10^{+126}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2}}{\ell}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 7: 70.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ t_1 := \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 - -0.5 \cdot \left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{if}\;d \leq -1.3 \cdot 10^{+83}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -1.15 \cdot 10^{-41}:\\ \;\;\;\;t_0 \cdot \left(1 - 0.5 \cdot \left(\frac{0.25}{\ell} \cdot \frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{d \cdot \frac{d}{h}}\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{-102}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot \left(M \cdot \left(M \cdot \left(D \cdot \frac{D}{d}\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq 8.5 \cdot 10^{+112}:\\ \;\;\;\;t_0 \cdot \left(1 - 0.5 \cdot \frac{\frac{h \cdot \left(D \cdot \left(M \cdot \left(M \cdot D\right)\right)\right)}{\frac{d \cdot d}{0.25}}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (sqrt (/ d l)) (sqrt (/ d h))))
        (t_1
         (*
          (* d (pow (* l h) -0.5))
          (- -1.0 (* -0.5 (* (pow (* 0.5 (* M (/ D d))) 2.0) (/ h l)))))))
   (if (<= d -1.3e+83)
     t_1
     (if (<= d -1.15e-41)
       (*
        t_0
        (- 1.0 (* 0.5 (* (/ 0.25 l) (/ (* M (* D (* M D))) (* d (/ d h)))))))
       (if (<= d -5e-310)
         t_1
         (if (<= d 1.45e-102)
           (* (/ (sqrt h) (pow l 1.5)) (* -0.125 (* M (* M (* D (/ D d))))))
           (if (<= d 8.5e+112)
             (*
              t_0
              (-
               1.0
               (* 0.5 (/ (/ (* h (* D (* M (* M D)))) (/ (* d d) 0.25)) l))))
             (* d (/ (sqrt (/ 1.0 h)) (sqrt l))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / l)) * sqrt((d / h));
	double t_1 = (d * pow((l * h), -0.5)) * (-1.0 - (-0.5 * (pow((0.5 * (M * (D / d))), 2.0) * (h / l))));
	double tmp;
	if (d <= -1.3e+83) {
		tmp = t_1;
	} else if (d <= -1.15e-41) {
		tmp = t_0 * (1.0 - (0.5 * ((0.25 / l) * ((M * (D * (M * D))) / (d * (d / h))))));
	} else if (d <= -5e-310) {
		tmp = t_1;
	} else if (d <= 1.45e-102) {
		tmp = (sqrt(h) / pow(l, 1.5)) * (-0.125 * (M * (M * (D * (D / d)))));
	} else if (d <= 8.5e+112) {
		tmp = t_0 * (1.0 - (0.5 * (((h * (D * (M * (M * D)))) / ((d * d) / 0.25)) / l)));
	} else {
		tmp = d * (sqrt((1.0 / 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 = sqrt((d / l)) * sqrt((d / h))
    t_1 = (d * ((l * h) ** (-0.5d0))) * ((-1.0d0) - ((-0.5d0) * (((0.5d0 * (m * (d_1 / d))) ** 2.0d0) * (h / l))))
    if (d <= (-1.3d+83)) then
        tmp = t_1
    else if (d <= (-1.15d-41)) then
        tmp = t_0 * (1.0d0 - (0.5d0 * ((0.25d0 / l) * ((m * (d_1 * (m * d_1))) / (d * (d / h))))))
    else if (d <= (-5d-310)) then
        tmp = t_1
    else if (d <= 1.45d-102) then
        tmp = (sqrt(h) / (l ** 1.5d0)) * ((-0.125d0) * (m * (m * (d_1 * (d_1 / d)))))
    else if (d <= 8.5d+112) then
        tmp = t_0 * (1.0d0 - (0.5d0 * (((h * (d_1 * (m * (m * d_1)))) / ((d * d) / 0.25d0)) / l)))
    else
        tmp = d * (sqrt((1.0d0 / 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 / l)) * Math.sqrt((d / h));
	double t_1 = (d * Math.pow((l * h), -0.5)) * (-1.0 - (-0.5 * (Math.pow((0.5 * (M * (D / d))), 2.0) * (h / l))));
	double tmp;
	if (d <= -1.3e+83) {
		tmp = t_1;
	} else if (d <= -1.15e-41) {
		tmp = t_0 * (1.0 - (0.5 * ((0.25 / l) * ((M * (D * (M * D))) / (d * (d / h))))));
	} else if (d <= -5e-310) {
		tmp = t_1;
	} else if (d <= 1.45e-102) {
		tmp = (Math.sqrt(h) / Math.pow(l, 1.5)) * (-0.125 * (M * (M * (D * (D / d)))));
	} else if (d <= 8.5e+112) {
		tmp = t_0 * (1.0 - (0.5 * (((h * (D * (M * (M * D)))) / ((d * d) / 0.25)) / l)));
	} else {
		tmp = d * (Math.sqrt((1.0 / h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / l)) * math.sqrt((d / h))
	t_1 = (d * math.pow((l * h), -0.5)) * (-1.0 - (-0.5 * (math.pow((0.5 * (M * (D / d))), 2.0) * (h / l))))
	tmp = 0
	if d <= -1.3e+83:
		tmp = t_1
	elif d <= -1.15e-41:
		tmp = t_0 * (1.0 - (0.5 * ((0.25 / l) * ((M * (D * (M * D))) / (d * (d / h))))))
	elif d <= -5e-310:
		tmp = t_1
	elif d <= 1.45e-102:
		tmp = (math.sqrt(h) / math.pow(l, 1.5)) * (-0.125 * (M * (M * (D * (D / d)))))
	elif d <= 8.5e+112:
		tmp = t_0 * (1.0 - (0.5 * (((h * (D * (M * (M * D)))) / ((d * d) / 0.25)) / l)))
	else:
		tmp = d * (math.sqrt((1.0 / h)) / math.sqrt(l))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)))
	t_1 = Float64(Float64(d * (Float64(l * h) ^ -0.5)) * Float64(-1.0 - Float64(-0.5 * Float64((Float64(0.5 * Float64(M * Float64(D / d))) ^ 2.0) * Float64(h / l)))))
	tmp = 0.0
	if (d <= -1.3e+83)
		tmp = t_1;
	elseif (d <= -1.15e-41)
		tmp = Float64(t_0 * Float64(1.0 - Float64(0.5 * Float64(Float64(0.25 / l) * Float64(Float64(M * Float64(D * Float64(M * D))) / Float64(d * Float64(d / h)))))));
	elseif (d <= -5e-310)
		tmp = t_1;
	elseif (d <= 1.45e-102)
		tmp = Float64(Float64(sqrt(h) / (l ^ 1.5)) * Float64(-0.125 * Float64(M * Float64(M * Float64(D * Float64(D / d))))));
	elseif (d <= 8.5e+112)
		tmp = Float64(t_0 * Float64(1.0 - Float64(0.5 * Float64(Float64(Float64(h * Float64(D * Float64(M * Float64(M * D)))) / Float64(Float64(d * d) / 0.25)) / l))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / l)) * sqrt((d / h));
	t_1 = (d * ((l * h) ^ -0.5)) * (-1.0 - (-0.5 * (((0.5 * (M * (D / d))) ^ 2.0) * (h / l))));
	tmp = 0.0;
	if (d <= -1.3e+83)
		tmp = t_1;
	elseif (d <= -1.15e-41)
		tmp = t_0 * (1.0 - (0.5 * ((0.25 / l) * ((M * (D * (M * D))) / (d * (d / h))))));
	elseif (d <= -5e-310)
		tmp = t_1;
	elseif (d <= 1.45e-102)
		tmp = (sqrt(h) / (l ^ 1.5)) * (-0.125 * (M * (M * (D * (D / d)))));
	elseif (d <= 8.5e+112)
		tmp = t_0 * (1.0 - (0.5 * (((h * (D * (M * (M * D)))) / ((d * d) / 0.25)) / l)));
	else
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(-0.5 * N[(N[Power[N[(0.5 * N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.3e+83], t$95$1, If[LessEqual[d, -1.15e-41], N[(t$95$0 * N[(1.0 - N[(0.5 * N[(N[(0.25 / l), $MachinePrecision] * N[(N[(M * N[(D * N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], t$95$1, If[LessEqual[d, 1.45e-102], N[(N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision] * N[(-0.125 * N[(M * N[(M * N[(D * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 8.5e+112], N[(t$95$0 * N[(1.0 - N[(0.5 * N[(N[(N[(h * N[(D * N[(M * N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] / 0.25), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -1.3000000000000001e83 or -1.15000000000000005e-41 < d < -4.999999999999985e-310

