Henrywood and Agarwal, Equation (12)

Percentage Accurate: 67.2% → 87.6%

Time: 34.6s

Alternatives: 22

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.2% 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: 87.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{h \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)}{\ell}\\ t_1 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{t\_1}{\sqrt{-h}} \cdot \left(\frac{t\_1}{\sqrt{-\ell}} \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(t\_0 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if l < -1.000000000000002e-309

   1. Initial program 72.4%

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

   3. Simplified 73.7%

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

      1. associate-*l/ 75.4%

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

   6. Applied egg-rr 75.4%

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

      1. frac-2neg 75.4%

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

      2. sqrt-div 79.4%

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

   8. Applied egg-rr 79.4%

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

      1. frac-2neg 79.4%

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

      2. sqrt-div 89.6%

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

   10. Applied egg-rr 89.6%

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

   if -1.000000000000002e-309 < l

   1. Initial program 67.2%

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

   3. Simplified 66.4%

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

      1. associate-*l/ 68.3%

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

   6. Applied egg-rr 68.3%

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

      1. sqrt-div 84.3%

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

   8. Applied egg-rr 84.3%

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

      1. sqrt-div 89.1%

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

   10. Applied egg-rr 89.1%

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

4. Final simplification 89.4%

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

Alternative 2: 83.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{h \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)}{\ell}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(t\_0 \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(t\_0 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if l < -1.000000000000002e-309

   1. Initial program 72.4%

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

   3. Simplified 73.7%

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

      1. associate-*l/ 75.4%

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

   6. Applied egg-rr 75.4%

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

      1. frac-2neg 79.4%

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

      2. sqrt-div 89.6%

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

   8. Applied egg-rr 82.6%

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

   if -1.000000000000002e-309 < l

   1. Initial program 67.2%

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

   3. Simplified 66.4%

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

      1. associate-*l/ 68.3%

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

   6. Applied egg-rr 68.3%

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

      1. sqrt-div 84.3%

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

   8. Applied egg-rr 84.3%

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

      1. sqrt-div 89.1%

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

   10. Applied egg-rr 89.1%

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

4. Final simplification 85.7%

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

Alternative 3: 78.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{h \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)}{\ell}\\ \mathbf{if}\;d \leq -5 \cdot 10^{-137}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot t\_0\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \frac{1}{\frac{\sqrt{h}}{\sqrt{d}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (* h (* (pow (* (* M 0.5) (/ D d)) 2.0) -0.5)) l))))
   (if (<= d -5e-137)
     (* (* (/ (sqrt (- d)) (sqrt (- l))) t_0) (sqrt (/ d h)))
     (if (<= d -1e-309)
       (*
        (* d (sqrt (/ (/ 1.0 h) l)))
        (- -1.0 (* -0.5 (* (/ h l) (pow (* D (* M (/ 0.5 d))) 2.0)))))
       (* (* t_0 (sqrt (/ d l))) (/ 1.0 (/ (sqrt h) (sqrt d))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 + ((h * (pow(((M * 0.5) * (D / d)), 2.0) * -0.5)) / l);
	double tmp;
	if (d <= -5e-137) {
		tmp = ((sqrt(-d) / sqrt(-l)) * t_0) * sqrt((d / h));
	} else if (d <= -1e-309) {
		tmp = (d * sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 * ((h / l) * pow((D * (M * (0.5 / d))), 2.0))));
	} else {
		tmp = (t_0 * sqrt((d / l))) * (1.0 / (sqrt(h) / sqrt(d)));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + ((h * ((((m * 0.5d0) * (d_1 / d)) ** 2.0d0) * (-0.5d0))) / l)
    if (d <= (-5d-137)) then
        tmp = ((sqrt(-d) / sqrt(-l)) * t_0) * sqrt((d / h))
    else if (d <= (-1d-309)) then
        tmp = (d * sqrt(((1.0d0 / h) / l))) * ((-1.0d0) - ((-0.5d0) * ((h / l) * ((d_1 * (m * (0.5d0 / d))) ** 2.0d0))))
    else
        tmp = (t_0 * sqrt((d / l))) * (1.0d0 / (sqrt(h) / sqrt(d)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 + ((h * (Math.pow(((M * 0.5) * (D / d)), 2.0) * -0.5)) / l);
	double tmp;
	if (d <= -5e-137) {
		tmp = ((Math.sqrt(-d) / Math.sqrt(-l)) * t_0) * Math.sqrt((d / h));
	} else if (d <= -1e-309) {
		tmp = (d * Math.sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 * ((h / l) * Math.pow((D * (M * (0.5 / d))), 2.0))));
	} else {
		tmp = (t_0 * Math.sqrt((d / l))) * (1.0 / (Math.sqrt(h) / Math.sqrt(d)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = 1.0 + ((h * (math.pow(((M * 0.5) * (D / d)), 2.0) * -0.5)) / l)
	tmp = 0
	if d <= -5e-137:
		tmp = ((math.sqrt(-d) / math.sqrt(-l)) * t_0) * math.sqrt((d / h))
	elif d <= -1e-309:
		tmp = (d * math.sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 * ((h / l) * math.pow((D * (M * (0.5 / d))), 2.0))))
	else:
		tmp = (t_0 * math.sqrt((d / l))) * (1.0 / (math.sqrt(h) / math.sqrt(d)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(1.0 + Float64(Float64(h * Float64((Float64(Float64(M * 0.5) * Float64(D / d)) ^ 2.0) * -0.5)) / l))
	tmp = 0.0
	if (d <= -5e-137)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))) * t_0) * sqrt(Float64(d / h)));
	elseif (d <= -1e-309)
		tmp = Float64(Float64(d * sqrt(Float64(Float64(1.0 / h) / l))) * Float64(-1.0 - Float64(-0.5 * Float64(Float64(h / l) * (Float64(D * Float64(M * Float64(0.5 / d))) ^ 2.0)))));
	else
		tmp = Float64(Float64(t_0 * sqrt(Float64(d / l))) * Float64(1.0 / Float64(sqrt(h) / sqrt(d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = 1.0 + ((h * ((((M * 0.5) * (D / d)) ^ 2.0) * -0.5)) / l);
	tmp = 0.0;
	if (d <= -5e-137)
		tmp = ((sqrt(-d) / sqrt(-l)) * t_0) * sqrt((d / h));
	elseif (d <= -1e-309)
		tmp = (d * sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 * ((h / l) * ((D * (M * (0.5 / d))) ^ 2.0))));
	else
		tmp = (t_0 * sqrt((d / l))) * (1.0 / (sqrt(h) / sqrt(d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(1.0 + N[(N[(h * N[(N[Power[N[(N[(M * 0.5), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -5e-137], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1e-309], N[(N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[Sqrt[h], $MachinePrecision] / N[Sqrt[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.00000000000000001e-137

