Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.5% → 80.6%

Time: 22.9s

Alternatives: 19

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -3.5 \cdot 10^{+36}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{t_0}{\sqrt{-\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{d \cdot \frac{2}{D}}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(\frac{t_0}{\sqrt{-h}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (- d))))
   (if (<= l -3.5e+36)
     (*
      (sqrt (/ d h))
      (*
       (/ t_0 (sqrt (- l)))
       (+ 1.0 (* (/ h l) (* (pow (/ M (* d (/ 2.0 D))) 2.0) -0.5)))))
     (if (<= l -1e-309)
       (*
        (* (/ t_0 (sqrt (- h))) (/ 1.0 (sqrt (/ l d))))
        (- 1.0 (* 0.5 (/ (* h (pow (* (* M 0.5) (/ D d)) 2.0)) l))))
       (*
        (/ (sqrt d) (sqrt h))
        (*
         (/ (sqrt d) (sqrt l))
         (+ 1.0 (* (/ h l) (* -0.5 (pow (* (/ D d) (/ M 2.0)) 2.0))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(-d);
	double tmp;
	if (l <= -3.5e+36) {
		tmp = sqrt((d / h)) * ((t_0 / sqrt(-l)) * (1.0 + ((h / l) * (pow((M / (d * (2.0 / D))), 2.0) * -0.5))));
	} else if (l <= -1e-309) {
		tmp = ((t_0 / sqrt(-h)) * (1.0 / sqrt((l / d)))) * (1.0 - (0.5 * ((h * pow(((M * 0.5) * (D / d)), 2.0)) / l)));
	} else {
		tmp = (sqrt(d) / sqrt(h)) * ((sqrt(d) / sqrt(l)) * (1.0 + ((h / l) * (-0.5 * pow(((D / d) * (M / 2.0)), 2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(-d)
    if (l <= (-3.5d+36)) then
        tmp = sqrt((d / h)) * ((t_0 / sqrt(-l)) * (1.0d0 + ((h / l) * (((m / (d * (2.0d0 / d_1))) ** 2.0d0) * (-0.5d0)))))
    else if (l <= (-1d-309)) then
        tmp = ((t_0 / sqrt(-h)) * (1.0d0 / sqrt((l / d)))) * (1.0d0 - (0.5d0 * ((h * (((m * 0.5d0) * (d_1 / d)) ** 2.0d0)) / l)))
    else
        tmp = (sqrt(d) / sqrt(h)) * ((sqrt(d) / sqrt(l)) * (1.0d0 + ((h / l) * ((-0.5d0) * (((d_1 / d) * (m / 2.0d0)) ** 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt(-d);
	double tmp;
	if (l <= -3.5e+36) {
		tmp = Math.sqrt((d / h)) * ((t_0 / Math.sqrt(-l)) * (1.0 + ((h / l) * (Math.pow((M / (d * (2.0 / D))), 2.0) * -0.5))));
	} else if (l <= -1e-309) {
		tmp = ((t_0 / Math.sqrt(-h)) * (1.0 / Math.sqrt((l / d)))) * (1.0 - (0.5 * ((h * Math.pow(((M * 0.5) * (D / d)), 2.0)) / l)));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * ((Math.sqrt(d) / Math.sqrt(l)) * (1.0 + ((h / l) * (-0.5 * Math.pow(((D / d) * (M / 2.0)), 2.0)))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt(-d)
	tmp = 0
	if l <= -3.5e+36:
		tmp = math.sqrt((d / h)) * ((t_0 / math.sqrt(-l)) * (1.0 + ((h / l) * (math.pow((M / (d * (2.0 / D))), 2.0) * -0.5))))
	elif l <= -1e-309:
		tmp = ((t_0 / math.sqrt(-h)) * (1.0 / math.sqrt((l / d)))) * (1.0 - (0.5 * ((h * math.pow(((M * 0.5) * (D / d)), 2.0)) / l)))
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * ((math.sqrt(d) / math.sqrt(l)) * (1.0 + ((h / l) * (-0.5 * math.pow(((D / d) * (M / 2.0)), 2.0)))))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(-d))
	tmp = 0.0
	if (l <= -3.5e+36)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(t_0 / sqrt(Float64(-l))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(M / Float64(d * Float64(2.0 / D))) ^ 2.0) * -0.5)))));
	elseif (l <= -1e-309)
		tmp = Float64(Float64(Float64(t_0 / sqrt(Float64(-h))) * Float64(1.0 / sqrt(Float64(l / d)))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(Float64(M * 0.5) * Float64(D / d)) ^ 2.0)) / l))));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(Float64(sqrt(d) / sqrt(l)) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(Float64(D / d) * Float64(M / 2.0)) ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(-d);
	tmp = 0.0;
	if (l <= -3.5e+36)
		tmp = sqrt((d / h)) * ((t_0 / sqrt(-l)) * (1.0 + ((h / l) * (((M / (d * (2.0 / D))) ^ 2.0) * -0.5))));
	elseif (l <= -1e-309)
		tmp = ((t_0 / sqrt(-h)) * (1.0 / sqrt((l / d)))) * (1.0 - (0.5 * ((h * (((M * 0.5) * (D / d)) ^ 2.0)) / l)));
	else
		tmp = (sqrt(d) / sqrt(h)) * ((sqrt(d) / sqrt(l)) * (1.0 + ((h / l) * (-0.5 * (((D / d) * (M / 2.0)) ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[l, -3.5e+36], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(M / N[(d * N[(2.0 / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1e-309], N[(N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(N[(M * 0.5), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.4999999999999998e36

   1. Initial program 62.5%

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

   3. Simplified 66.0%

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

      1. frac-times 62.5%

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

      2. *-commutative 62.5%

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

      3. frac-times 66.0%

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

      4. clear-num 66.0%

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

      5. frac-times 66.0%

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

      6. *-un-lft-identity 66.0%

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

   5. Applied egg-rr 66.0%

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

      1. frac-2neg 66.0%

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

      2. sqrt-div 73.7%

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

   7. Applied egg-rr 73.7%

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

   if -3.4999999999999998e36 < l < -1.000000000000002e-309

   1. Initial program 80.0%

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

   3. Simplified 78.8%

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

      1. associate-*r/ 82.9%

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

      2. frac-times 84.1%

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

      3. *-commutative 84.1%

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

      4. frac-times 83.3%

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

      5. div-inv 83.3%

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

      6. metadata-eval 83.3%

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

   5. Applied egg-rr 83.3%

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

      1. clear-num 83.3%

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

      2. sqrt-div 84.7%

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

      3. metadata-eval 84.7%

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

   7. Applied egg-rr 84.7%

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

      1. frac-2neg 84.7%

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

      2. sqrt-div 92.7%

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

   9. Applied egg-rr 92.7%

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

   if -1.000000000000002e-309 < l

   1. Initial program 70.4%

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

   3. Simplified 71.1%

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

      1. sqrt-div 75.7%

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

   5. Applied egg-rr 75.7%

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

      1. sqrt-div 85.4%

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

   7. Applied egg-rr 85.4%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.5 \cdot 10^{+36}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{d \cdot \frac{2}{D}}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)\\ \end{array} \]