   1. Initial program 56.6%

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

   3. Simplified 57.2%

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

      1. associate-*r/ 56.3%

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

      2. div-inv 56.3%

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

      3. metadata-eval 56.3%

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

   5. Applied egg-rr 56.3%

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

      1. sqrt-div 0.0%

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

      2. metadata-eval 0.0%

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

      3. div-inv 0.0%

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

      4. associate-*r/ 0.0%

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

      5. pow1 0.0%

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

   7. Applied egg-rr 51.7%

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

      1. unpow1 51.7%

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

      2. *-commutative 51.7%

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

      3. *-commutative 51.7%

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

      4. associate-*r* 51.7%

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

   9. Simplified 51.7%

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

   10. Taylor expanded in d around -inf 75.3%

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

       1. mul-1-neg 75.3%

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

       2. distribute-rgt-neg-in 75.3%

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

       3. unpow-1 75.3%

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

       4. metadata-eval 75.3%

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

       5. pow-sqr 75.3%

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

       6. rem-sqrt-square 75.3%

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

       7. rem-square-sqrt 75.2%

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

       8. fabs-sqr 75.2%

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

       9. rem-square-sqrt 75.3%

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

   12. Simplified 75.3%

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

   if -1.3000000000000001e83 < d < -1.15000000000000005e-41

   1. Initial program 84.8%

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

   3. Simplified 84.8%

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

      1. associate-*r/ 89.8%

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

      2. div-inv 89.8%

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

      3. metadata-eval 89.8%

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

   5. Applied egg-rr 89.8%

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

      1. pow1 89.8%

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

      2. *-commutative 89.8%

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

      3. metadata-eval 89.8%

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

      4. div-inv 89.8%

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

      5. *-commutative 89.8%

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

      6. div-inv 89.8%

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

      7. metadata-eval 89.8%

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

   7. Applied egg-rr 89.8%

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

      1. unpow1 89.8%

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

      2. associate-*l/ 89.8%

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

      3. associate-*r* 89.8%

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

      4. associate-*l/ 89.8%

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

      5. *-commutative 89.8%

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

      6. associate-*r/ 89.8%

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

      7. associate-/l* 89.8%

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

   9. Simplified 89.8%

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

       1. div-inv 89.8%

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

   11. Applied egg-rr 89.8%

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

   12. Taylor expanded in h around 0 60.1%

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

       1. associate-*r/ 60.1%

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

       2. *-commutative 60.1%

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

       3. times-frac 60.3%

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

       4. associate-*r* 65.2%

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

       5. unpow2 65.2%

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

       6. associate-*r* 80.1%

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

       7. unpow2 80.1%

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

       8. associate-/l* 84.9%

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

       9. unpow2 84.9%

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

       10. associate-*r* 70.1%

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

       11. unpow2 70.1%

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

       12. *-commutative 70.1%

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

       13. unpow2 70.1%

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

       14. associate-*l* 75.1%

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

       15. *-commutative 75.1%

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

       16. unpow2 75.1%

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

       17. associate-*l* 80.0%

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

       18. unpow2 80.0%

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

       19. associate-*l/ 80.0%

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

       20. *-commutative 80.0%

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

   14. Simplified 80.0%

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

   if -4.999999999999985e-310 < d < 1.44999999999999993e-102

   1. Initial program 35.3%

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

   3. Simplified 35.2%

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

   4. Taylor expanded in d around 0 41.2%

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

      1. associate-*r* 41.2%

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

      2. *-commutative 41.2%

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

      3. associate-/l* 41.1%

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

      4. associate-/r/ 43.3%

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

      5. unpow2 43.3%

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

      6. unpow2 43.3%

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

   6. Simplified 43.3%

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

   7. Taylor expanded in D around 0 41.2%

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

      1. unpow2 41.2%

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

      2. unpow2 41.2%

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

      3. unpow2 41.2%

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

      4. associate-*l/ 43.3%

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

      5. *-commutative 43.3%

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

      6. unpow2 43.3%

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

      7. associate-*l* 45.4%

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

      8. associate-/l* 45.8%

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

      9. associate-/r/ 45.7%

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

   9. Simplified 45.7%

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

       1. sqrt-div 47.8%

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

   11. Applied egg-rr 47.8%

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

       1. sqr-pow 47.8%

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

       2. rem-sqrt-square 57.7%

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

       3. sqr-pow 57.7%

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

       4. fabs-sqr 57.7%

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

       5. sqr-pow 57.7%

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

       6. metadata-eval 57.7%

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

   13. Simplified 57.7%

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

   if 1.44999999999999993e-102 < d < 8.50000000000000047e112

   1. Initial program 84.7%

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

   3. Simplified 84.7%

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

      1. associate-*r/ 92.2%

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

      2. div-inv 92.2%

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

      3. metadata-eval 92.2%

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

   5. Applied egg-rr 92.2%

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

   6. Taylor expanded in M around 0 76.7%

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

      1. associate-*r/ 76.7%

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

      2. unpow2 76.7%

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

      3. unpow2 76.7%

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

      4. associate-*r* 76.7%

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

      5. *-commutative 76.7%

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

      6. associate-/l* 76.7%

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

      7. associate-*r* 84.5%

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

      8. associate-*r* 86.4%

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

      9. associate-*r* 80.5%

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

      10. *-commutative 80.5%

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

      11. unswap-sqr 92.2%

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

      12. associate-*l* 90.4%

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

      13. unpow2 90.4%

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

   8. Simplified 90.4%

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

   if 8.50000000000000047e112 < d

   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 70.5%

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

   4. Taylor expanded in d around inf 68.3%

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

      1. associate-/r* 69.0%

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

   6. Simplified 69.0%

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

      1. sqrt-div 79.5%

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

   8. Applied egg-rr 79.5%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.3 \cdot 10^{+83}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 - -0.5 \cdot \left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -1.15 \cdot 10^{-41}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{0.25}{\ell} \cdot \frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{d \cdot \frac{d}{h}}\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 - -0.5 \cdot \left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{-102}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot \left(M \cdot \left(M \cdot \left(D \cdot \frac{D}{d}\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq 8.5 \cdot 10^{+112}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \frac{\frac{h \cdot \left(D \cdot \left(M \cdot \left(M \cdot D\right)\right)\right)}{\frac{d \cdot d}{0.25}}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 8: 64.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.2 \cdot 10^{-272}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 - -0.5 \cdot \left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+126}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(D \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \left(h \cdot M\right)}{\ell \cdot d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -6.2e-272)
   (*
    (* d (pow (* l h) -0.5))
    (- -1.0 (* -0.5 (* (pow (* 0.5 (* M (/ D d))) 2.0) (/ h l)))))
   (if (<= l 6e+126)
     (*
      (* (sqrt (/ d l)) (sqrt (/ d h)))
      (- 1.0 (* 0.5 (* 0.25 (* (* D (/ D d)) (/ (* M (* h M)) (* l d)))))))
     (* d (* (pow h -0.5) (pow l -0.5))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -6.2e-272) {
		tmp = (d * pow((l * h), -0.5)) * (-1.0 - (-0.5 * (pow((0.5 * (M * (D / d))), 2.0) * (h / l))));
	} else if (l <= 6e+126) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.5 * (0.25 * ((D * (D / d)) * ((M * (h * M)) / (l * d))))));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-6.2d-272)) then
        tmp = (d * ((l * h) ** (-0.5d0))) * ((-1.0d0) - ((-0.5d0) * (((0.5d0 * (m * (d_1 / d))) ** 2.0d0) * (h / l))))
    else if (l <= 6d+126) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - (0.5d0 * (0.25d0 * ((d_1 * (d_1 / d)) * ((m * (h * m)) / (l * d))))))
    else
        tmp = d * ((h ** (-0.5d0)) * (l ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -6.2e-272) {
		tmp = (d * Math.pow((l * h), -0.5)) * (-1.0 - (-0.5 * (Math.pow((0.5 * (M * (D / d))), 2.0) * (h / l))));
	} else if (l <= 6e+126) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (0.5 * (0.25 * ((D * (D / d)) * ((M * (h * M)) / (l * d))))));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -6.2e-272:
		tmp = (d * math.pow((l * h), -0.5)) * (-1.0 - (-0.5 * (math.pow((0.5 * (M * (D / d))), 2.0) * (h / l))))
	elif l <= 6e+126:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (0.5 * (0.25 * ((D * (D / d)) * ((M * (h * M)) / (l * d))))))
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -6.2e-272)
		tmp = Float64(Float64(d * (Float64(l * h) ^ -0.5)) * Float64(-1.0 - Float64(-0.5 * Float64((Float64(0.5 * Float64(M * Float64(D / d))) ^ 2.0) * Float64(h / l)))));
	elseif (l <= 6e+126)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(0.5 * Float64(0.25 * Float64(Float64(D * Float64(D / d)) * Float64(Float64(M * Float64(h * M)) / Float64(l * d)))))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -6.2e-272)
		tmp = (d * ((l * h) ^ -0.5)) * (-1.0 - (-0.5 * (((0.5 * (M * (D / d))) ^ 2.0) * (h / l))));
	elseif (l <= 6e+126)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.5 * (0.25 * ((D * (D / d)) * ((M * (h * M)) / (l * d))))));
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -6.2e-272], N[(N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(-0.5 * N[(N[Power[N[(0.5 * N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6e+126], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(0.25 * N[(N[(D * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(M * N[(h * M), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.20000000000000059e-272