   1. Initial program 85.9%

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

   3. Simplified 88.0%

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

      1. associate-*l/ 90.4%

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

   6. Applied egg-rr 90.4%

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

      1. frac-2neg 90.4%

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

      2. sqrt-div 92.3%

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

   8. Applied egg-rr 92.3%

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

   if -5.00000000000000001e-137 < d < -1.000000000000002e-309

   1. Initial program 43.3%

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

   3. Simplified 43.1%

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

      1. add-sqr-sqrt 43.2%

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

      2. pow2 43.2%

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

      3. sqrt-prod 43.1%

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

      4. unpow2 43.1%

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

      5. sqrt-prod 19.7%

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

      6. add-sqr-sqrt 43.3%

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

      7. div-inv 43.3%

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

      8. metadata-eval 43.3%

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

   6. Applied egg-rr 43.3%

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

      1. *-commutative 43.3%

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

      2. /-rgt-identity 43.3%

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

      3. associate-/l* 43.3%

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

      4. metadata-eval 43.3%

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

      5. times-frac 43.5%

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

      6. associate-*r/ 43.3%

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

      7. associate-*l* 43.3%

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

      8. *-lft-identity 43.3%

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

      9. associate-/l* 43.3%

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

      10. associate-/r/ 43.3%

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

      11. *-commutative 43.3%

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

      12. associate-/r* 43.3%

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

      13. metadata-eval 43.3%

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

   8. Simplified 43.3%

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

      1. expm1-log1p-u 12.7%

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

      2. expm1-udef 5.0%

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

   10. Applied egg-rr 5.1%

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

       1. expm1-def 8.4%

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

       2. expm1-log1p 31.8%

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

       3. rem-log-exp 29.2%

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

       4. rem-log-exp 31.8%

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

       5. *-commutative 31.8%

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

   12. Simplified 31.8%

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

   13. Taylor expanded in d around -inf 68.1%

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

       1. mul-1-neg 68.1%

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

       2. *-commutative 68.1%

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

       3. distribute-rgt-neg-in 68.1%

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

       4. associate-/r* 68.2%

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

   15. Simplified 68.2%

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

   if -1.000000000000002e-309 < d

   1. Initial program 67.2%

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

   3. Simplified 66.4%

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

      1. associate-*l/ 68.3%

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

   6. Applied egg-rr 68.3%

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

      1. clear-num 37.2%

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

      2. sqrt-div 38.2%

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

      3. metadata-eval 38.2%

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

   8. Applied egg-rr 67.2%

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

      1. sqrt-div 84.3%

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

   10. Applied egg-rr 84.3%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{-137}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 + \frac{h \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)}{\ell}\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + \frac{h \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)}{\ell}\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \frac{1}{\frac{\sqrt{h}}{\sqrt{d}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -7.5 \cdot 10^{-137}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t\_0 \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\left(1 + \frac{h \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)}{\ell}\right) \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l))))
   (if (<= d -7.5e-137)
     (*
      (sqrt (/ d h))
      (* t_0 (+ 1.0 (/ (* h (* -0.5 (pow (* M (* D (/ 0.5 d))) 2.0))) l))))
     (if (<= d -1e-309)
       (*
        (* d (sqrt (/ (/ 1.0 h) l)))
        (- -1.0 (* -0.5 (* (/ h l) (pow (* D (* M (/ 0.5 d))) 2.0)))))
       (*
        (/ (sqrt d) (sqrt h))
        (*
         (+ 1.0 (/ (* h (* (pow (* (* M 0.5) (/ D d)) 2.0) -0.5)) l))
         t_0))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / l));
	double tmp;
	if (d <= -7.5e-137) {
		tmp = sqrt((d / h)) * (t_0 * (1.0 + ((h * (-0.5 * pow((M * (D * (0.5 / d))), 2.0))) / l)));
	} else if (d <= -1e-309) {
		tmp = (d * sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 * ((h / l) * pow((D * (M * (0.5 / d))), 2.0))));
	} else {
		tmp = (sqrt(d) / sqrt(h)) * ((1.0 + ((h * (pow(((M * 0.5) * (D / d)), 2.0) * -0.5)) / l)) * 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((d / l))
    if (d <= (-7.5d-137)) then
        tmp = sqrt((d / h)) * (t_0 * (1.0d0 + ((h * ((-0.5d0) * ((m * (d_1 * (0.5d0 / d))) ** 2.0d0))) / l)))
    else if (d <= (-1d-309)) then
        tmp = (d * sqrt(((1.0d0 / h) / l))) * ((-1.0d0) - ((-0.5d0) * ((h / l) * ((d_1 * (m * (0.5d0 / d))) ** 2.0d0))))
    else
        tmp = (sqrt(d) / sqrt(h)) * ((1.0d0 + ((h * ((((m * 0.5d0) * (d_1 / d)) ** 2.0d0) * (-0.5d0))) / l)) * 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((d / l));
	double tmp;
	if (d <= -7.5e-137) {
		tmp = Math.sqrt((d / h)) * (t_0 * (1.0 + ((h * (-0.5 * Math.pow((M * (D * (0.5 / d))), 2.0))) / l)));
	} else if (d <= -1e-309) {
		tmp = (d * Math.sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 * ((h / l) * Math.pow((D * (M * (0.5 / d))), 2.0))));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * ((1.0 + ((h * (Math.pow(((M * 0.5) * (D / d)), 2.0) * -0.5)) / l)) * t_0);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / l))
	tmp = 0
	if d <= -7.5e-137:
		tmp = math.sqrt((d / h)) * (t_0 * (1.0 + ((h * (-0.5 * math.pow((M * (D * (0.5 / d))), 2.0))) / l)))
	elif d <= -1e-309:
		tmp = (d * math.sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 * ((h / l) * math.pow((D * (M * (0.5 / d))), 2.0))))
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * ((1.0 + ((h * (math.pow(((M * 0.5) * (D / d)), 2.0) * -0.5)) / l)) * t_0)
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / l))
	tmp = 0.0
	if (d <= -7.5e-137)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(t_0 * Float64(1.0 + Float64(Float64(h * Float64(-0.5 * (Float64(M * Float64(D * Float64(0.5 / d))) ^ 2.0))) / l))));
	elseif (d <= -1e-309)
		tmp = Float64(Float64(d * sqrt(Float64(Float64(1.0 / h) / l))) * Float64(-1.0 - Float64(-0.5 * Float64(Float64(h / l) * (Float64(D * Float64(M * Float64(0.5 / d))) ^ 2.0)))));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(Float64(1.0 + Float64(Float64(h * Float64((Float64(Float64(M * 0.5) * Float64(D / d)) ^ 2.0) * -0.5)) / l)) * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / l));
	tmp = 0.0;
	if (d <= -7.5e-137)
		tmp = sqrt((d / h)) * (t_0 * (1.0 + ((h * (-0.5 * ((M * (D * (0.5 / d))) ^ 2.0))) / l)));
	elseif (d <= -1e-309)
		tmp = (d * sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 * ((h / l) * ((D * (M * (0.5 / d))) ^ 2.0))));
	else
		tmp = (sqrt(d) / sqrt(h)) * ((1.0 + ((h * ((((M * 0.5) * (D / d)) ^ 2.0) * -0.5)) / l)) * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -7.5e-137], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(1.0 + N[(N[(h * N[(-0.5 * N[Power[N[(M * N[(D * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1e-309], N[(N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(h * N[(N[Power[N[(N[(M * 0.5), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -7.4999999999999995e-137