Alternative 2: 79.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}\\ \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+36}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{t_0}{\sqrt{-\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{d \cdot \frac{2}{D}}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(\frac{t_0}{\sqrt{-h}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot t_1}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot t_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (- d))) (t_1 (pow (* (* M 0.5) (/ D d)) 2.0)))
   (if (<= l -4.5e+36)
     (*
      (sqrt (/ d h))
      (*
       (/ t_0 (sqrt (- l)))
       (+ 1.0 (* (/ h l) (* (pow (/ M (* d (/ 2.0 D))) 2.0) -0.5)))))
     (if (<= l -1e-309)
       (*
        (* (/ t_0 (sqrt (- h))) (/ 1.0 (sqrt (/ l d))))
        (- 1.0 (* 0.5 (/ (* h t_1) l))))
       (* (/ d (* (sqrt h) (sqrt l))) (- 1.0 (* (/ h l) (* 0.5 t_1))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(-d);
	double t_1 = pow(((M * 0.5) * (D / d)), 2.0);
	double tmp;
	if (l <= -4.5e+36) {
		tmp = sqrt((d / h)) * ((t_0 / sqrt(-l)) * (1.0 + ((h / l) * (pow((M / (d * (2.0 / D))), 2.0) * -0.5))));
	} else if (l <= -1e-309) {
		tmp = ((t_0 / sqrt(-h)) * (1.0 / sqrt((l / d)))) * (1.0 - (0.5 * ((h * t_1) / l)));
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - ((h / l) * (0.5 * t_1)));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(-d)
    t_1 = ((m * 0.5d0) * (d_1 / d)) ** 2.0d0
    if (l <= (-4.5d+36)) then
        tmp = sqrt((d / h)) * ((t_0 / sqrt(-l)) * (1.0d0 + ((h / l) * (((m / (d * (2.0d0 / d_1))) ** 2.0d0) * (-0.5d0)))))
    else if (l <= (-1d-309)) then
        tmp = ((t_0 / sqrt(-h)) * (1.0d0 / sqrt((l / d)))) * (1.0d0 - (0.5d0 * ((h * t_1) / l)))
    else
        tmp = (d / (sqrt(h) * sqrt(l))) * (1.0d0 - ((h / l) * (0.5d0 * t_1)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt(-d);
	double t_1 = Math.pow(((M * 0.5) * (D / d)), 2.0);
	double tmp;
	if (l <= -4.5e+36) {
		tmp = Math.sqrt((d / h)) * ((t_0 / Math.sqrt(-l)) * (1.0 + ((h / l) * (Math.pow((M / (d * (2.0 / D))), 2.0) * -0.5))));
	} else if (l <= -1e-309) {
		tmp = ((t_0 / Math.sqrt(-h)) * (1.0 / Math.sqrt((l / d)))) * (1.0 - (0.5 * ((h * t_1) / l)));
	} else {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * (1.0 - ((h / l) * (0.5 * t_1)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt(-d)
	t_1 = math.pow(((M * 0.5) * (D / d)), 2.0)
	tmp = 0
	if l <= -4.5e+36:
		tmp = math.sqrt((d / h)) * ((t_0 / math.sqrt(-l)) * (1.0 + ((h / l) * (math.pow((M / (d * (2.0 / D))), 2.0) * -0.5))))
	elif l <= -1e-309:
		tmp = ((t_0 / math.sqrt(-h)) * (1.0 / math.sqrt((l / d)))) * (1.0 - (0.5 * ((h * t_1) / l)))
	else:
		tmp = (d / (math.sqrt(h) * math.sqrt(l))) * (1.0 - ((h / l) * (0.5 * t_1)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(-d))
	t_1 = Float64(Float64(M * 0.5) * Float64(D / d)) ^ 2.0
	tmp = 0.0
	if (l <= -4.5e+36)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(t_0 / sqrt(Float64(-l))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(M / Float64(d * Float64(2.0 / D))) ^ 2.0) * -0.5)))));
	elseif (l <= -1e-309)
		tmp = Float64(Float64(Float64(t_0 / sqrt(Float64(-h))) * Float64(1.0 / sqrt(Float64(l / d)))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * t_1) / l))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(-d);
	t_1 = ((M * 0.5) * (D / d)) ^ 2.0;
	tmp = 0.0;
	if (l <= -4.5e+36)
		tmp = sqrt((d / h)) * ((t_0 / sqrt(-l)) * (1.0 + ((h / l) * (((M / (d * (2.0 / D))) ^ 2.0) * -0.5))));
	elseif (l <= -1e-309)
		tmp = ((t_0 / sqrt(-h)) * (1.0 / sqrt((l / d)))) * (1.0 - (0.5 * ((h * t_1) / l)));
	else
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - ((h / l) * (0.5 * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(N[(M * 0.5), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[l, -4.5e+36], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(M / N[(d * N[(2.0 / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1e-309], N[(N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * t$95$1), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.49999999999999997e36

   1. Initial program 62.5%

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

   3. Simplified 66.0%

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

      1. frac-times 62.5%

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

      2. *-commutative 62.5%

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

      3. frac-times 66.0%

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

      4. clear-num 66.0%

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

      5. frac-times 66.0%

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

      6. *-un-lft-identity 66.0%

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

   5. Applied egg-rr 66.0%

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

      1. frac-2neg 66.0%

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

      2. sqrt-div 73.7%

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

   7. Applied egg-rr 73.7%

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

   if -4.49999999999999997e36 < l < -1.000000000000002e-309

   1. Initial program 80.0%

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

   3. Simplified 78.8%

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

      1. associate-*r/ 82.9%

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

      2. frac-times 84.1%

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

      3. *-commutative 84.1%

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

      4. frac-times 83.3%

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

      5. div-inv 83.3%

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

      6. metadata-eval 83.3%

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

   5. Applied egg-rr 83.3%

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

      1. clear-num 83.3%

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

      2. sqrt-div 84.7%

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

      3. metadata-eval 84.7%

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

   7. Applied egg-rr 84.7%

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

      1. frac-2neg 84.7%

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

      2. sqrt-div 92.7%

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

   9. Applied egg-rr 92.7%

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

   if -1.000000000000002e-309 < l

   1. Initial program 70.4%

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

   3. Simplified 71.1%

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

   5. Applied egg-rr 82.3%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+36}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{d \cdot \frac{2}{D}}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 3: 79.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}\\ \mathbf{if}\;\ell \leq -1.8 \cdot 10^{+115}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{t_0}{\sqrt{-\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{d \cdot \frac{2}{D}}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot t_1}{\ell}\right) \cdot \left(\frac{t_0}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot t_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (- d))) (t_1 (pow (* (* M 0.5) (/ D d)) 2.0)))
   (if (<= l -1.8e+115)
     (*
      (sqrt (/ d h))
      (*
       (/ t_0 (sqrt (- l)))
       (+ 1.0 (* (/ h l) (* (pow (/ M (* d (/ 2.0 D))) 2.0) -0.5)))))
     (if (<= l -1e-309)
       (*
        (- 1.0 (* 0.5 (/ (* h t_1) l)))
        (* (/ t_0 (sqrt (- h))) (sqrt (/ d l))))
       (* (/ d (* (sqrt h) (sqrt l))) (- 1.0 (* (/ h l) (* 0.5 t_1))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(-d);
	double t_1 = pow(((M * 0.5) * (D / d)), 2.0);
	double tmp;
	if (l <= -1.8e+115) {
		tmp = sqrt((d / h)) * ((t_0 / sqrt(-l)) * (1.0 + ((h / l) * (pow((M / (d * (2.0 / D))), 2.0) * -0.5))));
	} else if (l <= -1e-309) {
		tmp = (1.0 - (0.5 * ((h * t_1) / l))) * ((t_0 / sqrt(-h)) * sqrt((d / l)));
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - ((h / l) * (0.5 * t_1)));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(-d)
    t_1 = ((m * 0.5d0) * (d_1 / d)) ** 2.0d0
    if (l <= (-1.8d+115)) then
        tmp = sqrt((d / h)) * ((t_0 / sqrt(-l)) * (1.0d0 + ((h / l) * (((m / (d * (2.0d0 / d_1))) ** 2.0d0) * (-0.5d0)))))
    else if (l <= (-1d-309)) then
        tmp = (1.0d0 - (0.5d0 * ((h * t_1) / l))) * ((t_0 / sqrt(-h)) * sqrt((d / l)))
    else
        tmp = (d / (sqrt(h) * sqrt(l))) * (1.0d0 - ((h / l) * (0.5d0 * t_1)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt(-d);
	double t_1 = Math.pow(((M * 0.5) * (D / d)), 2.0);
	double tmp;
	if (l <= -1.8e+115) {
		tmp = Math.sqrt((d / h)) * ((t_0 / Math.sqrt(-l)) * (1.0 + ((h / l) * (Math.pow((M / (d * (2.0 / D))), 2.0) * -0.5))));
	} else if (l <= -1e-309) {
		tmp = (1.0 - (0.5 * ((h * t_1) / l))) * ((t_0 / Math.sqrt(-h)) * Math.sqrt((d / l)));
	} else {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * (1.0 - ((h / l) * (0.5 * t_1)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt(-d)
	t_1 = math.pow(((M * 0.5) * (D / d)), 2.0)
	tmp = 0
	if l <= -1.8e+115:
		tmp = math.sqrt((d / h)) * ((t_0 / math.sqrt(-l)) * (1.0 + ((h / l) * (math.pow((M / (d * (2.0 / D))), 2.0) * -0.5))))
	elif l <= -1e-309:
		tmp = (1.0 - (0.5 * ((h * t_1) / l))) * ((t_0 / math.sqrt(-h)) * math.sqrt((d / l)))
	else:
		tmp = (d / (math.sqrt(h) * math.sqrt(l))) * (1.0 - ((h / l) * (0.5 * t_1)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(-d))
	t_1 = Float64(Float64(M * 0.5) * Float64(D / d)) ^ 2.0
	tmp = 0.0
	if (l <= -1.8e+115)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(t_0 / sqrt(Float64(-l))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(M / Float64(d * Float64(2.0 / D))) ^ 2.0) * -0.5)))));
	elseif (l <= -1e-309)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h * t_1) / l))) * Float64(Float64(t_0 / sqrt(Float64(-h))) * sqrt(Float64(d / l))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(-d);
	t_1 = ((M * 0.5) * (D / d)) ^ 2.0;
	tmp = 0.0;
	if (l <= -1.8e+115)
		tmp = sqrt((d / h)) * ((t_0 / sqrt(-l)) * (1.0 + ((h / l) * (((M / (d * (2.0 / D))) ^ 2.0) * -0.5))));
	elseif (l <= -1e-309)
		tmp = (1.0 - (0.5 * ((h * t_1) / l))) * ((t_0 / sqrt(-h)) * sqrt((d / l)));
	else
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - ((h / l) * (0.5 * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(N[(M * 0.5), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[l, -1.8e+115], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(M / N[(d * N[(2.0 / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1e-309], N[(N[(1.0 - N[(0.5 * N[(N[(h * t$95$1), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.8e115