   1. Initial program 60.0%

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

   3. Simplified 60.4%

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

      1. associate-*r/ 60.7%

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

      2. div-inv 60.7%

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

      3. metadata-eval 60.7%

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

   5. Applied egg-rr 60.7%

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

      1. sqrt-div 0.0%

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

      2. metadata-eval 0.0%

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

      3. div-inv 0.0%

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

      4. associate-*r/ 0.0%

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

      5. pow1 0.0%

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

   7. Applied egg-rr 52.2%

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

      1. unpow1 52.2%

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

      2. *-commutative 52.2%

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

      3. *-commutative 52.2%

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

      4. associate-*r* 52.2%

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

   9. Simplified 52.2%

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

   10. Taylor expanded in d around -inf 71.4%

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

       1. mul-1-neg 71.4%

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

       2. distribute-rgt-neg-in 71.4%

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

       3. unpow-1 71.4%

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

       4. metadata-eval 71.4%

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

       5. pow-sqr 71.4%

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

       6. rem-sqrt-square 71.4%

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

       7. rem-square-sqrt 71.3%

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

       8. fabs-sqr 71.3%

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

       9. rem-square-sqrt 71.4%

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

   12. Simplified 71.4%

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

   if -6.20000000000000059e-272 < l < 6.0000000000000005e126

   1. Initial program 71.5%

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

   3. Simplified 71.5%

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

   4. Taylor expanded in M around 0 53.9%

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

      1. associate-*r/ 53.9%

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

      2. unpow2 53.9%

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

      3. unpow2 53.9%

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

      4. unpow2 53.9%

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

   6. Simplified 53.9%

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

   7. Taylor expanded in D around 0 53.9%

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

      1. unpow2 53.9%

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

      2. unpow2 53.9%

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

      3. associate-*r* 55.7%

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

      4. unpow2 55.7%

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

      5. associate-*r* 60.9%

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

      6. times-frac 64.4%

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

      7. associate-/l* 65.4%

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

      8. associate-/r/ 65.4%

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

   9. Simplified 65.4%

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

   if 6.0000000000000005e126 < l

   1. Initial program 44.4%

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

   3. Simplified 44.3%

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

      1. associate-*r/ 44.1%

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

      2. div-inv 44.1%

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

      3. metadata-eval 44.1%

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

   5. Applied egg-rr 44.1%

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

   6. Taylor expanded in d around inf 51.6%

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

      1. unpow-1 51.6%

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

      2. sqr-pow 51.7%

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

      3. rem-sqrt-square 51.7%

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

      4. metadata-eval 51.7%

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

      5. sqr-pow 51.4%

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

      6. fabs-sqr 51.4%

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

      7. sqr-pow 51.7%

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

   8. Simplified 51.7%

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

      1. unpow-prod-down 64.6%

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

   10. Applied egg-rr 64.6%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.2 \cdot 10^{-272}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 - -0.5 \cdot \left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+126}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(D \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \left(h \cdot M\right)}{\ell \cdot d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 9: 59.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + -0.5 \cdot \left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{if}\;d \leq -2.6 \cdot 10^{-270}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 2.05 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot \left(M \cdot \left(M \cdot \left(D \cdot \frac{D}{d}\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq 3.1 \cdot 10^{+125}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (+ 1.0 (* -0.5 (* (pow (* 0.5 (* M (/ D d))) 2.0) (/ h l))))
          (sqrt (* (/ d l) (/ d h))))))
   (if (<= d -2.6e-270)
     t_0
     (if (<= d 2.05e-54)
       (* (/ (sqrt h) (pow l 1.5)) (* -0.125 (* M (* M (* D (/ D d))))))
       (if (<= d 3.1e+125) t_0 (* d (* (pow h -0.5) (pow l -0.5))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (1.0 + (-0.5 * (pow((0.5 * (M * (D / d))), 2.0) * (h / l)))) * sqrt(((d / l) * (d / h)));
	double tmp;
	if (d <= -2.6e-270) {
		tmp = t_0;
	} else if (d <= 2.05e-54) {
		tmp = (sqrt(h) / pow(l, 1.5)) * (-0.125 * (M * (M * (D * (D / d)))));
	} else if (d <= 3.1e+125) {
		tmp = t_0;
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + ((-0.5d0) * (((0.5d0 * (m * (d_1 / d))) ** 2.0d0) * (h / l)))) * sqrt(((d / l) * (d / h)))
    if (d <= (-2.6d-270)) then
        tmp = t_0
    else if (d <= 2.05d-54) then
        tmp = (sqrt(h) / (l ** 1.5d0)) * ((-0.125d0) * (m * (m * (d_1 * (d_1 / d)))))
    else if (d <= 3.1d+125) then
        tmp = t_0
    else
        tmp = d * ((h ** (-0.5d0)) * (l ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (1.0 + (-0.5 * (Math.pow((0.5 * (M * (D / d))), 2.0) * (h / l)))) * Math.sqrt(((d / l) * (d / h)));
	double tmp;
	if (d <= -2.6e-270) {
		tmp = t_0;
	} else if (d <= 2.05e-54) {
		tmp = (Math.sqrt(h) / Math.pow(l, 1.5)) * (-0.125 * (M * (M * (D * (D / d)))));
	} else if (d <= 3.1e+125) {
		tmp = t_0;
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (1.0 + (-0.5 * (math.pow((0.5 * (M * (D / d))), 2.0) * (h / l)))) * math.sqrt(((d / l) * (d / h)))
	tmp = 0
	if d <= -2.6e-270:
		tmp = t_0
	elif d <= 2.05e-54:
		tmp = (math.sqrt(h) / math.pow(l, 1.5)) * (-0.125 * (M * (M * (D * (D / d)))))
	elif d <= 3.1e+125:
		tmp = t_0
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(1.0 + Float64(-0.5 * Float64((Float64(0.5 * Float64(M * Float64(D / d))) ^ 2.0) * Float64(h / l)))) * sqrt(Float64(Float64(d / l) * Float64(d / h))))
	tmp = 0.0
	if (d <= -2.6e-270)
		tmp = t_0;
	elseif (d <= 2.05e-54)
		tmp = Float64(Float64(sqrt(h) / (l ^ 1.5)) * Float64(-0.125 * Float64(M * Float64(M * Float64(D * Float64(D / d))))));
	elseif (d <= 3.1e+125)
		tmp = t_0;
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (1.0 + (-0.5 * (((0.5 * (M * (D / d))) ^ 2.0) * (h / l)))) * sqrt(((d / l) * (d / h)));
	tmp = 0.0;
	if (d <= -2.6e-270)
		tmp = t_0;
	elseif (d <= 2.05e-54)
		tmp = (sqrt(h) / (l ^ 1.5)) * (-0.125 * (M * (M * (D * (D / d)))));
	elseif (d <= 3.1e+125)
		tmp = t_0;
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(1.0 + N[(-0.5 * N[(N[Power[N[(0.5 * N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.6e-270], t$95$0, If[LessEqual[d, 2.05e-54], N[(N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision] * N[(-0.125 * N[(M * N[(M * N[(D * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.1e+125], t$95$0, N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.6000000000000002e-270 or 2.05e-54 < d < 3.1e125