   1. Initial program 85.9%

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

   3. Simplified 88.0%

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

      1. associate-*l/ 90.4%

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

   6. Applied egg-rr 90.4%

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

   7. Taylor expanded in M around 0 89.4%

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

      1. *-commutative 89.4%

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

      2. associate-*l/ 89.4%

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

      3. associate-*r/ 89.4%

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

      4. associate-*r* 90.2%

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

      5. *-commutative 90.2%

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

      6. associate-*l* 90.4%

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

   9. Simplified 90.4%

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

   if -7.4999999999999995e-137 < d < -1.000000000000002e-309

   1. Initial program 43.3%

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

   3. Simplified 43.1%

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

      1. add-sqr-sqrt 43.2%

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

      2. pow2 43.2%

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

      3. sqrt-prod 43.1%

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

      4. unpow2 43.1%

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

      5. sqrt-prod 19.7%

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

      6. add-sqr-sqrt 43.3%

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

      7. div-inv 43.3%

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

      8. metadata-eval 43.3%

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

   6. Applied egg-rr 43.3%

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

      1. *-commutative 43.3%

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

      2. /-rgt-identity 43.3%

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

      3. associate-/l* 43.3%

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

      4. metadata-eval 43.3%

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

      5. times-frac 43.5%

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

      6. associate-*r/ 43.3%

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

      7. associate-*l* 43.3%

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

      8. *-lft-identity 43.3%

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

      9. associate-/l* 43.3%

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

      10. associate-/r/ 43.3%

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

      11. *-commutative 43.3%

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

      12. associate-/r* 43.3%

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

      13. metadata-eval 43.3%

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

   8. Simplified 43.3%

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

      1. expm1-log1p-u 12.7%

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

      2. expm1-udef 5.0%

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

   10. Applied egg-rr 5.1%

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

       1. expm1-def 8.4%

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

       2. expm1-log1p 31.8%

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

       3. rem-log-exp 29.2%

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

       4. rem-log-exp 31.8%

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

       5. *-commutative 31.8%

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

   12. Simplified 31.8%

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

   13. Taylor expanded in d around -inf 68.1%

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

       1. mul-1-neg 68.1%

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

       2. *-commutative 68.1%

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

       3. distribute-rgt-neg-in 68.1%

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

       4. associate-/r* 68.2%

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

   15. Simplified 68.2%

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

   if -1.000000000000002e-309 < d

   1. Initial program 67.2%

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

   3. Simplified 66.4%

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

      1. associate-*l/ 68.3%

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

   6. Applied egg-rr 68.3%

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

      1. sqrt-div 84.3%

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

   8. Applied egg-rr 84.3%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.5 \cdot 10^{-137}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\left(1 + \frac{h \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)}{\ell}\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{h \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)}{\ell}\\ \mathbf{if}\;d \leq -1.5 \cdot 10^{-134}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot t\_0\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(t\_0 \cdot \sqrt{\frac{d}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (* h (* (pow (* (* M 0.5) (/ D d)) 2.0) -0.5)) l))))
   (if (<= d -1.5e-134)
     (* (* (/ (sqrt (- d)) (sqrt (- l))) t_0) (sqrt (/ d h)))
     (if (<= d -1e-309)
       (*
        (* d (sqrt (/ (/ 1.0 h) l)))
        (- -1.0 (* -0.5 (* (/ h l) (pow (* D (* M (/ 0.5 d))) 2.0)))))
       (* (/ (sqrt d) (sqrt h)) (* t_0 (sqrt (/ d l))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 + ((h * (pow(((M * 0.5) * (D / d)), 2.0) * -0.5)) / l);
	double tmp;
	if (d <= -1.5e-134) {
		tmp = ((sqrt(-d) / sqrt(-l)) * t_0) * sqrt((d / h));
	} else if (d <= -1e-309) {
		tmp = (d * sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 * ((h / l) * pow((D * (M * (0.5 / d))), 2.0))));
	} else {
		tmp = (sqrt(d) / sqrt(h)) * (t_0 * sqrt((d / l)));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + ((h * ((((m * 0.5d0) * (d_1 / d)) ** 2.0d0) * (-0.5d0))) / l)
    if (d <= (-1.5d-134)) then
        tmp = ((sqrt(-d) / sqrt(-l)) * t_0) * sqrt((d / h))
    else if (d <= (-1d-309)) then
        tmp = (d * sqrt(((1.0d0 / h) / l))) * ((-1.0d0) - ((-0.5d0) * ((h / l) * ((d_1 * (m * (0.5d0 / d))) ** 2.0d0))))
    else
        tmp = (sqrt(d) / sqrt(h)) * (t_0 * sqrt((d / l)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 + ((h * (Math.pow(((M * 0.5) * (D / d)), 2.0) * -0.5)) / l);
	double tmp;
	if (d <= -1.5e-134) {
		tmp = ((Math.sqrt(-d) / Math.sqrt(-l)) * t_0) * Math.sqrt((d / h));
	} else if (d <= -1e-309) {
		tmp = (d * Math.sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 * ((h / l) * Math.pow((D * (M * (0.5 / d))), 2.0))));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * (t_0 * Math.sqrt((d / l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = 1.0 + ((h * (math.pow(((M * 0.5) * (D / d)), 2.0) * -0.5)) / l)
	tmp = 0
	if d <= -1.5e-134:
		tmp = ((math.sqrt(-d) / math.sqrt(-l)) * t_0) * math.sqrt((d / h))
	elif d <= -1e-309:
		tmp = (d * math.sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 * ((h / l) * math.pow((D * (M * (0.5 / d))), 2.0))))
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * (t_0 * math.sqrt((d / l)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(1.0 + Float64(Float64(h * Float64((Float64(Float64(M * 0.5) * Float64(D / d)) ^ 2.0) * -0.5)) / l))
	tmp = 0.0
	if (d <= -1.5e-134)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))) * t_0) * sqrt(Float64(d / h)));
	elseif (d <= -1e-309)
		tmp = Float64(Float64(d * sqrt(Float64(Float64(1.0 / h) / l))) * Float64(-1.0 - Float64(-0.5 * Float64(Float64(h / l) * (Float64(D * Float64(M * Float64(0.5 / d))) ^ 2.0)))));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(t_0 * sqrt(Float64(d / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = 1.0 + ((h * ((((M * 0.5) * (D / d)) ^ 2.0) * -0.5)) / l);
	tmp = 0.0;
	if (d <= -1.5e-134)
		tmp = ((sqrt(-d) / sqrt(-l)) * t_0) * sqrt((d / h));
	elseif (d <= -1e-309)
		tmp = (d * sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 * ((h / l) * ((D * (M * (0.5 / d))) ^ 2.0))));
	else
		tmp = (sqrt(d) / sqrt(h)) * (t_0 * sqrt((d / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(1.0 + N[(N[(h * N[(N[Power[N[(N[(M * 0.5), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.5e-134], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1e-309], N[(N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.5e-134