   1. Initial program 58.2%

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

   3. Simplified 63.0%

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

      1. frac-times 58.2%

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

      2. *-commutative 58.2%

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

      3. frac-times 63.1%

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

      4. clear-num 63.1%

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

      5. frac-times 63.0%

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

      6. *-un-lft-identity 63.0%

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

   5. Applied egg-rr 63.0%

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

      1. frac-2neg 63.0%

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

      2. sqrt-div 73.3%

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

   7. Applied egg-rr 73.3%

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

   if -1.8e115 < l < -1.000000000000002e-309

   1. Initial program 78.9%

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

   3. Simplified 77.9%

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

      1. associate-*r/ 81.2%

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

      2. frac-times 82.2%

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

      3. *-commutative 82.2%

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

      4. frac-times 81.5%

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

      5. div-inv 81.5%

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

      6. metadata-eval 81.5%

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

   5. Applied egg-rr 81.5%

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

      1. frac-2neg 81.7%

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

      2. sqrt-div 89.4%

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

   7. Applied egg-rr 89.2%

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

   if -1.000000000000002e-309 < l

   1. Initial program 70.4%

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

   3. Simplified 71.1%

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

   5. Applied egg-rr 82.3%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.8 \cdot 10^{+115}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{d \cdot \frac{2}{D}}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}\right) \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 4: 75.9% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -5e-310)
   (*
    (sqrt (/ d h))
    (*
     (/ (sqrt (- d)) (sqrt (- l)))
     (+ 1.0 (* (/ h l) (* -0.5 (pow (* (/ D d) (/ M 2.0)) 2.0))))))
   (*
    (/ d (* (sqrt h) (sqrt l)))
    (- 1.0 (* (/ h l) (* 0.5 (pow (* (* M 0.5) (/ D d)) 2.0)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -5e-310) {
		tmp = sqrt((d / h)) * ((sqrt(-d) / sqrt(-l)) * (1.0 + ((h / l) * (-0.5 * pow(((D / d) * (M / 2.0)), 2.0)))));
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - ((h / l) * (0.5 * pow(((M * 0.5) * (D / d)), 2.0))));
	}
	return tmp;
}

Fortran

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

Java

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

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= -5e-310:
		tmp = math.sqrt((d / h)) * ((math.sqrt(-d) / math.sqrt(-l)) * (1.0 + ((h / l) * (-0.5 * math.pow(((D / d) * (M / 2.0)), 2.0)))))
	else:
		tmp = (d / (math.sqrt(h) * math.sqrt(l))) * (1.0 - ((h / l) * (0.5 * math.pow(((M * 0.5) * (D / d)), 2.0))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -5e-310)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(Float64(D / d) * Float64(M / 2.0)) ^ 2.0))))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(M * 0.5) * Float64(D / d)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -5e-310)
		tmp = sqrt((d / h)) * ((sqrt(-d) / sqrt(-l)) * (1.0 + ((h / l) * (-0.5 * (((D / d) * (M / 2.0)) ^ 2.0)))));
	else
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - ((h / l) * (0.5 * (((M * 0.5) * (D / d)) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -5e-310], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(N[(M * 0.5), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -4.999999999999985e-310

   1. Initial program 72.8%

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

   3. Simplified 74.3%

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

      1. frac-2neg 74.2%

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

      2. sqrt-div 79.6%

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

   5. Applied egg-rr 79.6%

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

   if -4.999999999999985e-310 < d

   1. Initial program 70.4%

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

   3. Simplified 71.1%

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

   5. Applied egg-rr 82.3%

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

4. Final simplification 81.0%

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

Alternative 5: 75.9% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -5e-310)
   (*
    (sqrt (/ d h))
    (*
     (/ (sqrt (- d)) (sqrt (- l)))
     (+ 1.0 (* (/ h l) (* (pow (/ M (* d (/ 2.0 D))) 2.0) -0.5)))))
   (*
    (/ d (* (sqrt h) (sqrt l)))
    (- 1.0 (* (/ h l) (* 0.5 (pow (* (* M 0.5) (/ D d)) 2.0)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -5e-310) {
		tmp = sqrt((d / h)) * ((sqrt(-d) / sqrt(-l)) * (1.0 + ((h / l) * (pow((M / (d * (2.0 / D))), 2.0) * -0.5))));
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - ((h / l) * (0.5 * pow(((M * 0.5) * (D / d)), 2.0))));
	}
	return tmp;
}

Fortran

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

Java

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

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= -5e-310:
		tmp = math.sqrt((d / h)) * ((math.sqrt(-d) / math.sqrt(-l)) * (1.0 + ((h / l) * (math.pow((M / (d * (2.0 / D))), 2.0) * -0.5))))
	else:
		tmp = (d / (math.sqrt(h) * math.sqrt(l))) * (1.0 - ((h / l) * (0.5 * math.pow(((M * 0.5) * (D / d)), 2.0))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -5e-310)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(M / Float64(d * Float64(2.0 / D))) ^ 2.0) * -0.5)))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(M * 0.5) * Float64(D / d)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -5e-310)
		tmp = sqrt((d / h)) * ((sqrt(-d) / sqrt(-l)) * (1.0 + ((h / l) * (((M / (d * (2.0 / D))) ^ 2.0) * -0.5))));
	else
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - ((h / l) * (0.5 * (((M * 0.5) * (D / d)) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -5e-310], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(M / N[(d * N[(2.0 / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(N[(M * 0.5), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -4.999999999999985e-310