   1. Initial program 71.9%

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

   3. Simplified 72.2%

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

      1. associate-*r/ 75.1%

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

      2. div-inv 75.1%

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

      3. metadata-eval 75.1%

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

   5. Applied egg-rr 75.1%

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

      1. sqrt-div 29.4%

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

      2. metadata-eval 29.4%

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

      3. div-inv 29.4%

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

      4. associate-*r/ 26.8%

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

      5. pow1 26.8%

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

   7. Applied egg-rr 63.0%

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

      1. unpow1 63.0%

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

      2. *-commutative 63.0%

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

      3. *-commutative 63.0%

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

      4. associate-*r* 63.0%

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

   9. Simplified 63.0%

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

   if -2.6000000000000002e-270 < d < 2.05e-54

   1. Initial program 38.5%

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

   3. Simplified 38.3%

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

   4. Taylor expanded in d around 0 38.3%

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

      1. associate-*r* 38.3%

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

      2. *-commutative 38.3%

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

      3. associate-/l* 38.1%

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

      4. associate-/r/ 39.8%

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

      5. unpow2 39.8%

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

      6. unpow2 39.8%

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

   6. Simplified 39.8%

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

   7. Taylor expanded in D around 0 38.3%

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

      1. unpow2 38.3%

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

      2. unpow2 38.3%

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

      3. unpow2 38.3%

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

      4. associate-*l/ 39.8%

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

      5. *-commutative 39.8%

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

      6. unpow2 39.8%

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

      7. associate-*l* 41.4%

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

      8. associate-/l* 41.7%

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

      9. associate-/r/ 41.7%

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

   9. Simplified 41.7%

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

       1. sqrt-div 43.1%

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

   11. Applied egg-rr 43.1%

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

       1. sqr-pow 43.1%

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

       2. rem-sqrt-square 51.9%

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

       3. sqr-pow 51.9%

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

       4. fabs-sqr 51.9%

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

       5. sqr-pow 51.9%

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

       6. metadata-eval 51.9%

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

   13. Simplified 51.9%

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

   if 3.1e125 < d

   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 69.9%

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

      1. associate-*r/ 72.6%

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

      2. div-inv 72.6%

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

      3. metadata-eval 72.6%

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

   5. Applied egg-rr 72.6%

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

   6. Taylor expanded in d around inf 69.8%

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

      1. unpow-1 69.8%

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

      2. sqr-pow 70.0%

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

      3. rem-sqrt-square 70.0%

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

      4. metadata-eval 70.0%

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

      5. sqr-pow 69.6%

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

      6. fabs-sqr 69.6%

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

      7. sqr-pow 70.0%

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

   8. Simplified 70.0%

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

      1. unpow-prod-down 80.0%

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

   10. Applied egg-rr 80.0%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.6 \cdot 10^{-270}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;d \leq 2.05 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot \left(M \cdot \left(M \cdot \left(D \cdot \frac{D}{d}\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq 3.1 \cdot 10^{+125}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 10: 61.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \frac{h \cdot \left(0.25 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{if}\;d \leq -2.6 \cdot 10^{-270}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 2.05 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot \left(M \cdot \left(M \cdot \left(D \cdot \frac{D}{d}\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq 5 \cdot 10^{+127}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (sqrt (* (/ d l) (/ d h)))
          (+ 1.0 (* -0.5 (/ (* h (* 0.25 (pow (* M (/ D d)) 2.0))) l))))))
   (if (<= d -2.6e-270)
     t_0
     (if (<= d 2.05e-54)
       (* (/ (sqrt h) (pow l 1.5)) (* -0.125 (* M (* M (* D (/ D d))))))
       (if (<= d 5e+127) t_0 (* d (* (pow h -0.5) (pow l -0.5))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * ((h * (0.25 * pow((M * (D / d)), 2.0))) / l)));
	double tmp;
	if (d <= -2.6e-270) {
		tmp = t_0;
	} else if (d <= 2.05e-54) {
		tmp = (sqrt(h) / pow(l, 1.5)) * (-0.125 * (M * (M * (D * (D / d)))));
	} else if (d <= 5e+127) {
		tmp = t_0;
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((d / l) * (d / h))) * (1.0d0 + ((-0.5d0) * ((h * (0.25d0 * ((m * (d_1 / d)) ** 2.0d0))) / l)))
    if (d <= (-2.6d-270)) then
        tmp = t_0
    else if (d <= 2.05d-54) then
        tmp = (sqrt(h) / (l ** 1.5d0)) * ((-0.125d0) * (m * (m * (d_1 * (d_1 / d)))))
    else if (d <= 5d+127) then
        tmp = t_0
    else
        tmp = d * ((h ** (-0.5d0)) * (l ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * ((h * (0.25 * Math.pow((M * (D / d)), 2.0))) / l)));
	double tmp;
	if (d <= -2.6e-270) {
		tmp = t_0;
	} else if (d <= 2.05e-54) {
		tmp = (Math.sqrt(h) / Math.pow(l, 1.5)) * (-0.125 * (M * (M * (D * (D / d)))));
	} else if (d <= 5e+127) {
		tmp = t_0;
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * ((h * (0.25 * math.pow((M * (D / d)), 2.0))) / l)))
	tmp = 0
	if d <= -2.6e-270:
		tmp = t_0
	elif d <= 2.05e-54:
		tmp = (math.sqrt(h) / math.pow(l, 1.5)) * (-0.125 * (M * (M * (D * (D / d)))))
	elif d <= 5e+127:
		tmp = t_0
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h * Float64(0.25 * (Float64(M * Float64(D / d)) ^ 2.0))) / l))))
	tmp = 0.0
	if (d <= -2.6e-270)
		tmp = t_0;
	elseif (d <= 2.05e-54)
		tmp = Float64(Float64(sqrt(h) / (l ^ 1.5)) * Float64(-0.125 * Float64(M * Float64(M * Float64(D * Float64(D / d))))));
	elseif (d <= 5e+127)
		tmp = t_0;
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * ((h * (0.25 * ((M * (D / d)) ^ 2.0))) / l)));
	tmp = 0.0;
	if (d <= -2.6e-270)
		tmp = t_0;
	elseif (d <= 2.05e-54)
		tmp = (sqrt(h) / (l ^ 1.5)) * (-0.125 * (M * (M * (D * (D / d)))));
	elseif (d <= 5e+127)
		tmp = t_0;
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[(h * N[(0.25 * N[Power[N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.6e-270], t$95$0, If[LessEqual[d, 2.05e-54], N[(N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision] * N[(-0.125 * N[(M * N[(M * N[(D * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5e+127], t$95$0, N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.6000000000000002e-270 or 2.05e-54 < d < 5.0000000000000004e127