   1. Initial program 85.9%

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

   3. Simplified 88.0%

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

      1. associate-*l/ 90.4%

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

   6. Applied egg-rr 90.4%

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

      1. frac-2neg 90.4%

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

      2. sqrt-div 92.3%

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

   8. Applied egg-rr 92.3%

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

   if -1.5e-134 < d < -1.000000000000002e-309

   1. Initial program 43.3%

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

   3. Simplified 43.1%

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

      1. add-sqr-sqrt 43.2%

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

      2. pow2 43.2%

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

      3. sqrt-prod 43.1%

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

      4. unpow2 43.1%

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

      5. sqrt-prod 19.7%

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

      6. add-sqr-sqrt 43.3%

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

      7. div-inv 43.3%

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

      8. metadata-eval 43.3%

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

   6. Applied egg-rr 43.3%

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

      1. *-commutative 43.3%

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

      2. /-rgt-identity 43.3%

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

      3. associate-/l* 43.3%

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

      4. metadata-eval 43.3%

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

      5. times-frac 43.5%

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

      6. associate-*r/ 43.3%

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

      7. associate-*l* 43.3%

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

      8. *-lft-identity 43.3%

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

      9. associate-/l* 43.3%

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

      10. associate-/r/ 43.3%

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

      11. *-commutative 43.3%

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

      12. associate-/r* 43.3%

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

      13. metadata-eval 43.3%

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

   8. Simplified 43.3%

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

      1. expm1-log1p-u 12.7%

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

      2. expm1-udef 5.0%

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

   10. Applied egg-rr 5.1%

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

       1. expm1-def 8.4%

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

       2. expm1-log1p 31.8%

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

       3. rem-log-exp 29.2%

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

       4. rem-log-exp 31.8%

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

       5. *-commutative 31.8%

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

   12. Simplified 31.8%

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

   13. Taylor expanded in d around -inf 68.1%

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

       1. mul-1-neg 68.1%

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

       2. *-commutative 68.1%

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

       3. distribute-rgt-neg-in 68.1%

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

       4. associate-/r* 68.2%

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

   15. Simplified 68.2%

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

   if -1.000000000000002e-309 < d

   1. Initial program 67.2%

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

   3. Simplified 66.4%

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

      1. associate-*l/ 68.3%

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

   6. Applied egg-rr 68.3%

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

      1. sqrt-div 84.3%

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

   8. Applied egg-rr 84.3%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.5 \cdot 10^{-134}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 + \frac{h \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)}{\ell}\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\left(1 + \frac{h \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)}{\ell}\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + \frac{h \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)}{\ell}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{1}{\frac{\sqrt{h}}{\sqrt{d}}}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if l < -1.000000000000002e-309

   1. Initial program 72.4%

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

   3. Simplified 73.7%

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

      1. associate-*l/ 75.4%

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

   6. Applied egg-rr 75.4%

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

      1. frac-2neg 79.4%

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

      2. sqrt-div 89.6%

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

   8. Applied egg-rr 82.6%

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

   if -1.000000000000002e-309 < l

   1. Initial program 67.2%

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

   3. Simplified 66.4%

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

      1. associate-*l/ 68.3%

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

   6. Applied egg-rr 68.3%

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

      1. clear-num 37.2%

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

      2. sqrt-div 38.2%

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

      3. metadata-eval 38.2%

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

   8. Applied egg-rr 67.2%

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

      1. sqrt-div 84.3%

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

   10. Applied egg-rr 84.3%

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

4. Final simplification 83.4%

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

Alternative 7: 76.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.8 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}, 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -2.8e-133)
   (*
    (sqrt (/ d h))
    (*
     (sqrt (/ d l))
     (+ 1.0 (/ (* h (* -0.5 (pow (* M (* D (/ 0.5 d))) 2.0))) l))))
   (if (<= d -1e-309)
     (*
      (* d (sqrt (/ (/ 1.0 h) l)))
      (- -1.0 (* -0.5 (* (/ h l) (pow (* D (* M (/ 0.5 d))) 2.0)))))
     (*
      (fma (/ h l) (* -0.5 (pow (* M (/ D (* d 2.0))) 2.0)) 1.0)
      (/ d (* (sqrt h) (sqrt l)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -2.8e-133) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 + ((h * (-0.5 * pow((M * (D * (0.5 / d))), 2.0))) / l)));
	} else if (d <= -1e-309) {
		tmp = (d * sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 * ((h / l) * pow((D * (M * (0.5 / d))), 2.0))));
	} else {
		tmp = fma((h / l), (-0.5 * pow((M * (D / (d * 2.0))), 2.0)), 1.0) * (d / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -2.8e-133)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(Float64(h * Float64(-0.5 * (Float64(M * Float64(D * Float64(0.5 / d))) ^ 2.0))) / l))));
	elseif (d <= -1e-309)
		tmp = Float64(Float64(d * sqrt(Float64(Float64(1.0 / h) / l))) * Float64(-1.0 - Float64(-0.5 * Float64(Float64(h / l) * (Float64(D * Float64(M * Float64(0.5 / d))) ^ 2.0)))));
	else
		tmp = Float64(fma(Float64(h / l), Float64(-0.5 * (Float64(M * Float64(D / Float64(d * 2.0))) ^ 2.0)), 1.0) * Float64(d / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -2.8e-133], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h * N[(-0.5 * N[Power[N[(M * N[(D * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1e-309], N[(N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(M * N[(D / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.7999999999999999e-133