   1. Initial program 72.8%

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

   3. Simplified 74.3%

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

      1. frac-times 72.8%

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

      2. *-commutative 72.8%

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

      3. frac-times 73.6%

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

      4. clear-num 73.6%

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

      5. frac-times 74.2%

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

      6. *-un-lft-identity 74.2%

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

   5. Applied egg-rr 74.2%

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

      1. frac-2neg 74.2%

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

      2. sqrt-div 79.6%

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

   7. Applied egg-rr 79.6%

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

   if -4.999999999999985e-310 < d

   1. Initial program 70.4%

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

   3. Simplified 71.1%

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

   5. Applied egg-rr 82.3%

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

4. Final simplification 80.9%

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

Alternative 6: 77.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot t_0}{\ell}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot t_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (pow (* (* M 0.5) (/ D d)) 2.0)))
   (if (<= l -1e-309)
     (*
      (- 1.0 (* 0.5 (/ (* h t_0) l)))
      (* (sqrt (/ d h)) (/ (sqrt (- d)) (sqrt (- l)))))
     (* (/ d (* (sqrt h) (sqrt l))) (- 1.0 (* (/ h l) (* 0.5 t_0)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = pow(((M * 0.5) * (D / d)), 2.0);
	double tmp;
	if (l <= -1e-309) {
		tmp = (1.0 - (0.5 * ((h * t_0) / l))) * (sqrt((d / h)) * (sqrt(-d) / sqrt(-l)));
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - ((h / l) * (0.5 * t_0)));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((m * 0.5d0) * (d_1 / d)) ** 2.0d0
    if (l <= (-1d-309)) then
        tmp = (1.0d0 - (0.5d0 * ((h * t_0) / l))) * (sqrt((d / h)) * (sqrt(-d) / sqrt(-l)))
    else
        tmp = (d / (sqrt(h) * sqrt(l))) * (1.0d0 - ((h / l) * (0.5d0 * t_0)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.pow(((M * 0.5) * (D / d)), 2.0);
	double tmp;
	if (l <= -1e-309) {
		tmp = (1.0 - (0.5 * ((h * t_0) / l))) * (Math.sqrt((d / h)) * (Math.sqrt(-d) / Math.sqrt(-l)));
	} else {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * (1.0 - ((h / l) * (0.5 * t_0)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.pow(((M * 0.5) * (D / d)), 2.0)
	tmp = 0
	if l <= -1e-309:
		tmp = (1.0 - (0.5 * ((h * t_0) / l))) * (math.sqrt((d / h)) * (math.sqrt(-d) / math.sqrt(-l)))
	else:
		tmp = (d / (math.sqrt(h) * math.sqrt(l))) * (1.0 - ((h / l) * (0.5 * t_0)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(M * 0.5) * Float64(D / d)) ^ 2.0
	tmp = 0.0
	if (l <= -1e-309)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h * t_0) / l))) * Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-l)))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = ((M * 0.5) * (D / d)) ^ 2.0;
	tmp = 0.0;
	if (l <= -1e-309)
		tmp = (1.0 - (0.5 * ((h * t_0) / l))) * (sqrt((d / h)) * (sqrt(-d) / sqrt(-l)));
	else
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - ((h / l) * (0.5 * t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Power[N[(N[(M * 0.5), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[l, -1e-309], N[(N[(1.0 - N[(0.5 * N[(N[(h * t$95$0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -1.000000000000002e-309

   1. Initial program 72.8%

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

   3. Simplified 73.5%

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

      1. associate-*r/ 73.7%

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

      2. frac-times 74.4%

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

      3. *-commutative 74.4%

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

      4. frac-times 73.9%

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

      5. div-inv 73.9%

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

      6. metadata-eval 73.9%

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

   5. Applied egg-rr 73.9%

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

      1. frac-2neg 74.2%

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

      2. sqrt-div 79.6%

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

   7. Applied egg-rr 80.7%

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

   if -1.000000000000002e-309 < l

   1. Initial program 70.4%

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

   3. Simplified 71.1%

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

   5. Applied egg-rr 82.3%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 7: 72.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= h -1e-309)
   (*
    (sqrt (/ d h))
    (*
     (+ 1.0 (* (/ h l) (* -0.5 (pow (* (/ D d) (/ M 2.0)) 2.0))))
     (sqrt (/ d l))))
   (*
    (/ (/ d (sqrt l)) (sqrt h))
    (- 1.0 (* (/ h l) (* 0.5 (pow (* D (* 0.5 (/ M d))) 2.0)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -1e-309) {
		tmp = sqrt((d / h)) * ((1.0 + ((h / l) * (-0.5 * pow(((D / d) * (M / 2.0)), 2.0)))) * sqrt((d / l)));
	} else {
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - ((h / l) * (0.5 * pow((D * (0.5 * (M / d))), 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (h <= (-1d-309)) then
        tmp = sqrt((d / h)) * ((1.0d0 + ((h / l) * ((-0.5d0) * (((d_1 / d) * (m / 2.0d0)) ** 2.0d0)))) * sqrt((d / l)))
    else
        tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0d0 - ((h / l) * (0.5d0 * ((d_1 * (0.5d0 * (m / d))) ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -1e-309) {
		tmp = Math.sqrt((d / h)) * ((1.0 + ((h / l) * (-0.5 * Math.pow(((D / d) * (M / 2.0)), 2.0)))) * Math.sqrt((d / l)));
	} else {
		tmp = ((d / Math.sqrt(l)) / Math.sqrt(h)) * (1.0 - ((h / l) * (0.5 * Math.pow((D * (0.5 * (M / d))), 2.0))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if h <= -1e-309:
		tmp = math.sqrt((d / h)) * ((1.0 + ((h / l) * (-0.5 * math.pow(((D / d) * (M / 2.0)), 2.0)))) * math.sqrt((d / l)))
	else:
		tmp = ((d / math.sqrt(l)) / math.sqrt(h)) * (1.0 - ((h / l) * (0.5 * math.pow((D * (0.5 * (M / d))), 2.0))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= -1e-309)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(Float64(D / d) * Float64(M / 2.0)) ^ 2.0)))) * sqrt(Float64(d / l))));
	else
		tmp = Float64(Float64(Float64(d / sqrt(l)) / sqrt(h)) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(D * Float64(0.5 * Float64(M / d))) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= -1e-309)
		tmp = sqrt((d / h)) * ((1.0 + ((h / l) * (-0.5 * (((D / d) * (M / 2.0)) ^ 2.0)))) * sqrt((d / l)));
	else
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - ((h / l) * (0.5 * ((D * (0.5 * (M / d))) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[h, -1e-309], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(D * N[(0.5 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -1.000000000000002e-309

   1. Initial program 72.8%

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

   3. Simplified 74.3%

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

   if -1.000000000000002e-309 < h

   1. Initial program 70.4%

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

   3. Simplified 71.1%

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

   5. Applied egg-rr 33.8%

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

      1. expm1-def 45.8%

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

      2. expm1-log1p 82.3%

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

      3. associate-/r* 82.2%

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

      4. *-commutative 82.2%

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

      5. associate-*r/ 81.5%

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

      6. associate-*l* 81.5%

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

      7. *-commutative 81.5%

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

      8. associate-*l/ 82.2%

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

      9. *-commutative 82.2%

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

      10. associate-*l* 82.2%

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

   7. Simplified 82.2%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 8: 72.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D}{\frac{d \cdot 2}{M}}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= h -1e-309)
   (*
    (sqrt (/ d h))
    (*
     (sqrt (/ d l))
     (+ 1.0 (* (/ h l) (* -0.5 (pow (/ D (/ (* d 2.0) M)) 2.0))))))
   (*
    (/ (/ d (sqrt l)) (sqrt h))
    (- 1.0 (* (/ h l) (* 0.5 (pow (* D (* 0.5 (/ M d))) 2.0)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -1e-309) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 + ((h / l) * (-0.5 * pow((D / ((d * 2.0) / M)), 2.0)))));
	} else {
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - ((h / l) * (0.5 * pow((D * (0.5 * (M / d))), 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (h <= (-1d-309)) then
        tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 + ((h / l) * ((-0.5d0) * ((d_1 / ((d * 2.0d0) / m)) ** 2.0d0)))))
    else
        tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0d0 - ((h / l) * (0.5d0 * ((d_1 * (0.5d0 * (m / d))) ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -1e-309) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 + ((h / l) * (-0.5 * Math.pow((D / ((d * 2.0) / M)), 2.0)))));
	} else {
		tmp = ((d / Math.sqrt(l)) / Math.sqrt(h)) * (1.0 - ((h / l) * (0.5 * Math.pow((D * (0.5 * (M / d))), 2.0))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if h <= -1e-309:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 + ((h / l) * (-0.5 * math.pow((D / ((d * 2.0) / M)), 2.0)))))
	else:
		tmp = ((d / math.sqrt(l)) / math.sqrt(h)) * (1.0 - ((h / l) * (0.5 * math.pow((D * (0.5 * (M / d))), 2.0))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= -1e-309)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(D / Float64(Float64(d * 2.0) / M)) ^ 2.0))))));
	else
		tmp = Float64(Float64(Float64(d / sqrt(l)) / sqrt(h)) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(D * Float64(0.5 * Float64(M / d))) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= -1e-309)
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 + ((h / l) * (-0.5 * ((D / ((d * 2.0) / M)) ^ 2.0)))));
	else
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - ((h / l) * (0.5 * ((D * (0.5 * (M / d))) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[h, -1e-309], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(D / N[(N[(d * 2.0), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(D * N[(0.5 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -1.000000000000002e-309