   1. Initial program 71.9%

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

   3. Simplified 72.2%

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

      1. associate-*r/ 75.1%

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

      2. div-inv 75.1%

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

      3. metadata-eval 75.1%

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

   5. Applied egg-rr 75.1%

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

      1. sqrt-div 29.4%

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

      2. metadata-eval 29.4%

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

      3. div-inv 29.4%

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

      4. associate-*r/ 26.8%

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

      5. pow1 26.8%

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

   7. Applied egg-rr 63.0%

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

      1. unpow1 63.0%

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

      2. *-commutative 63.0%

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

      3. *-commutative 63.0%

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

      4. associate-*r* 63.0%

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

   9. Simplified 63.0%

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

       1. associate-*r/ 65.9%

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

       2. unpow-prod-down 65.9%

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

       3. metadata-eval 65.9%

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

   11. Applied egg-rr 65.9%

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

   if -2.6000000000000002e-270 < d < 2.05e-54

   1. Initial program 38.5%

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

   3. Simplified 38.3%

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

   4. Taylor expanded in d around 0 38.3%

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

      1. associate-*r* 38.3%

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

      2. *-commutative 38.3%

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

      3. associate-/l* 38.1%

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

      4. associate-/r/ 39.8%

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

      5. unpow2 39.8%

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

      6. unpow2 39.8%

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

   6. Simplified 39.8%

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

   7. Taylor expanded in D around 0 38.3%

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

      1. unpow2 38.3%

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

      2. unpow2 38.3%

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

      3. unpow2 38.3%

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

      4. associate-*l/ 39.8%

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

      5. *-commutative 39.8%

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

      6. unpow2 39.8%

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

      7. associate-*l* 41.4%

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

      8. associate-/l* 41.7%

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

      9. associate-/r/ 41.7%

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

   9. Simplified 41.7%

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

       1. sqrt-div 43.1%

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

   11. Applied egg-rr 43.1%

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

       1. sqr-pow 43.1%

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

       2. rem-sqrt-square 51.9%

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

       3. sqr-pow 51.9%

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

       4. fabs-sqr 51.9%

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

       5. sqr-pow 51.9%

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

       6. metadata-eval 51.9%

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

   13. Simplified 51.9%

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

   if 5.0000000000000004e127 < d

   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 69.9%

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

      1. associate-*r/ 72.6%

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

      2. div-inv 72.6%

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

      3. metadata-eval 72.6%

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

   5. Applied egg-rr 72.6%

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

   6. Taylor expanded in d around inf 69.8%

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

      1. unpow-1 69.8%

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

      2. sqr-pow 70.0%

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

      3. rem-sqrt-square 70.0%

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

      4. metadata-eval 70.0%

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

      5. sqr-pow 69.6%

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

      6. fabs-sqr 69.6%

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

      7. sqr-pow 70.0%

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

   8. Simplified 70.0%

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

      1. unpow-prod-down 80.0%

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

   10. Applied egg-rr 80.0%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.6 \cdot 10^{-270}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \frac{h \cdot \left(0.25 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{elif}\;d \leq 2.05 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot \left(M \cdot \left(M \cdot \left(D \cdot \frac{D}{d}\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq 5 \cdot 10^{+127}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \frac{h \cdot \left(0.25 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 11: 67.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := M \cdot \frac{D}{d}\\ \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 - -0.5 \cdot \left({\left(0.5 \cdot t_0\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq 2.05 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot \left(M \cdot \left(M \cdot \left(D \cdot \frac{D}{d}\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq 8.6 \cdot 10^{+128}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \frac{h \cdot \left(0.25 \cdot {t_0}^{2}\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* M (/ D d))))
   (if (<= d -5e-310)
     (*
      (* d (pow (* l h) -0.5))
      (- -1.0 (* -0.5 (* (pow (* 0.5 t_0) 2.0) (/ h l)))))
     (if (<= d 2.05e-54)
       (* (/ (sqrt h) (pow l 1.5)) (* -0.125 (* M (* M (* D (/ D d))))))
       (if (<= d 8.6e+128)
         (*
          (sqrt (* (/ d l) (/ d h)))
          (+ 1.0 (* -0.5 (/ (* h (* 0.25 (pow t_0 2.0))) l))))
         (* d (* (pow h -0.5) (pow l -0.5))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = M * (D / d);
	double tmp;
	if (d <= -5e-310) {
		tmp = (d * pow((l * h), -0.5)) * (-1.0 - (-0.5 * (pow((0.5 * t_0), 2.0) * (h / l))));
	} else if (d <= 2.05e-54) {
		tmp = (sqrt(h) / pow(l, 1.5)) * (-0.125 * (M * (M * (D * (D / d)))));
	} else if (d <= 8.6e+128) {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * ((h * (0.25 * pow(t_0, 2.0))) / l)));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = m * (d_1 / d)
    if (d <= (-5d-310)) then
        tmp = (d * ((l * h) ** (-0.5d0))) * ((-1.0d0) - ((-0.5d0) * (((0.5d0 * t_0) ** 2.0d0) * (h / l))))
    else if (d <= 2.05d-54) then
        tmp = (sqrt(h) / (l ** 1.5d0)) * ((-0.125d0) * (m * (m * (d_1 * (d_1 / d)))))
    else if (d <= 8.6d+128) then
        tmp = sqrt(((d / l) * (d / h))) * (1.0d0 + ((-0.5d0) * ((h * (0.25d0 * (t_0 ** 2.0d0))) / l)))
    else
        tmp = d * ((h ** (-0.5d0)) * (l ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = M * (D / d);
	double tmp;
	if (d <= -5e-310) {
		tmp = (d * Math.pow((l * h), -0.5)) * (-1.0 - (-0.5 * (Math.pow((0.5 * t_0), 2.0) * (h / l))));
	} else if (d <= 2.05e-54) {
		tmp = (Math.sqrt(h) / Math.pow(l, 1.5)) * (-0.125 * (M * (M * (D * (D / d)))));
	} else if (d <= 8.6e+128) {
		tmp = Math.sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * ((h * (0.25 * Math.pow(t_0, 2.0))) / l)));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = M * (D / d)
	tmp = 0
	if d <= -5e-310:
		tmp = (d * math.pow((l * h), -0.5)) * (-1.0 - (-0.5 * (math.pow((0.5 * t_0), 2.0) * (h / l))))
	elif d <= 2.05e-54:
		tmp = (math.sqrt(h) / math.pow(l, 1.5)) * (-0.125 * (M * (M * (D * (D / d)))))
	elif d <= 8.6e+128:
		tmp = math.sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * ((h * (0.25 * math.pow(t_0, 2.0))) / l)))
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(M * Float64(D / d))
	tmp = 0.0
	if (d <= -5e-310)
		tmp = Float64(Float64(d * (Float64(l * h) ^ -0.5)) * Float64(-1.0 - Float64(-0.5 * Float64((Float64(0.5 * t_0) ^ 2.0) * Float64(h / l)))));
	elseif (d <= 2.05e-54)
		tmp = Float64(Float64(sqrt(h) / (l ^ 1.5)) * Float64(-0.125 * Float64(M * Float64(M * Float64(D * Float64(D / d))))));
	elseif (d <= 8.6e+128)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h * Float64(0.25 * (t_0 ^ 2.0))) / l))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = M * (D / d);
	tmp = 0.0;
	if (d <= -5e-310)
		tmp = (d * ((l * h) ^ -0.5)) * (-1.0 - (-0.5 * (((0.5 * t_0) ^ 2.0) * (h / l))));
	elseif (d <= 2.05e-54)
		tmp = (sqrt(h) / (l ^ 1.5)) * (-0.125 * (M * (M * (D * (D / d)))));
	elseif (d <= 8.6e+128)
		tmp = sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * ((h * (0.25 * (t_0 ^ 2.0))) / l)));
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -5e-310], N[(N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(-0.5 * N[(N[Power[N[(0.5 * t$95$0), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.05e-54], N[(N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision] * N[(-0.125 * N[(M * N[(M * N[(D * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 8.6e+128], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[(h * N[(0.25 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -4.999999999999985e-310