   1. Initial program 85.9%

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

   3. Simplified 88.0%

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

      1. associate-*l/ 90.4%

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

   6. Applied egg-rr 90.4%

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

   7. Taylor expanded in M around 0 89.4%

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

      1. *-commutative 89.4%

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

      2. associate-*l/ 89.4%

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

      3. associate-*r/ 89.4%

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

      4. associate-*r* 90.2%

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

      5. *-commutative 90.2%

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

      6. associate-*l* 90.4%

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

   9. Simplified 90.4%

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

   if -2.7999999999999999e-133 < d < -1.000000000000002e-309

   1. Initial program 43.3%

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

   3. Simplified 43.1%

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

      1. add-sqr-sqrt 43.2%

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

      2. pow2 43.2%

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

      3. sqrt-prod 43.1%

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

      4. unpow2 43.1%

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

      5. sqrt-prod 19.7%

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

      6. add-sqr-sqrt 43.3%

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

      7. div-inv 43.3%

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

      8. metadata-eval 43.3%

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

   6. Applied egg-rr 43.3%

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

      1. *-commutative 43.3%

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

      2. /-rgt-identity 43.3%

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

      3. associate-/l* 43.3%

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

      4. metadata-eval 43.3%

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

      5. times-frac 43.5%

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

      6. associate-*r/ 43.3%

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

      7. associate-*l* 43.3%

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

      8. *-lft-identity 43.3%

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

      9. associate-/l* 43.3%

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

      10. associate-/r/ 43.3%

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

      11. *-commutative 43.3%

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

      12. associate-/r* 43.3%

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

      13. metadata-eval 43.3%

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

   8. Simplified 43.3%

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

      1. expm1-log1p-u 12.7%

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

      2. expm1-udef 5.0%

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

   10. Applied egg-rr 5.1%

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

       1. expm1-def 8.4%

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

       2. expm1-log1p 31.8%

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

       3. rem-log-exp 29.2%

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

       4. rem-log-exp 31.8%

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

       5. *-commutative 31.8%

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

   12. Simplified 31.8%

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

   13. Taylor expanded in d around -inf 68.1%

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

       1. mul-1-neg 68.1%

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

       2. *-commutative 68.1%

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

       3. distribute-rgt-neg-in 68.1%

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

       4. associate-/r* 68.2%

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

   15. Simplified 68.2%

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

   if -1.000000000000002e-309 < d

   1. Initial program 67.2%

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

   3. Simplified 66.5%

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

      1. expm1-log1p-u 35.3%

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

      2. expm1-udef 24.5%

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

   6. Applied egg-rr 30.1%

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

      1. expm1-def 42.5%

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

      2. expm1-log1p 81.8%

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

      3. *-commutative 81.8%

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

   8. Simplified 81.8%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.8 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}, 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{if}\;d \leq -6.5 \cdot 10^{-135}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 4 \cdot 10^{-304}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{-267}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(\frac{{D}^{2}}{d} \cdot {M}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (sqrt (/ d h))
          (*
           (sqrt (/ d l))
           (+ 1.0 (/ (* h (* -0.5 (pow (* M (* D (/ 0.5 d))) 2.0))) l))))))
   (if (<= d -6.5e-135)
     t_0
     (if (<= d 4e-304)
       (*
        (* d (sqrt (/ (/ 1.0 h) l)))
        (- -1.0 (* -0.5 (* (/ h l) (pow (* D (* M (/ 0.5 d))) 2.0)))))
       (if (<= d 2.8e-267)
         (*
          (sqrt (/ h (pow l 3.0)))
          (* -0.125 (* (/ (pow D 2.0) d) (pow M 2.0))))
         t_0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h)) * (sqrt((d / l)) * (1.0 + ((h * (-0.5 * pow((M * (D * (0.5 / d))), 2.0))) / l)));
	double tmp;
	if (d <= -6.5e-135) {
		tmp = t_0;
	} else if (d <= 4e-304) {
		tmp = (d * sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 * ((h / l) * pow((D * (M * (0.5 / d))), 2.0))));
	} else if (d <= 2.8e-267) {
		tmp = sqrt((h / pow(l, 3.0))) * (-0.125 * ((pow(D, 2.0) / d) * pow(M, 2.0)));
	} else {
		tmp = 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((d / h)) * (sqrt((d / l)) * (1.0d0 + ((h * ((-0.5d0) * ((m * (d_1 * (0.5d0 / d))) ** 2.0d0))) / l)))
    if (d <= (-6.5d-135)) then
        tmp = t_0
    else if (d <= 4d-304) then
        tmp = (d * sqrt(((1.0d0 / h) / l))) * ((-1.0d0) - ((-0.5d0) * ((h / l) * ((d_1 * (m * (0.5d0 / d))) ** 2.0d0))))
    else if (d <= 2.8d-267) then
        tmp = sqrt((h / (l ** 3.0d0))) * ((-0.125d0) * (((d_1 ** 2.0d0) / d) * (m ** 2.0d0)))
    else
        tmp = 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((d / h)) * (Math.sqrt((d / l)) * (1.0 + ((h * (-0.5 * Math.pow((M * (D * (0.5 / d))), 2.0))) / l)));
	double tmp;
	if (d <= -6.5e-135) {
		tmp = t_0;
	} else if (d <= 4e-304) {
		tmp = (d * Math.sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 * ((h / l) * Math.pow((D * (M * (0.5 / d))), 2.0))));
	} else if (d <= 2.8e-267) {
		tmp = Math.sqrt((h / Math.pow(l, 3.0))) * (-0.125 * ((Math.pow(D, 2.0) / d) * Math.pow(M, 2.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 + ((h * (-0.5 * math.pow((M * (D * (0.5 / d))), 2.0))) / l)))
	tmp = 0
	if d <= -6.5e-135:
		tmp = t_0
	elif d <= 4e-304:
		tmp = (d * math.sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 * ((h / l) * math.pow((D * (M * (0.5 / d))), 2.0))))
	elif d <= 2.8e-267:
		tmp = math.sqrt((h / math.pow(l, 3.0))) * (-0.125 * ((math.pow(D, 2.0) / d) * math.pow(M, 2.0)))
	else:
		tmp = t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(Float64(h * Float64(-0.5 * (Float64(M * Float64(D * Float64(0.5 / d))) ^ 2.0))) / l))))
	tmp = 0.0
	if (d <= -6.5e-135)
		tmp = t_0;
	elseif (d <= 4e-304)
		tmp = Float64(Float64(d * sqrt(Float64(Float64(1.0 / h) / l))) * Float64(-1.0 - Float64(-0.5 * Float64(Float64(h / l) * (Float64(D * Float64(M * Float64(0.5 / d))) ^ 2.0)))));
	elseif (d <= 2.8e-267)
		tmp = Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(-0.125 * Float64(Float64((D ^ 2.0) / d) * (M ^ 2.0))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / h)) * (sqrt((d / l)) * (1.0 + ((h * (-0.5 * ((M * (D * (0.5 / d))) ^ 2.0))) / l)));
	tmp = 0.0;
	if (d <= -6.5e-135)
		tmp = t_0;
	elseif (d <= 4e-304)
		tmp = (d * sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 * ((h / l) * ((D * (M * (0.5 / d))) ^ 2.0))));
	elseif (d <= 2.8e-267)
		tmp = sqrt((h / (l ^ 3.0))) * (-0.125 * (((D ^ 2.0) / d) * (M ^ 2.0)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h * N[(-0.5 * N[Power[N[(M * N[(D * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -6.5e-135], t$95$0, If[LessEqual[d, 4e-304], N[(N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.8e-267], N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.125 * N[(N[(N[Power[D, 2.0], $MachinePrecision] / d), $MachinePrecision] * N[Power[M, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -6.50000000000000056e-135 or 2.80000000000000004e-267 < d