   1. Initial program 72.8%

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

   3. Simplified 74.3%

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

      1. clear-num 74.3%

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

      2. frac-times 73.6%

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

      3. *-un-lft-identity 73.6%

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

      4. associate-*l/ 73.5%

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

      5. *-commutative 73.5%

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

   5. Applied egg-rr 73.5%

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

   if -1.000000000000002e-309 < h

   1. Initial program 70.4%

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

   3. Simplified 71.1%

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

   5. Applied egg-rr 33.8%

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

      1. expm1-def 45.8%

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

      2. expm1-log1p 82.3%

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

      3. associate-/r* 82.2%

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

      4. *-commutative 82.2%

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

      5. associate-*r/ 81.5%

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

      6. associate-*l* 81.5%

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

      7. *-commutative 81.5%

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

      8. associate-*l/ 82.2%

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

      9. *-commutative 82.2%

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

      10. associate-*l* 82.2%

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

   7. Simplified 82.2%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D}{\frac{d \cdot 2}{M}}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 9: 72.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq 9 \cdot 10^{-253}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= h 9e-253)
   (*
    (- 1.0 (* 0.5 (/ (* h (pow (* (* M 0.5) (/ D d)) 2.0)) l)))
    (* (sqrt (/ d h)) (sqrt (/ d l))))
   (*
    (/ (/ d (sqrt l)) (sqrt h))
    (- 1.0 (* (/ h l) (* 0.5 (pow (* D (* 0.5 (/ M d))) 2.0)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= 9e-253) {
		tmp = (1.0 - (0.5 * ((h * pow(((M * 0.5) * (D / d)), 2.0)) / l))) * (sqrt((d / h)) * sqrt((d / l)));
	} else {
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - ((h / l) * (0.5 * pow((D * (0.5 * (M / d))), 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (h <= 9d-253) then
        tmp = (1.0d0 - (0.5d0 * ((h * (((m * 0.5d0) * (d_1 / d)) ** 2.0d0)) / l))) * (sqrt((d / h)) * sqrt((d / l)))
    else
        tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0d0 - ((h / l) * (0.5d0 * ((d_1 * (0.5d0 * (m / d))) ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= 9e-253) {
		tmp = (1.0 - (0.5 * ((h * Math.pow(((M * 0.5) * (D / d)), 2.0)) / l))) * (Math.sqrt((d / h)) * Math.sqrt((d / l)));
	} else {
		tmp = ((d / Math.sqrt(l)) / Math.sqrt(h)) * (1.0 - ((h / l) * (0.5 * Math.pow((D * (0.5 * (M / d))), 2.0))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if h <= 9e-253:
		tmp = (1.0 - (0.5 * ((h * math.pow(((M * 0.5) * (D / d)), 2.0)) / l))) * (math.sqrt((d / h)) * math.sqrt((d / l)))
	else:
		tmp = ((d / math.sqrt(l)) / math.sqrt(h)) * (1.0 - ((h / l) * (0.5 * math.pow((D * (0.5 * (M / d))), 2.0))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= 9e-253)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(Float64(M * 0.5) * Float64(D / d)) ^ 2.0)) / l))) * Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))));
	else
		tmp = Float64(Float64(Float64(d / sqrt(l)) / sqrt(h)) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(D * Float64(0.5 * Float64(M / d))) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= 9e-253)
		tmp = (1.0 - (0.5 * ((h * (((M * 0.5) * (D / d)) ^ 2.0)) / l))) * (sqrt((d / h)) * sqrt((d / l)));
	else
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - ((h / l) * (0.5 * ((D * (0.5 * (M / d))) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[h, 9e-253], N[(N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(N[(M * 0.5), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(D * N[(0.5 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 9.00000000000000057e-253

   1. Initial program 72.8%

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

   3. Simplified 73.5%

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

      1. associate-*r/ 74.4%

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

      2. frac-times 75.0%

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

      3. *-commutative 75.0%

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

      4. frac-times 74.6%

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

      5. div-inv 74.6%

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

      6. metadata-eval 74.6%

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

   5. Applied egg-rr 74.6%

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

   if 9.00000000000000057e-253 < h

   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 71.0%

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

   5. Applied egg-rr 33.6%

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

      1. expm1-def 45.0%

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

      2. expm1-log1p 83.2%

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

      3. associate-/r* 83.0%

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

      4. *-commutative 83.0%

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

      5. associate-*r/ 82.3%

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

      6. associate-*l* 82.3%

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

      7. *-commutative 82.3%

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

      8. associate-*l/ 83.1%

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

      9. *-commutative 83.1%

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

      10. associate-*l* 83.1%

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

   7. Simplified 83.1%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 9 \cdot 10^{-253}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 10: 73.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq 9 \cdot 10^{-253}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= h 9e-253)
   (*
    (/ (sqrt (/ d h)) (sqrt (/ l d)))
    (+ 1.0 (* -0.5 (* h (/ (pow (* 0.5 (* M (/ D d))) 2.0) l)))))
   (*
    (/ (/ d (sqrt l)) (sqrt h))
    (- 1.0 (* (/ h l) (* 0.5 (pow (* D (* 0.5 (/ M d))) 2.0)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= 9e-253) {
		tmp = (sqrt((d / h)) / sqrt((l / d))) * (1.0 + (-0.5 * (h * (pow((0.5 * (M * (D / d))), 2.0) / l))));
	} else {
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - ((h / l) * (0.5 * pow((D * (0.5 * (M / d))), 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (h <= 9d-253) then
        tmp = (sqrt((d / h)) / sqrt((l / d))) * (1.0d0 + ((-0.5d0) * (h * (((0.5d0 * (m * (d_1 / d))) ** 2.0d0) / l))))
    else
        tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0d0 - ((h / l) * (0.5d0 * ((d_1 * (0.5d0 * (m / d))) ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= 9e-253) {
		tmp = (Math.sqrt((d / h)) / Math.sqrt((l / d))) * (1.0 + (-0.5 * (h * (Math.pow((0.5 * (M * (D / d))), 2.0) / l))));
	} else {
		tmp = ((d / Math.sqrt(l)) / Math.sqrt(h)) * (1.0 - ((h / l) * (0.5 * Math.pow((D * (0.5 * (M / d))), 2.0))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if h <= 9e-253:
		tmp = (math.sqrt((d / h)) / math.sqrt((l / d))) * (1.0 + (-0.5 * (h * (math.pow((0.5 * (M * (D / d))), 2.0) / l))))
	else:
		tmp = ((d / math.sqrt(l)) / math.sqrt(h)) * (1.0 - ((h / l) * (0.5 * math.pow((D * (0.5 * (M / d))), 2.0))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= 9e-253)
		tmp = Float64(Float64(sqrt(Float64(d / h)) / sqrt(Float64(l / d))) * Float64(1.0 + Float64(-0.5 * Float64(h * Float64((Float64(0.5 * Float64(M * Float64(D / d))) ^ 2.0) / l)))));
	else
		tmp = Float64(Float64(Float64(d / sqrt(l)) / sqrt(h)) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(D * Float64(0.5 * Float64(M / d))) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= 9e-253)
		tmp = (sqrt((d / h)) / sqrt((l / d))) * (1.0 + (-0.5 * (h * (((0.5 * (M * (D / d))) ^ 2.0) / l))));
	else
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - ((h / l) * (0.5 * ((D * (0.5 * (M / d))) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[h, 9e-253], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(h * N[(N[Power[N[(0.5 * N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(D * N[(0.5 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 9.00000000000000057e-253