   1. Initial program 61.9%

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

   3. Simplified 62.3%

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

      1. associate-*r/ 62.5%

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

      2. div-inv 62.5%

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

      3. metadata-eval 62.5%

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

   5. Applied egg-rr 62.5%

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

      1. sqrt-div 0.0%

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

      2. metadata-eval 0.0%

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

      3. div-inv 0.0%

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

      4. associate-*r/ 0.0%

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

      5. pow1 0.0%

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

   7. Applied egg-rr 54.4%

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

      1. unpow1 54.4%

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

      2. *-commutative 54.4%

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

      3. *-commutative 54.4%

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

      4. associate-*r* 54.4%

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

   9. Simplified 54.4%

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

   10. Taylor expanded in d around -inf 72.7%

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

       1. mul-1-neg 72.7%

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

       2. distribute-rgt-neg-in 72.7%

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

       3. unpow-1 72.7%

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

       4. metadata-eval 72.7%

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

       5. pow-sqr 72.8%

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

       6. rem-sqrt-square 72.8%

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

       7. rem-square-sqrt 72.6%

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

       8. fabs-sqr 72.6%

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

       9. rem-square-sqrt 72.8%

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

   12. Simplified 72.8%

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

   if -4.999999999999985e-310 < d < 2.05e-54

   1. Initial program 42.7%

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

   3. Simplified 42.5%

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

   4. Taylor expanded in d around 0 44.2%

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

      1. associate-*r* 44.2%

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

      2. *-commutative 44.2%

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

      3. associate-/l* 44.1%

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

      4. associate-/r/ 46.0%

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

      5. unpow2 46.0%

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

      6. unpow2 46.0%

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

   6. Simplified 46.0%

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

   7. Taylor expanded in D around 0 44.2%

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

      1. unpow2 44.2%

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

      2. unpow2 44.2%

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

      3. unpow2 44.2%

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

      4. associate-*l/ 46.0%

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

      5. *-commutative 46.0%

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

      6. unpow2 46.0%

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

      7. associate-*l* 47.9%

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

      8. associate-/l* 48.2%

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

      9. associate-/r/ 48.2%

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

   9. Simplified 48.2%

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

       1. sqrt-div 49.9%

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

   11. Applied egg-rr 49.9%

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

       1. sqr-pow 49.9%

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

       2. rem-sqrt-square 60.1%

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

       3. sqr-pow 60.1%

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

       4. fabs-sqr 60.1%

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

       5. sqr-pow 60.1%

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

       6. metadata-eval 60.1%

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

   13. Simplified 60.1%

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

   if 2.05e-54 < d < 8.59999999999999949e128

   1. Initial program 83.5%

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

   3. Simplified 83.5%

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

      1. associate-*r/ 91.7%

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

      2. div-inv 91.7%

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

      3. metadata-eval 91.7%

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

   5. Applied egg-rr 91.7%

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

      1. sqrt-div 91.5%

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

      2. metadata-eval 91.5%

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

      3. div-inv 91.5%

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

      4. associate-*r/ 83.3%

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

      5. pow1 83.3%

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

   7. Applied egg-rr 71.0%

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

      1. unpow1 71.0%

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

      2. *-commutative 71.0%

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

      3. *-commutative 71.0%

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

      4. associate-*r* 71.0%

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

   9. Simplified 71.0%

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

       1. associate-*r/ 79.2%

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

       2. unpow-prod-down 79.2%

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

       3. metadata-eval 79.2%

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

   11. Applied egg-rr 79.2%

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

   if 8.59999999999999949e128 < d

   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 69.9%

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

      1. associate-*r/ 72.6%

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

      2. div-inv 72.6%

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

      3. metadata-eval 72.6%

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

   5. Applied egg-rr 72.6%

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

   6. Taylor expanded in d around inf 69.8%

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

      1. unpow-1 69.8%

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

      2. sqr-pow 70.0%

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

      3. rem-sqrt-square 70.0%

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

      4. metadata-eval 70.0%

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

      5. sqr-pow 69.6%

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

      6. fabs-sqr 69.6%

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

      7. sqr-pow 70.0%

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

   8. Simplified 70.0%

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

      1. unpow-prod-down 80.0%

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

   10. Applied egg-rr 80.0%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 - -0.5 \cdot \left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq 2.05 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot \left(M \cdot \left(M \cdot \left(D \cdot \frac{D}{d}\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq 8.6 \cdot 10^{+128}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \frac{h \cdot \left(0.25 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 12: 48.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\ell \cdot h}\\ \mathbf{if}\;d \leq -6.2 \cdot 10^{-127}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq 1.65 \cdot 10^{-284}:\\ \;\;\;\;d \cdot \sqrt{\sqrt[3]{t_0 \cdot \left(t_0 \cdot t_0\right)}}\\ \mathbf{elif}\;d \leq 4.6 \cdot 10^{+70}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot \left(M \cdot \left(M \cdot \left(D \cdot \frac{D}{d}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* l h))))
   (if (<= d -6.2e-127)
     (* (sqrt (/ d l)) (sqrt (/ d h)))
     (if (<= d 1.65e-284)
       (* d (sqrt (cbrt (* t_0 (* t_0 t_0)))))
       (if (<= d 4.6e+70)
         (* (/ (sqrt h) (pow l 1.5)) (* -0.125 (* M (* M (* D (/ D d))))))
         (* d (* (pow h -0.5) (pow l -0.5))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 / (l * h);
	double tmp;
	if (d <= -6.2e-127) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (d <= 1.65e-284) {
		tmp = d * sqrt(cbrt((t_0 * (t_0 * t_0))));
	} else if (d <= 4.6e+70) {
		tmp = (sqrt(h) / pow(l, 1.5)) * (-0.125 * (M * (M * (D * (D / d)))));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 / (l * h);
	double tmp;
	if (d <= -6.2e-127) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else if (d <= 1.65e-284) {
		tmp = d * Math.sqrt(Math.cbrt((t_0 * (t_0 * t_0))));
	} else if (d <= 4.6e+70) {
		tmp = (Math.sqrt(h) / Math.pow(l, 1.5)) * (-0.125 * (M * (M * (D * (D / d)))));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(1.0 / Float64(l * h))
	tmp = 0.0
	if (d <= -6.2e-127)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (d <= 1.65e-284)
		tmp = Float64(d * sqrt(cbrt(Float64(t_0 * Float64(t_0 * t_0)))));
	elseif (d <= 4.6e+70)
		tmp = Float64(Float64(sqrt(h) / (l ^ 1.5)) * Float64(-0.125 * Float64(M * Float64(M * Float64(D * Float64(D / d))))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -6.2e-127], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.65e-284], N[(d * N[Sqrt[N[Power[N[(t$95$0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.6e+70], N[(N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision] * N[(-0.125 * N[(M * N[(M * N[(D * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -6.2e-127

   1. Initial program 70.2%

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

   3. Simplified 68.9%

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

   4. Taylor expanded in M around 0 46.4%

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

   if -6.2e-127 < d < 1.65000000000000004e-284

   1. Initial program 36.4%

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

   3. Simplified 36.4%

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

   4. Taylor expanded in d around inf 19.9%

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

      1. add-cbrt-cube 25.7%

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

   6. Applied egg-rr 25.7%

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

   if 1.65000000000000004e-284 < d < 4.59999999999999987e70

   1. Initial program 59.7%

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

   3. Simplified 59.6%

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

   4. Taylor expanded in d around 0 45.9%

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

      1. associate-*r* 45.9%

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

      2. *-commutative 45.9%

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

      3. associate-/l* 44.7%

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

      4. associate-/r/ 45.9%

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

      5. unpow2 45.9%

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

      6. unpow2 45.9%

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

   6. Simplified 45.9%

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

   7. Taylor expanded in D around 0 45.9%

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

      1. unpow2 45.9%

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

      2. unpow2 45.9%

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

      3. unpow2 45.9%

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

      4. associate-*l/ 45.9%

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

      5. *-commutative 45.9%

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

      6. unpow2 45.9%

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

      7. associate-*l* 47.2%

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

      8. associate-/l* 47.4%

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

      9. associate-/r/ 47.4%

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

   9. Simplified 47.4%

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

       1. sqrt-div 50.8%

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

   11. Applied egg-rr 50.8%

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

       1. sqr-pow 50.8%

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

       2. rem-sqrt-square 58.6%

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

       3. sqr-pow 58.6%

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

       4. fabs-sqr 58.6%

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

       5. sqr-pow 58.6%

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

       6. metadata-eval 58.6%

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

   13. Simplified 58.6%

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

   if 4.59999999999999987e70 < d

   1. Initial program 74.2%

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

   3. Simplified 74.3%

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

      1. associate-*r/ 76.4%

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

      2. div-inv 76.4%

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

      3. metadata-eval 76.4%

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

   5. Applied egg-rr 76.4%

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

   6. Taylor expanded in d around inf 67.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. unpow-1 67.0%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

      2. sqr-pow 67.0%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \]

      3. rem-sqrt-square 67.0%

         \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \]

      4. metadata-eval 67.0%

         \[\leadsto d \cdot \left|{\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right| \]

      5. sqr-pow 66.8%

         \[\leadsto d \cdot \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)}}\right| \]

      6. fabs-sqr 66.8%

         \[\leadsto d \cdot \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)}\right)} \]

      7. sqr-pow 67.0%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   8. Simplified 67.0%

      \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
   9. Step-by-step derivation

      1. unpow-prod-down 78.4%

         \[\leadsto d \cdot \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \]