   1. Initial program 77.9%

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

   3. Simplified 78.4%

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

      1. associate-*l/ 80.6%

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

   6. Applied egg-rr 80.6%

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

   7. Taylor expanded in M around 0 80.6%

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

      1. *-commutative 80.6%

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

      2. associate-*l/ 80.6%

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

      3. associate-*r/ 80.6%

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

      4. associate-*r* 80.5%

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

      5. *-commutative 80.5%

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

      6. associate-*l* 80.6%

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

   9. Simplified 80.6%

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

   if -6.50000000000000056e-135 < d < 3.99999999999999988e-304

   1. Initial program 42.3%

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

   3. Simplified 42.1%

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

      1. add-sqr-sqrt 42.2%

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

      2. pow2 42.2%

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

      3. sqrt-prod 42.2%

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

      4. unpow2 42.2%

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

      5. sqrt-prod 19.3%

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

      6. add-sqr-sqrt 42.3%

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

      7. div-inv 42.3%

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

      8. metadata-eval 42.3%

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

   6. Applied egg-rr 42.3%

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

      1. *-commutative 42.3%

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

      2. /-rgt-identity 42.3%

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

      3. associate-/l* 42.3%

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

      4. metadata-eval 42.3%

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

      5. times-frac 42.5%

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

      6. associate-*r/ 42.3%

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

      7. associate-*l* 42.3%

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

      8. *-lft-identity 42.3%

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

      9. associate-/l* 42.3%

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

      10. associate-/r/ 42.3%

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

      11. *-commutative 42.3%

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

      12. associate-/r* 42.3%

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

      13. metadata-eval 42.3%

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

   8. Simplified 42.3%

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

      1. expm1-log1p-u 12.4%

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

      2. expm1-udef 4.9%

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

   10. Applied egg-rr 4.9%

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

       1. expm1-def 8.2%

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

       2. expm1-log1p 31.1%

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

       3. rem-log-exp 28.6%

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

       4. rem-log-exp 31.1%

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

       5. *-commutative 31.1%

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

   12. Simplified 31.1%

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

   13. Taylor expanded in d around -inf 66.5%

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

       1. mul-1-neg 66.5%

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

       2. *-commutative 66.5%

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

       3. distribute-rgt-neg-in 66.5%

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

       4. associate-/r* 66.6%

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

   15. Simplified 66.6%

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

   if 3.99999999999999988e-304 < d < 2.80000000000000004e-267

   1. Initial program 30.7%

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

   3. Simplified 30.7%

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

   5. Taylor expanded in d around 0 70.7%

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

      1. associate-*r* 70.7%

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

      2. *-commutative 70.7%

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

      3. associate-/l* 70.7%

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

      4. associate-/r/ 70.7%

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

   7. Simplified 70.7%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.5 \cdot 10^{-135}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq 4 \cdot 10^{-304}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{-267}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(\frac{{D}^{2}}{d} \cdot {M}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}\right)}{\ell}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)\\ \mathbf{if}\;\ell \leq -1.75 \cdot 10^{+148}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -1.9 \cdot 10^{-280}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 2.9 \cdot 10^{+51}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (sqrt (/ d h))
          (*
           (sqrt (/ d l))
           (+ 1.0 (* (/ h l) (* -0.5 (pow (* (/ D d) (/ M 2.0)) 2.0))))))))
   (if (<= l -1.75e+148)
     t_0
     (if (<= l -1.9e-280)
       (*
        (* d (sqrt (/ (/ 1.0 h) l)))
        (- -1.0 (* -0.5 (* (/ h l) (pow (* D (* M (/ 0.5 d))) 2.0)))))
       (if (<= l 2.9e+51) t_0 (* d (* (pow l -0.5) (pow h -0.5))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h)) * (sqrt((d / l)) * (1.0 + ((h / l) * (-0.5 * pow(((D / d) * (M / 2.0)), 2.0)))));
	double tmp;
	if (l <= -1.75e+148) {
		tmp = t_0;
	} else if (l <= -1.9e-280) {
		tmp = (d * sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 * ((h / l) * pow((D * (M * (0.5 / d))), 2.0))));
	} else if (l <= 2.9e+51) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 + ((h / l) * ((-0.5d0) * (((d_1 / d) * (m / 2.0d0)) ** 2.0d0)))))
    if (l <= (-1.75d+148)) then
        tmp = t_0
    else if (l <= (-1.9d-280)) then
        tmp = (d * sqrt(((1.0d0 / h) / l))) * ((-1.0d0) - ((-0.5d0) * ((h / l) * ((d_1 * (m * (0.5d0 / d))) ** 2.0d0))))
    else if (l <= 2.9d+51) then
        tmp = t_0
    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 t_0 = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 + ((h / l) * (-0.5 * Math.pow(((D / d) * (M / 2.0)), 2.0)))));
	double tmp;
	if (l <= -1.75e+148) {
		tmp = t_0;
	} else if (l <= -1.9e-280) {
		tmp = (d * Math.sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 * ((h / l) * Math.pow((D * (M * (0.5 / d))), 2.0))));
	} else if (l <= 2.9e+51) {
		tmp = t_0;
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 + ((h / l) * (-0.5 * math.pow(((D / d) * (M / 2.0)), 2.0)))))
	tmp = 0
	if l <= -1.75e+148:
		tmp = t_0
	elif l <= -1.9e-280:
		tmp = (d * math.sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 * ((h / l) * math.pow((D * (M * (0.5 / d))), 2.0))))
	elif l <= 2.9e+51:
		tmp = t_0
	else:
		tmp = d * (math.pow(l, -0.5) * math.pow(h, -0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(Float64(D / d) * Float64(M / 2.0)) ^ 2.0))))))
	tmp = 0.0
	if (l <= -1.75e+148)
		tmp = t_0;
	elseif (l <= -1.9e-280)
		tmp = Float64(Float64(d * sqrt(Float64(Float64(1.0 / h) / l))) * Float64(-1.0 - Float64(-0.5 * Float64(Float64(h / l) * (Float64(D * Float64(M * Float64(0.5 / d))) ^ 2.0)))));
	elseif (l <= 2.9e+51)
		tmp = t_0;
	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)
	t_0 = sqrt((d / h)) * (sqrt((d / l)) * (1.0 + ((h / l) * (-0.5 * (((D / d) * (M / 2.0)) ^ 2.0)))));
	tmp = 0.0;
	if (l <= -1.75e+148)
		tmp = t_0;
	elseif (l <= -1.9e-280)
		tmp = (d * sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 * ((h / l) * ((D * (M * (0.5 / d))) ^ 2.0))));
	elseif (l <= 2.9e+51)
		tmp = t_0;
	else
		tmp = d * ((l ^ -0.5) * (h ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = 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[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.75e+148], t$95$0, If[LessEqual[l, -1.9e-280], N[(N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.9e+51], t$95$0, 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 l < -1.7499999999999999e148 or -1.9000000000000001e-280 < l < 2.8999999999999998e51