   1. Initial program 72.8%

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

   3. Simplified 73.5%

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

      1. associate-*r/ 74.4%

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

      2. frac-times 75.0%

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

      3. *-commutative 75.0%

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

      4. frac-times 74.6%

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

      5. div-inv 74.6%

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

      6. metadata-eval 74.6%

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

   5. Applied egg-rr 74.6%

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

      1. clear-num 73.8%

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

      2. sqrt-div 74.6%

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

      3. metadata-eval 74.6%

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

   7. Applied egg-rr 74.6%

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

      1. expm1-log1p-u 40.8%

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

      2. expm1-udef 26.6%

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

   9. Applied egg-rr 25.8%

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

       1. expm1-def 39.9%

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

       2. expm1-log1p 74.1%

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

       3. associate-/r/ 76.0%

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

       4. associate-*l* 76.0%

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

   11. Simplified 76.0%

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

   if 9.00000000000000057e-253 < h

   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 71.0%

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

   5. Applied egg-rr 33.6%

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

      1. expm1-def 45.0%

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

      2. expm1-log1p 83.2%

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

      3. associate-/r* 83.0%

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

      4. *-commutative 83.0%

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

      5. associate-*r/ 82.3%

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

      6. associate-*l* 82.3%

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

      7. *-commutative 82.3%

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

      8. associate-*l/ 83.1%

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

      9. *-commutative 83.1%

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

      10. associate-*l* 83.1%

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

   7. Simplified 83.1%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 9 \cdot 10^{-253}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 11: 73.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq 9 \cdot 10^{-253}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= h 9e-253)
   (*
    (/ (sqrt (/ d h)) (sqrt (/ l d)))
    (+ 1.0 (* -0.5 (* h (/ (pow (* 0.5 (* M (/ D d))) 2.0) l)))))
   (*
    (/ d (* (sqrt h) (sqrt l)))
    (- 1.0 (* (/ h l) (* 0.5 (pow (* (* M 0.5) (/ D d)) 2.0)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= 9e-253) {
		tmp = (sqrt((d / h)) / sqrt((l / d))) * (1.0 + (-0.5 * (h * (pow((0.5 * (M * (D / d))), 2.0) / l))));
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - ((h / l) * (0.5 * pow(((M * 0.5) * (D / d)), 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (h <= 9d-253) then
        tmp = (sqrt((d / h)) / sqrt((l / d))) * (1.0d0 + ((-0.5d0) * (h * (((0.5d0 * (m * (d_1 / d))) ** 2.0d0) / l))))
    else
        tmp = (d / (sqrt(h) * sqrt(l))) * (1.0d0 - ((h / l) * (0.5d0 * (((m * 0.5d0) * (d_1 / d)) ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= 9e-253) {
		tmp = (Math.sqrt((d / h)) / Math.sqrt((l / d))) * (1.0 + (-0.5 * (h * (Math.pow((0.5 * (M * (D / d))), 2.0) / l))));
	} else {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * (1.0 - ((h / l) * (0.5 * Math.pow(((M * 0.5) * (D / d)), 2.0))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if h <= 9e-253:
		tmp = (math.sqrt((d / h)) / math.sqrt((l / d))) * (1.0 + (-0.5 * (h * (math.pow((0.5 * (M * (D / d))), 2.0) / l))))
	else:
		tmp = (d / (math.sqrt(h) * math.sqrt(l))) * (1.0 - ((h / l) * (0.5 * math.pow(((M * 0.5) * (D / d)), 2.0))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= 9e-253)
		tmp = Float64(Float64(sqrt(Float64(d / h)) / sqrt(Float64(l / d))) * Float64(1.0 + Float64(-0.5 * Float64(h * Float64((Float64(0.5 * Float64(M * Float64(D / d))) ^ 2.0) / l)))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(M * 0.5) * Float64(D / d)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= 9e-253)
		tmp = (sqrt((d / h)) / sqrt((l / d))) * (1.0 + (-0.5 * (h * (((0.5 * (M * (D / d))) ^ 2.0) / l))));
	else
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - ((h / l) * (0.5 * (((M * 0.5) * (D / d)) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[h, 9e-253], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(h * N[(N[Power[N[(0.5 * N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(N[(M * 0.5), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 9.00000000000000057e-253

   1. Initial program 72.8%

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

   3. Simplified 73.5%

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

      1. associate-*r/ 74.4%

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

      2. frac-times 75.0%

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

      3. *-commutative 75.0%

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

      4. frac-times 74.6%

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

      5. div-inv 74.6%

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

      6. metadata-eval 74.6%

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

   5. Applied egg-rr 74.6%

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

      1. clear-num 73.8%

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

      2. sqrt-div 74.6%

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

      3. metadata-eval 74.6%

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

   7. Applied egg-rr 74.6%

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

      1. expm1-log1p-u 40.8%

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

      2. expm1-udef 26.6%

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

   9. Applied egg-rr 25.8%

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

       1. expm1-def 39.9%

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

       2. expm1-log1p 74.1%

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

       3. associate-/r/ 76.0%

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

       4. associate-*l* 76.0%

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

   11. Simplified 76.0%

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

   if 9.00000000000000057e-253 < h

   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 71.0%

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

   5. Applied egg-rr 83.2%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 9 \cdot 10^{-253}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 12: 67.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\frac{\ell}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -2e-309)
   (*
    (sqrt (* (/ d h) (/ d l)))
    (+ 1.0 (* -0.5 (/ (pow (* (* M 0.5) (/ D d)) 2.0) (/ l h)))))
   (*
    (/ (/ d (sqrt l)) (sqrt h))
    (- 1.0 (* (/ h l) (* 0.5 (pow (* D (* 0.5 (/ M d))) 2.0)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -2e-309) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * (pow(((M * 0.5) * (D / d)), 2.0) / (l / h))));
	} else {
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - ((h / l) * (0.5 * pow((D * (0.5 * (M / d))), 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-2d-309)) then
        tmp = sqrt(((d / h) * (d / l))) * (1.0d0 + ((-0.5d0) * ((((m * 0.5d0) * (d_1 / d)) ** 2.0d0) / (l / h))))
    else
        tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0d0 - ((h / l) * (0.5d0 * ((d_1 * (0.5d0 * (m / d))) ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -2e-309) {
		tmp = Math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * (Math.pow(((M * 0.5) * (D / d)), 2.0) / (l / h))));
	} else {
		tmp = ((d / Math.sqrt(l)) / Math.sqrt(h)) * (1.0 - ((h / l) * (0.5 * Math.pow((D * (0.5 * (M / d))), 2.0))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -2e-309:
		tmp = math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * (math.pow(((M * 0.5) * (D / d)), 2.0) / (l / h))))
	else:
		tmp = ((d / math.sqrt(l)) / math.sqrt(h)) * (1.0 - ((h / l) * (0.5 * math.pow((D * (0.5 * (M / d))), 2.0))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -2e-309)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 + Float64(-0.5 * Float64((Float64(Float64(M * 0.5) * Float64(D / d)) ^ 2.0) / Float64(l / h)))));
	else
		tmp = Float64(Float64(Float64(d / sqrt(l)) / sqrt(h)) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(D * Float64(0.5 * Float64(M / d))) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -2e-309)
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((((M * 0.5) * (D / d)) ^ 2.0) / (l / h))));
	else
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - ((h / l) * (0.5 * ((D * (0.5 * (M / d))) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -2e-309], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[Power[N[(N[(M * 0.5), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(D * N[(0.5 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.9999999999999988e-309

   1. Initial program 72.8%

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

   3. Simplified 73.5%

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

      1. associate-*r/ 73.7%

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

      2. frac-times 74.4%

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

      3. *-commutative 74.4%

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

      4. frac-times 73.9%

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

      5. div-inv 73.9%

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

      6. metadata-eval 73.9%

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

   5. Applied egg-rr 73.9%

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

      1. expm1-log1p-u 38.8%

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

      2. expm1-udef 24.9%

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

   7. Applied egg-rr 21.3%

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

      1. expm1-def 31.4%

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

      2. expm1-log1p 65.9%

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

      3. *-commutative 65.9%

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

   9. Simplified 65.9%

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

   if -1.9999999999999988e-309 < l

   1. Initial program 70.4%

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

   3. Simplified 71.1%

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

   5. Applied egg-rr 33.8%

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

      1. expm1-def 45.8%

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

      2. expm1-log1p 82.3%

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

      3. associate-/r* 82.2%

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

      4. *-commutative 82.2%

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

      5. associate-*r/ 81.5%

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

      6. associate-*l* 81.5%

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

      7. *-commutative 81.5%

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

      8. associate-*l/ 82.2%

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

      9. *-commutative 82.2%

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

      10. associate-*l* 82.2%

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

   7. Simplified 82.2%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\frac{\ell}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 13: 48.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 4.8 \cdot 10^{-48}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\frac{\ell}{h}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= D 4.8e-48)
   (* (sqrt (/ d h)) (sqrt (/ d l)))
   (*
    (sqrt (* (/ d h) (/ d l)))
    (+ 1.0 (* -0.5 (/ (pow (* (* M 0.5) (/ D d)) 2.0) (/ l h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (D <= 4.8e-48) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * (pow(((M * 0.5) * (D / d)), 2.0) / (l / h))));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d_1 <= 4.8d-48) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else
        tmp = sqrt(((d / h) * (d / l))) * (1.0d0 + ((-0.5d0) * ((((m * 0.5d0) * (d_1 / d)) ** 2.0d0) / (l / h))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (D <= 4.8e-48) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else {
		tmp = Math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * (Math.pow(((M * 0.5) * (D / d)), 2.0) / (l / h))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if D <= 4.8e-48:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	else:
		tmp = math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * (math.pow(((M * 0.5) * (D / d)), 2.0) / (l / h))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (D <= 4.8e-48)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	else
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 + Float64(-0.5 * Float64((Float64(Float64(M * 0.5) * Float64(D / d)) ^ 2.0) / Float64(l / h)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (D <= 4.8e-48)
		tmp = sqrt((d / h)) * sqrt((d / l));
	else
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((((M * 0.5) * (D / d)) ^ 2.0) / (l / h))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[D, 4.8e-48], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[Power[N[(N[(M * 0.5), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 4.8e-48