   10. Applied egg-rr 78.4%

       \[\leadsto d \cdot \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.2 \cdot 10^{-127}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq 1.65 \cdot 10^{-284}:\\ \;\;\;\;d \cdot \sqrt{\sqrt[3]{\frac{1}{\ell \cdot h} \cdot \left(\frac{1}{\ell \cdot h} \cdot \frac{1}{\ell \cdot h}\right)}}\\ \mathbf{elif}\;d \leq 4.6 \cdot 10^{+70}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot \left(M \cdot \left(M \cdot \left(D \cdot \frac{D}{d}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 13: 47.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 1.65 \cdot 10^{-284}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq 4.6 \cdot 10^{+70}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot \left(M \cdot \left(M \cdot \left(D \cdot \frac{D}{d}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d 1.65e-284)
   (* (sqrt (/ d l)) (sqrt (/ d h)))
   (if (<= d 4.6e+70)
     (* (/ (sqrt h) (pow l 1.5)) (* -0.125 (* M (* M (* D (/ D d))))))
     (* d (* (pow h -0.5) (pow l -0.5))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 1.65e-284) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (d <= 4.6e+70) {
		tmp = (sqrt(h) / pow(l, 1.5)) * (-0.125 * (M * (M * (D * (D / d)))));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= 1.65d-284) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else if (d <= 4.6d+70) then
        tmp = (sqrt(h) / (l ** 1.5d0)) * ((-0.125d0) * (m * (m * (d_1 * (d_1 / d)))))
    else
        tmp = d * ((h ** (-0.5d0)) * (l ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 1.65e-284) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else if (d <= 4.6e+70) {
		tmp = (Math.sqrt(h) / Math.pow(l, 1.5)) * (-0.125 * (M * (M * (D * (D / d)))));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= 1.65e-284:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	elif d <= 4.6e+70:
		tmp = (math.sqrt(h) / math.pow(l, 1.5)) * (-0.125 * (M * (M * (D * (D / d)))))
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= 1.65e-284)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (d <= 4.6e+70)
		tmp = Float64(Float64(sqrt(h) / (l ^ 1.5)) * Float64(-0.125 * Float64(M * Float64(M * Float64(D * Float64(D / d))))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= 1.65e-284)
		tmp = sqrt((d / l)) * sqrt((d / h));
	elseif (d <= 4.6e+70)
		tmp = (sqrt(h) / (l ^ 1.5)) * (-0.125 * (M * (M * (D * (D / d)))));
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, 1.65e-284], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.6e+70], N[(N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision] * N[(-0.125 * N[(M * N[(M * N[(D * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < 1.65000000000000004e-284

   1. Initial program 60.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 59.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]

   4. Taylor expanded in M around 0 36.7%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{1}\right) \]

   if 1.65000000000000004e-284 < d < 4.59999999999999987e70

   1. Initial program 59.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 59.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around 0 45.9%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 45.9%

         \[\leadsto \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative 45.9%

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

      3. associate-/l* 44.7%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}}\right) \]

      4. associate-/r/ 45.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{d} \cdot {M}^{2}\right)}\right) \]

      5. unpow2 45.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{d} \cdot {M}^{2}\right)\right) \]

      6. unpow2 45.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(\frac{D \cdot D}{d} \cdot \color{blue}{\left(M \cdot M\right)}\right)\right) \]

   6. Simplified 45.9%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)\right)} \]

   7. Taylor expanded in D around 0 45.9%

      \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{d}}\right) \]
   8. Step-by-step derivation

      1. unpow2 45.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{d}\right) \]

      2. unpow2 45.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}}{d}\right) \]

      3. unpow2 45.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{{M}^{2}}}{d}\right) \]

      4. associate-*l/ 45.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left(\frac{D \cdot D}{d} \cdot {M}^{2}\right)}\right) \]

      5. *-commutative 45.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left({M}^{2} \cdot \frac{D \cdot D}{d}\right)}\right) \]

      6. unpow2 45.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{D \cdot D}{d}\right)\right) \]

      7. associate-*l* 47.2%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left(M \cdot \left(M \cdot \frac{D \cdot D}{d}\right)\right)}\right) \]

      8. associate-/l* 47.4%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(M \cdot \left(M \cdot \color{blue}{\frac{D}{\frac{d}{D}}}\right)\right)\right) \]

      9. associate-/r/ 47.4%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(M \cdot \left(M \cdot \color{blue}{\left(\frac{D}{d} \cdot D\right)}\right)\right)\right) \]

   9. Simplified 47.4%

      \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left(M \cdot \left(M \cdot \left(\frac{D}{d} \cdot D\right)\right)\right)}\right) \]
   10. Step-by-step derivation

       1. sqrt-div 50.8%

          \[\leadsto \color{blue}{\frac{\sqrt{h}}{\sqrt{{\ell}^{3}}}} \cdot \left(-0.125 \cdot \left(M \cdot \left(M \cdot \left(\frac{D}{d} \cdot D\right)\right)\right)\right) \]

   11. Applied egg-rr 50.8%

       \[\leadsto \color{blue}{\frac{\sqrt{h}}{\sqrt{{\ell}^{3}}}} \cdot \left(-0.125 \cdot \left(M \cdot \left(M \cdot \left(\frac{D}{d} \cdot D\right)\right)\right)\right) \]
   12. Step-by-step derivation

       1. sqr-pow 50.8%

          \[\leadsto \frac{\sqrt{h}}{\sqrt{\color{blue}{{\ell}^{\left(\frac{3}{2}\right)} \cdot {\ell}^{\left(\frac{3}{2}\right)}}}} \cdot \left(-0.125 \cdot \left(M \cdot \left(M \cdot \left(\frac{D}{d} \cdot D\right)\right)\right)\right) \]

       2. rem-sqrt-square 58.6%

          \[\leadsto \frac{\sqrt{h}}{\color{blue}{\left|{\ell}^{\left(\frac{3}{2}\right)}\right|}} \cdot \left(-0.125 \cdot \left(M \cdot \left(M \cdot \left(\frac{D}{d} \cdot D\right)\right)\right)\right) \]

       3. sqr-pow 58.6%

          \[\leadsto \frac{\sqrt{h}}{\left|\color{blue}{{\ell}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot {\ell}^{\left(\frac{\frac{3}{2}}{2}\right)}}\right|} \cdot \left(-0.125 \cdot \left(M \cdot \left(M \cdot \left(\frac{D}{d} \cdot D\right)\right)\right)\right) \]

       4. fabs-sqr 58.6%

          \[\leadsto \frac{\sqrt{h}}{\color{blue}{{\ell}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot {\ell}^{\left(\frac{\frac{3}{2}}{2}\right)}}} \cdot \left(-0.125 \cdot \left(M \cdot \left(M \cdot \left(\frac{D}{d} \cdot D\right)\right)\right)\right) \]

       5. sqr-pow 58.6%

          \[\leadsto \frac{\sqrt{h}}{\color{blue}{{\ell}^{\left(\frac{3}{2}\right)}}} \cdot \left(-0.125 \cdot \left(M \cdot \left(M \cdot \left(\frac{D}{d} \cdot D\right)\right)\right)\right) \]

       6. metadata-eval 58.6%

          \[\leadsto \frac{\sqrt{h}}{{\ell}^{\color{blue}{1.5}}} \cdot \left(-0.125 \cdot \left(M \cdot \left(M \cdot \left(\frac{D}{d} \cdot D\right)\right)\right)\right) \]

   13. Simplified 58.6%

       \[\leadsto \color{blue}{\frac{\sqrt{h}}{{\ell}^{1.5}}} \cdot \left(-0.125 \cdot \left(M \cdot \left(M \cdot \left(\frac{D}{d} \cdot D\right)\right)\right)\right) \]

   if 4.59999999999999987e70 < d

   1. Initial program 74.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 74.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ 76.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right) \]

      2. div-inv 76.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}\right) \]

      3. metadata-eval 76.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}\right) \]

   5. Applied egg-rr 76.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right) \]

   6. Taylor expanded in d around inf 67.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. unpow-1 67.0%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

      2. sqr-pow 67.0%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \]

      3. rem-sqrt-square 67.0%

         \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \]

      4. metadata-eval 67.0%

         \[\leadsto d \cdot \left|{\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right| \]

      5. sqr-pow 66.8%

         \[\leadsto d \cdot \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)}}\right| \]

      6. fabs-sqr 66.8%

         \[\leadsto d \cdot \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)}\right)} \]

      7. sqr-pow 67.0%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   8. Simplified 67.0%

      \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
   9. Step-by-step derivation

      1. unpow-prod-down 78.4%

         \[\leadsto d \cdot \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \]

   10. Applied egg-rr 78.4%

       \[\leadsto d \cdot \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.65 \cdot 10^{-284}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq 4.6 \cdot 10^{+70}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot \left(M \cdot \left(M \cdot \left(D \cdot \frac{D}{d}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 14: 38.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 4.2 \cdot 10^{-68}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{\left(M \cdot \left(M \cdot \left(D \cdot D\right)\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= M 4.2e-68)
   (* (sqrt (/ d l)) (sqrt (/ d h)))
   (* -0.125 (/ (* (* M (* M (* D D))) (sqrt (/ h (pow l 3.0)))) d))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (M <= 4.2e-68) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = -0.125 * (((M * (M * (D * D))) * sqrt((h / pow(l, 3.0)))) / d);
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m <= 4.2d-68) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else
        tmp = (-0.125d0) * (((m * (m * (d_1 * d_1))) * sqrt((h / (l ** 3.0d0)))) / d)
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (M <= 4.2e-68) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else {
		tmp = -0.125 * (((M * (M * (D * D))) * Math.sqrt((h / Math.pow(l, 3.0)))) / d);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if M <= 4.2e-68:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	else:
		tmp = -0.125 * (((M * (M * (D * D))) * math.sqrt((h / math.pow(l, 3.0)))) / d)
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (M <= 4.2e-68)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(M * Float64(M * Float64(D * D))) * sqrt(Float64(h / (l ^ 3.0)))) / d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (M <= 4.2e-68)
		tmp = sqrt((d / l)) * sqrt((d / h));
	else
		tmp = -0.125 * (((M * (M * (D * D))) * sqrt((h / (l ^ 3.0)))) / d);
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[M, 4.2e-68], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-0.125 * N[(N[(N[(M * N[(M * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.20000000000000016e-68

   1. Initial program 61.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 60.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]

   4. Taylor expanded in M around 0 36.6%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{1}\right) \]

   if 4.20000000000000016e-68 < M

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 66.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around 0 24.3%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ 24.5%

         \[\leadsto -0.125 \cdot \color{blue}{\frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d}} \]

      2. *-commutative 24.5%

         \[\leadsto -0.125 \cdot \frac{\color{blue}{\left({M}^{2} \cdot {D}^{2}\right)} \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d} \]

      3. unpow2 24.5%

         \[\leadsto -0.125 \cdot \frac{\left(\color{blue}{\left(M \cdot M\right)} \cdot {D}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d} \]

      4. associate-*l* 24.6%

         \[\leadsto -0.125 \cdot \frac{\color{blue}{\left(M \cdot \left(M \cdot {D}^{2}\right)\right)} \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d} \]

      5. unpow2 24.6%

         \[\leadsto -0.125 \cdot \frac{\left(M \cdot \left(M \cdot \color{blue}{\left(D \cdot D\right)}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d} \]

   6. Simplified 24.6%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{\left(M \cdot \left(M \cdot \left(D \cdot D\right)\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4.2 \cdot 10^{-68}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{\left(M \cdot \left(M \cdot \left(D \cdot D\right)\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d}\\ \end{array} \]

Alternative 15: 31.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq -1 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= h -1e-310)
   (* d (sqrt (/ (/ 1.0 h) l)))
   (* d (* (pow h -0.5) (pow l -0.5)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -1e-310) {
		tmp = d * sqrt(((1.0 / h) / l));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (h <= (-1d-310)) then
        tmp = d * sqrt(((1.0d0 / h) / l))
    else
        tmp = d * ((h ** (-0.5d0)) * (l ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -1e-310) {
		tmp = d * Math.sqrt(((1.0 / h) / l));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if h <= -1e-310:
		tmp = d * math.sqrt(((1.0 / h) / l))
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= -1e-310)
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / h) / l)));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= -1e-310)
		tmp = d * sqrt(((1.0 / h) / l));
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[h, -1e-310], N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -9.999999999999969e-311

   1. Initial program 61.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 62.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 7.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. associate-/r* 7.9%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   6. Simplified 7.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]

   if -9.999999999999969e-311 < h

   1. Initial program 63.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 63.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ 67.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right) \]

      2. div-inv 67.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}\right) \]

      3. metadata-eval 67.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}\right) \]

   5. Applied egg-rr 67.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right) \]

   6. Taylor expanded in d around inf 39.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. unpow-1 39.6%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

      2. sqr-pow 39.6%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \]

      3. rem-sqrt-square 40.2%

         \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \]

      4. metadata-eval 40.2%

         \[\leadsto d \cdot \left|{\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right| \]

      5. sqr-pow 40.1%

         \[\leadsto d \cdot \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)}}\right| \]

      6. fabs-sqr 40.1%

         \[\leadsto d \cdot \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)}\right)} \]

      7. sqr-pow 40.2%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   8. Simplified 40.2%

      \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
   9. Step-by-step derivation

      1. unpow-prod-down 47.3%

         \[\leadsto d \cdot \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \]

   10. Applied egg-rr 47.3%

       \[\leadsto d \cdot \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 16: 45.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= h -1e-310)
   (* (sqrt (/ d l)) (sqrt (/ d h)))
   (* d (* (pow h -0.5) (pow l -0.5)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -1e-310) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (h <= (-1d-310)) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else
        tmp = d * ((h ** (-0.5d0)) * (l ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -1e-310) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if h <= -1e-310:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= -1e-310)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= -1e-310)
		tmp = sqrt((d / l)) * sqrt((d / h));
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[h, -1e-310], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -9.999999999999969e-311

   1. Initial program 61.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 60.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \frac{M}{\frac{d}{D}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]

   4. Taylor expanded in M around 0 37.3%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{1}\right) \]

   if -9.999999999999969e-311 < h

   1. Initial program 63.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 63.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ 67.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right) \]

      2. div-inv 67.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}\right) \]

      3. metadata-eval 67.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}\right) \]

   5. Applied egg-rr 67.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right) \]

   6. Taylor expanded in d around inf 39.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. unpow-1 39.6%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

      2. sqr-pow 39.6%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \]

      3. rem-sqrt-square 40.2%

         \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \]

      4. metadata-eval 40.2%

         \[\leadsto d \cdot \left|{\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right| \]

      5. sqr-pow 40.1%

         \[\leadsto d \cdot \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)}}\right| \]

      6. fabs-sqr 40.1%

         \[\leadsto d \cdot \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)}\right)} \]

      7. sqr-pow 40.2%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   8. Simplified 40.2%

      \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
   9. Step-by-step derivation

      1. unpow-prod-down 47.3%

         \[\leadsto d \cdot \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \]

   10. Applied egg-rr 47.3%

       \[\leadsto d \cdot \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 17: 27.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}} \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (* d (sqrt (/ (/ 1.0 h) l))))

C

double code(double d, double h, double l, double M, double D) {
	return d * sqrt(((1.0 / 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 * sqrt(((1.0d0 / h) / l))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return d * Math.sqrt(((1.0 / h) / l));
}

Python

def code(d, h, l, M, D):
	return d * math.sqrt(((1.0 / h) / l))

Julia

function code(d, h, l, M, D)
	return Float64(d * sqrt(Float64(Float64(1.0 / h) / l)))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = d * sqrt(((1.0 / h) / l));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.9%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 63.1%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

4. Taylor expanded in d around inf 26.2%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
5. Step-by-step derivation

   1. associate-/r* 26.7%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

6. Simplified 26.7%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]

7. Final simplification 26.7%

   \[\leadsto d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}} \]

Alternative 18: 26.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ d \cdot {\left(\ell \cdot h\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (* d (pow (* l h) -0.5)))

C

double code(double d, double h, double l, double M, double D) {
	return d * pow((l * h), -0.5);
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = d * ((l * h) ** (-0.5d0))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return d * Math.pow((l * h), -0.5);
}

Python

def code(d, h, l, M, D):
	return d * math.pow((l * h), -0.5)

Julia

function code(d, h, l, M, D)
	return Float64(d * (Float64(l * h) ^ -0.5))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = d * ((l * h) ^ -0.5);
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.9%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 63.1%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
4. Step-by-step derivation

   1. associate-*r/ 65.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right) \]

   2. div-inv 65.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}\right) \]

   3. metadata-eval 65.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}\right) \]

5. Applied egg-rr 65.2%

   \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right) \]

6. Taylor expanded in d around inf 26.2%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
7. Step-by-step derivation

   1. unpow-1 26.2%

      \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

   2. sqr-pow 26.2%

      \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \]

   3. rem-sqrt-square 26.6%

      \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \]

   4. metadata-eval 26.6%

      \[\leadsto d \cdot \left|{\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right| \]

   5. sqr-pow 26.5%

      \[\leadsto d \cdot \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)}}\right| \]

   6. fabs-sqr 26.5%

      \[\leadsto d \cdot \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)}\right)} \]

   7. sqr-pow 26.6%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

8. Simplified 26.6%

   \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

9. Final simplification 26.6%

   \[\leadsto d \cdot {\left(\ell \cdot h\right)}^{-0.5} \]

Reproduce

herbie shell --seed 2023272 
(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)))))