   1. Initial program 73.4%

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

   3. Simplified 73.9%

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

   4. Taylor expanded in h around 0 44.6%

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

   if 4.8e-48 < D

   1. Initial program 66.5%

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

   3. Simplified 68.0%

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

      1. associate-*r/ 66.7%

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

      2. frac-times 66.7%

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

      3. *-commutative 66.7%

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

      4. frac-times 66.7%

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

      5. div-inv 66.7%

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

      6. metadata-eval 66.7%

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

   5. Applied egg-rr 66.7%

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

      1. expm1-log1p-u 25.3%

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

      2. expm1-udef 19.6%

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

   7. Applied egg-rr 18.3%

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

      1. expm1-def 24.1%

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

      2. expm1-log1p 63.6%

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

      3. *-commutative 63.6%

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

   9. Simplified 63.6%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 4.8 \cdot 10^{-48}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\frac{\ell}{h}}\right)\\ \end{array} \]

Alternative 14: 46.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -6.5 \cdot 10^{-230}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{-117}:\\ \;\;\;\;d \cdot {\left(\frac{\frac{1}{h}}{\ell}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -6.5e-230)
   (* (- d) (sqrt (/ (/ 1.0 l) h)))
   (if (<= d 2.1e-117)
     (* d (pow (/ (/ 1.0 h) l) 0.5))
     (* d (* (pow l -0.5) (pow h -0.5))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -6.5e-230) {
		tmp = -d * sqrt(((1.0 / l) / h));
	} else if (d <= 2.1e-117) {
		tmp = d * pow(((1.0 / h) / l), 0.5);
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= (-6.5d-230)) then
        tmp = -d * sqrt(((1.0d0 / l) / h))
    else if (d <= 2.1d-117) then
        tmp = d * (((1.0d0 / h) / l) ** 0.5d0)
    else
        tmp = d * ((l ** (-0.5d0)) * (h ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -6.5e-230) {
		tmp = -d * Math.sqrt(((1.0 / l) / h));
	} else if (d <= 2.1e-117) {
		tmp = d * Math.pow(((1.0 / h) / l), 0.5);
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= -6.5e-230:
		tmp = -d * math.sqrt(((1.0 / l) / h))
	elif d <= 2.1e-117:
		tmp = d * math.pow(((1.0 / h) / l), 0.5)
	else:
		tmp = d * (math.pow(l, -0.5) * math.pow(h, -0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -6.5e-230)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / l) / h)));
	elseif (d <= 2.1e-117)
		tmp = Float64(d * (Float64(Float64(1.0 / h) / l) ^ 0.5));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -6.5e-230)
		tmp = -d * sqrt(((1.0 / l) / h));
	elseif (d <= 2.1e-117)
		tmp = d * (((1.0 / h) / l) ^ 0.5);
	else
		tmp = d * ((l ^ -0.5) * (h ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -6.5e-230], N[((-d) * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.1e-117], N[(d * N[Power[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -6.5000000000000004e-230

   1. Initial program 75.7%

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

   3. Simplified 76.6%

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

      1. associate-*r/ 76.8%

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

      2. frac-times 77.6%

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

      3. *-commutative 77.6%

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

      4. frac-times 77.0%

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

      5. div-inv 77.0%

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

      6. metadata-eval 77.0%

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

   5. Applied egg-rr 77.0%

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

      1. clear-num 76.9%

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

      2. sqrt-div 77.9%

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

      3. metadata-eval 77.9%

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

   7. Applied egg-rr 77.9%

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

   8. Taylor expanded in d around -inf 43.7%

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

      1. mul-1-neg 43.7%

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

      2. associate-/l/ 44.4%

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

   10. Simplified 44.4%

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

   if -6.5000000000000004e-230 < d < 2.0999999999999999e-117

   1. Initial program 40.9%

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

   3. Simplified 40.9%

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

   4. Taylor expanded in d around inf 25.6%

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

      1. pow1/2 25.6%

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

      2. associate-/r* 25.6%

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

   6. Applied egg-rr 25.6%

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

   if 2.0999999999999999e-117 < d

   1. Initial program 82.5%

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

   3. Simplified 83.6%

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

      1. associate-*r/ 83.9%

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

      2. frac-times 83.9%

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

      3. *-commutative 83.9%

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

      4. frac-times 83.9%

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

      5. div-inv 83.9%

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

      6. metadata-eval 83.9%

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

   5. Applied egg-rr 83.9%

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

   6. Taylor expanded in d around inf 53.5%

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

      1. unpow-1 53.5%

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

      2. metadata-eval 53.5%

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

      3. pow-sqr 53.6%

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

      4. rem-sqrt-square 53.6%

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

      5. rem-square-sqrt 53.3%

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

      6. fabs-sqr 53.3%

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

      7. rem-square-sqrt 53.6%

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

   8. Simplified 53.6%

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

      1. *-commutative 53.6%

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

      2. unpow-prod-down 62.5%

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

   10. Applied egg-rr 62.5%

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

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.5 \cdot 10^{-230}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{-117}:\\ \;\;\;\;d \cdot {\left(\frac{\frac{1}{h}}{\ell}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 15: 43.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{if}\;d \leq -6.3 \cdot 10^{-229}:\\ \;\;\;\;\left(-d\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ (/ 1.0 l) h))))
   (if (<= d -6.3e-229) (* (- d) t_0) (* d t_0))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(((1.0 / l) / h));
	double tmp;
	if (d <= -6.3e-229) {
		tmp = -d * t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((1.0d0 / l) / h))
    if (d <= (-6.3d-229)) then
        tmp = -d * t_0
    else
        tmp = d * t_0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt(((1.0 / l) / h));
	double tmp;
	if (d <= -6.3e-229) {
		tmp = -d * t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt(((1.0 / l) / h))
	tmp = 0
	if d <= -6.3e-229:
		tmp = -d * t_0
	else:
		tmp = d * t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(Float64(1.0 / l) / h))
	tmp = 0.0
	if (d <= -6.3e-229)
		tmp = Float64(Float64(-d) * t_0);
	else
		tmp = Float64(d * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(((1.0 / l) / h));
	tmp = 0.0;
	if (d <= -6.3e-229)
		tmp = -d * t_0;
	else
		tmp = d * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -6.3e-229], N[((-d) * t$95$0), $MachinePrecision], N[(d * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -6.29999999999999987e-229

   1. Initial program 75.7%

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

   3. Simplified 76.6%

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

      1. associate-*r/ 76.8%

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

      2. frac-times 77.6%

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

      3. *-commutative 77.6%

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

      4. frac-times 77.0%

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

      5. div-inv 77.0%

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

      6. metadata-eval 77.0%

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

   5. Applied egg-rr 77.0%

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

      1. clear-num 76.9%

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

      2. sqrt-div 77.9%

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

      3. metadata-eval 77.9%

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

   7. Applied egg-rr 77.9%

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

   8. Taylor expanded in d around -inf 43.7%

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

      1. mul-1-neg 43.7%

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

      2. associate-/l/ 44.4%

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

   10. Simplified 44.4%

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

   if -6.29999999999999987e-229 < d

   1. Initial program 68.4%

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

   3. Simplified 69.0%

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

   4. Taylor expanded in d around inf 44.0%

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

      1. *-commutative 44.0%

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

      2. associate-/r* 45.0%

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

   6. Simplified 45.0%

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

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.3 \cdot 10^{-229}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]

Alternative 16: 43.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{if}\;d \leq -5.8 \cdot 10^{-230}:\\ \;\;\;\;\left(-d\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (pow (* l h) -0.5)))
   (if (<= d -5.8e-230) (* (- d) t_0) (* d t_0))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = pow((l * h), -0.5);
	double tmp;
	if (d <= -5.8e-230) {
		tmp = -d * t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (l * h) ** (-0.5d0)
    if (d <= (-5.8d-230)) then
        tmp = -d * t_0
    else
        tmp = d * t_0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.pow((l * h), -0.5);
	double tmp;
	if (d <= -5.8e-230) {
		tmp = -d * t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.pow((l * h), -0.5)
	tmp = 0
	if d <= -5.8e-230:
		tmp = -d * t_0
	else:
		tmp = d * t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(l * h) ^ -0.5
	tmp = 0.0
	if (d <= -5.8e-230)
		tmp = Float64(Float64(-d) * t_0);
	else
		tmp = Float64(d * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (l * h) ^ -0.5;
	tmp = 0.0;
	if (d <= -5.8e-230)
		tmp = -d * t_0;
	else
		tmp = d * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[d, -5.8e-230], N[((-d) * t$95$0), $MachinePrecision], N[(d * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -5.80000000000000011e-230

   1. Initial program 75.7%

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

   3. Simplified 76.6%

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

      1. associate-*r/ 76.8%

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

      2. frac-times 77.6%

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

      3. *-commutative 77.6%

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

      4. frac-times 77.0%

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

      5. div-inv 77.0%

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

      6. metadata-eval 77.0%

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

   5. Applied egg-rr 77.0%

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

      1. clear-num 76.9%

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

      2. sqrt-div 77.9%

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

      3. metadata-eval 77.9%

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

   7. Applied egg-rr 77.9%

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

   8. Taylor expanded in d around -inf 43.7%

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

      1. mul-1-neg 43.7%

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

      2. *-commutative 43.7%

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

      3. distribute-rgt-neg-in 43.7%

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

      4. unpow1/2 43.7%

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

      5. rem-exp-log 41.3%

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

      6. exp-neg 41.3%

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

      7. exp-prod 41.3%

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

      8. *-commutative 41.3%

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

      9. distribute-rgt-neg-out 41.3%

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

      10. distribute-lft-neg-in 41.3%

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

      11. metadata-eval 41.3%

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

      12. log-prod 0.0%

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

      13. +-commutative 0.0%

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

      14. +-commutative 0.0%

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

      15. log-prod 41.3%

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

      16. *-commutative 41.3%

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

      17. exp-to-pow 43.7%

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

      18. *-commutative 43.7%

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

   10. Simplified 43.7%

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

   if -5.80000000000000011e-230 < d

   1. Initial program 68.4%

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

   3. Simplified 69.0%

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

      1. associate-*r/ 70.0%

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

      2. frac-times 70.0%

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

      3. *-commutative 70.0%

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

      4. frac-times 70.0%

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

      5. div-inv 70.0%

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

      6. metadata-eval 70.0%

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

   5. Applied egg-rr 70.0%

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

   6. Taylor expanded in d around inf 44.0%

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

      1. unpow-1 44.0%

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

      2. metadata-eval 44.0%

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

      3. pow-sqr 44.0%

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

      4. rem-sqrt-square 44.0%

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

      5. rem-square-sqrt 43.9%

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

      6. fabs-sqr 43.9%

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

      7. rem-square-sqrt 44.0%

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

   8. Simplified 44.0%

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

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.8 \cdot 10^{-230}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]

Alternative 17: 43.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -6.2 \cdot 10^{-234}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -6.2e-234)
   (* (- d) (pow (* l h) -0.5))
   (* d (sqrt (/ (/ 1.0 l) h)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -6.2e-234) {
		tmp = -d * pow((l * h), -0.5);
	} else {
		tmp = d * sqrt(((1.0 / l) / h));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= (-6.2d-234)) then
        tmp = -d * ((l * h) ** (-0.5d0))
    else
        tmp = d * sqrt(((1.0d0 / l) / h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -6.2e-234) {
		tmp = -d * Math.pow((l * h), -0.5);
	} else {
		tmp = d * Math.sqrt(((1.0 / l) / h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= -6.2e-234:
		tmp = -d * math.pow((l * h), -0.5)
	else:
		tmp = d * math.sqrt(((1.0 / l) / h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -6.2e-234)
		tmp = Float64(Float64(-d) * (Float64(l * h) ^ -0.5));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / l) / h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -6.2e-234)
		tmp = -d * ((l * h) ^ -0.5);
	else
		tmp = d * sqrt(((1.0 / l) / h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -6.2e-234], N[((-d) * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -6.2000000000000003e-234

   1. Initial program 75.7%

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

   3. Simplified 76.6%

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

      1. associate-*r/ 76.8%

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

      2. frac-times 77.6%

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

      3. *-commutative 77.6%

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

      4. frac-times 77.0%

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

      5. div-inv 77.0%

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

      6. metadata-eval 77.0%

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

   5. Applied egg-rr 77.0%

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

      1. clear-num 76.9%

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

      2. sqrt-div 77.9%

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

      3. metadata-eval 77.9%

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

   7. Applied egg-rr 77.9%

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

   8. Taylor expanded in d around -inf 43.7%

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

      1. mul-1-neg 43.7%

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

      2. *-commutative 43.7%

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

      3. distribute-rgt-neg-in 43.7%

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

      4. unpow1/2 43.7%

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

      5. rem-exp-log 41.3%

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

      6. exp-neg 41.3%

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

      7. exp-prod 41.3%

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

      8. *-commutative 41.3%

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

      9. distribute-rgt-neg-out 41.3%

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

      10. distribute-lft-neg-in 41.3%

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

      11. metadata-eval 41.3%

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

      12. log-prod 0.0%

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

      13. +-commutative 0.0%

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

      14. +-commutative 0.0%

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

      15. log-prod 41.3%

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

      16. *-commutative 41.3%

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

      17. exp-to-pow 43.7%

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

      18. *-commutative 43.7%

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

   10. Simplified 43.7%

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

   if -6.2000000000000003e-234 < d

   1. Initial program 68.4%

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

   3. Simplified 69.0%

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

   4. Taylor expanded in d around inf 44.0%

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

      1. *-commutative 44.0%

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

      2. associate-/r* 45.0%

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

   6. Simplified 45.0%

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

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.2 \cdot 10^{-234}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]

Alternative 18: 27.2% 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 71.6%

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

3. Simplified 72.3%

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

   1. associate-*r/ 73.0%

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

   2. frac-times 73.3%

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

   3. *-commutative 73.3%

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

   4. frac-times 73.1%

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

   5. div-inv 73.1%

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

   6. metadata-eval 73.1%

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

5. Applied egg-rr 73.1%

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

6. Taylor expanded in d around inf 28.3%

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

   1. unpow-1 28.3%

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

   2. metadata-eval 28.3%

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

   3. pow-sqr 28.3%

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

   4. rem-sqrt-square 28.3%

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

   5. rem-square-sqrt 28.2%

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

   6. fabs-sqr 28.2%

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

   7. rem-square-sqrt 28.3%

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

8. Simplified 28.3%

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

9. Final simplification 28.3%

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

Alternative 19: 4.5% accurate, 332.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 0.0)

C

double code(double d, double h, double l, double M, double D) {
	return 0.0;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = 0.0d0
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return 0.0;
}

Python

def code(d, h, l, M, D):
	return 0.0

Julia

function code(d, h, l, M, D)
	return 0.0
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = 0.0;
end

Wolfram

code[d_, h_, l_, M_, D_] := 0.0

Derivation

1. Initial program 71.6%

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

3. Simplified 72.3%

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

4. Taylor expanded in d around inf 28.3%

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

   1. expm1-log1p-u 22.9%

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

   2. expm1-udef 18.9%

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

   3. pow1/2 18.9%

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

   4. inv-pow 18.9%

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

   5. pow-pow 18.9%

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

   6. metadata-eval 18.9%

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

6. Applied egg-rr 18.9%

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

7. Taylor expanded in d around 0 4.2%

   \[\leadsto \color{blue}{1} - 1 \]

8. Final simplification 4.2%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023334 
(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)))))