   1. Initial program 74.1%

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

   3. Simplified 74.1%

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

   if -1.7499999999999999e148 < l < -1.9000000000000001e-280

   1. Initial program 73.5%

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

   3. Simplified 74.4%

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

      1. add-sqr-sqrt 74.4%

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

      2. pow2 74.4%

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

      3. sqrt-prod 74.4%

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

      4. unpow2 74.4%

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

      5. sqrt-prod 46.4%

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

      6. add-sqr-sqrt 74.4%

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

      7. div-inv 74.4%

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

      8. metadata-eval 74.4%

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

   6. Applied egg-rr 74.4%

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

      1. *-commutative 74.4%

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

      2. /-rgt-identity 74.4%

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

      3. associate-/l* 74.4%

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

      4. metadata-eval 74.4%

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

      5. times-frac 74.5%

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

      6. associate-*r/ 75.4%

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

      7. associate-*l* 75.4%

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

      8. *-lft-identity 75.4%

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

      9. associate-/l* 75.4%

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

      10. associate-/r/ 75.4%

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

      11. *-commutative 75.4%

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

      12. associate-/r* 75.4%

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

      13. metadata-eval 75.4%

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

   8. Simplified 75.4%

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

      1. expm1-log1p-u 41.2%

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

      2. expm1-udef 31.7%

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

   10. Applied egg-rr 29.0%

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

       1. expm1-def 36.5%

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

       2. expm1-log1p 66.0%

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

       3. rem-log-exp 58.8%

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

       4. rem-log-exp 66.0%

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

       5. *-commutative 66.0%

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

   12. Simplified 66.0%

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

   13. Taylor expanded in d around -inf 85.2%

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

       1. mul-1-neg 85.2%

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

       2. *-commutative 85.2%

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

       3. distribute-rgt-neg-in 85.2%

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

       4. associate-/r* 85.2%

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

   15. Simplified 85.2%

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

   if 2.8999999999999998e51 < l

   1. Initial program 49.4%

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

   3. Simplified 49.4%

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

      1. add-sqr-sqrt 49.4%

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

      2. pow2 49.4%

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

      3. sqrt-prod 49.4%

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

      4. unpow2 49.4%

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

      5. sqrt-prod 32.5%

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

      6. add-sqr-sqrt 49.8%

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

      7. div-inv 49.8%

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

      8. metadata-eval 49.8%

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

   6. Applied egg-rr 49.8%

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

      1. *-commutative 49.8%

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

      2. /-rgt-identity 49.8%

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

      3. associate-/l* 49.8%

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

      4. metadata-eval 49.8%

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

      5. times-frac 52.2%

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

      6. associate-*r/ 52.0%

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

      7. associate-*l* 52.0%

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

      8. *-lft-identity 52.0%

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

      9. associate-/l* 52.0%

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

      10. associate-/r/ 52.0%

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

      11. *-commutative 52.0%

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

      12. associate-/r* 52.0%

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

      13. metadata-eval 52.0%

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

   8. Simplified 52.0%

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

   9. Taylor expanded in d around inf 55.1%

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

       1. unpow-1 55.1%

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

       2. metadata-eval 55.1%

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

       3. pow-sqr 55.1%

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

       4. rem-sqrt-square 55.1%

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

       5. rem-square-sqrt 54.9%

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

       6. fabs-sqr 54.9%

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

       7. rem-square-sqrt 55.1%

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

   11. Simplified 55.1%

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

       1. *-commutative 55.1%

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

       2. unpow-prod-down 61.7%

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

   13. Applied egg-rr 61.7%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.75 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -1.9 \cdot 10^{-280}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 2.9 \cdot 10^{+51}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -7.2 \cdot 10^{-138} \lor \neg \left(d \leq 4 \cdot 10^{-304}\right):\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (or (<= d -7.2e-138) (not (<= d 4e-304)))
   (*
    (sqrt (/ d h))
    (*
     (sqrt (/ d l))
     (+ 1.0 (/ (* h (* -0.5 (pow (* M (* D (/ 0.5 d))) 2.0))) l))))
   (*
    (* d (sqrt (/ (/ 1.0 h) l)))
    (- -1.0 (* -0.5 (* (/ h l) (pow (* D (* M (/ 0.5 d))) 2.0)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if ((d <= -7.2e-138) || !(d <= 4e-304)) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 + ((h * (-0.5 * pow((M * (D * (0.5 / d))), 2.0))) / l)));
	} else {
		tmp = (d * sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 * ((h / l) * pow((D * (M * (0.5 / d))), 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if ((d <= (-7.2d-138)) .or. (.not. (d <= 4d-304))) then
        tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 + ((h * ((-0.5d0) * ((m * (d_1 * (0.5d0 / d))) ** 2.0d0))) / l)))
    else
        tmp = (d * sqrt(((1.0d0 / h) / l))) * ((-1.0d0) - ((-0.5d0) * ((h / l) * ((d_1 * (m * (0.5d0 / d))) ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if ((d <= -7.2e-138) || !(d <= 4e-304)) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 + ((h * (-0.5 * Math.pow((M * (D * (0.5 / d))), 2.0))) / l)));
	} else {
		tmp = (d * Math.sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 * ((h / l) * Math.pow((D * (M * (0.5 / d))), 2.0))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if (d <= -7.2e-138) or not (d <= 4e-304):
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 + ((h * (-0.5 * math.pow((M * (D * (0.5 / d))), 2.0))) / l)))
	else:
		tmp = (d * math.sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 * ((h / l) * math.pow((D * (M * (0.5 / d))), 2.0))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if ((d <= -7.2e-138) || !(d <= 4e-304))
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(Float64(h * Float64(-0.5 * (Float64(M * Float64(D * Float64(0.5 / d))) ^ 2.0))) / l))));
	else
		tmp = Float64(Float64(d * sqrt(Float64(Float64(1.0 / h) / l))) * Float64(-1.0 - Float64(-0.5 * Float64(Float64(h / l) * (Float64(D * Float64(M * Float64(0.5 / d))) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if ((d <= -7.2e-138) || ~((d <= 4e-304)))
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 + ((h * (-0.5 * ((M * (D * (0.5 / d))) ^ 2.0))) / l)));
	else
		tmp = (d * sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 * ((h / l) * ((D * (M * (0.5 / d))) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[Or[LessEqual[d, -7.2e-138], N[Not[LessEqual[d, 4e-304]], $MachinePrecision]], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h * N[(-0.5 * N[Power[N[(M * N[(D * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -7.20000000000000036e-138 or 3.99999999999999988e-304 < d

   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 76.1%

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

      1. associate-*l/ 78.2%

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

   6. Applied egg-rr 78.2%

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

   7. Taylor expanded in M around 0 78.2%

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

      1. *-commutative 78.2%

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

      2. associate-*l/ 78.2%

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

      3. associate-*r/ 78.2%

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

      4. associate-*r* 78.2%

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

      5. *-commutative 78.2%

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

      6. associate-*l* 78.2%

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

   9. Simplified 78.2%

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

   if -7.20000000000000036e-138 < d < 3.99999999999999988e-304

   1. Initial program 42.3%

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

   3. Simplified 42.1%

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

      1. add-sqr-sqrt 42.2%

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

      2. pow2 42.2%

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

      3. sqrt-prod 42.2%

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

      4. unpow2 42.2%

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

      5. sqrt-prod 19.3%

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

      6. add-sqr-sqrt 42.3%

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

      7. div-inv 42.3%

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

      8. metadata-eval 42.3%

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

   6. Applied egg-rr 42.3%

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

      1. *-commutative 42.3%

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

      2. /-rgt-identity 42.3%

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

      3. associate-/l* 42.3%

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

      4. metadata-eval 42.3%

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

      5. times-frac 42.5%

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

      6. associate-*r/ 42.3%

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

      7. associate-*l* 42.3%

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

      8. *-lft-identity 42.3%

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

      9. associate-/l* 42.3%

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

      10. associate-/r/ 42.3%

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

      11. *-commutative 42.3%

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

      12. associate-/r* 42.3%

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

      13. metadata-eval 42.3%

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

   8. Simplified 42.3%

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

      1. expm1-log1p-u 12.4%

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

      2. expm1-udef 4.9%

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

   10. Applied egg-rr 4.9%

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

       1. expm1-def 8.2%

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

       2. expm1-log1p 31.1%

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

       3. rem-log-exp 28.6%

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

       4. rem-log-exp 31.1%

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

       5. *-commutative 31.1%

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

   12. Simplified 31.1%

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

   13. Taylor expanded in d around -inf 66.5%

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

       1. mul-1-neg 66.5%

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

       2. *-commutative 66.5%

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

       3. distribute-rgt-neg-in 66.5%

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

       4. associate-/r* 66.6%

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

   15. Simplified 66.6%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.2 \cdot 10^{-138} \lor \neg \left(d \leq 4 \cdot 10^{-304}\right):\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.25 \cdot 10^{+200}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -2.6 \cdot 10^{-259}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 8.5 \cdot 10^{+50}:\\ \;\;\;\;\left(1 + -0.5 \cdot \frac{h \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}}{\ell}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1.25e+200)
   (* (/ (sqrt (- d)) (sqrt (- l))) (sqrt (/ d h)))
   (if (<= l -2.6e-259)
     (*
      (* d (sqrt (/ (/ 1.0 h) l)))
      (- -1.0 (* -0.5 (* (/ h l) (pow (* D (* M (/ 0.5 d))) 2.0)))))
     (if (<= l 8.5e+50)
       (*
        (+ 1.0 (* -0.5 (/ (* h (pow (* D (/ (* M 0.5) d)) 2.0)) l)))
        (sqrt (* (/ d l) (/ d h))))
       (* 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 (l <= -1.25e+200) {
		tmp = (sqrt(-d) / sqrt(-l)) * sqrt((d / h));
	} else if (l <= -2.6e-259) {
		tmp = (d * sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 * ((h / l) * pow((D * (M * (0.5 / d))), 2.0))));
	} else if (l <= 8.5e+50) {
		tmp = (1.0 + (-0.5 * ((h * pow((D * ((M * 0.5) / d)), 2.0)) / l))) * sqrt(((d / l) * (d / h)));
	} 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 (l <= (-1.25d+200)) then
        tmp = (sqrt(-d) / sqrt(-l)) * sqrt((d / h))
    else if (l <= (-2.6d-259)) then
        tmp = (d * sqrt(((1.0d0 / h) / l))) * ((-1.0d0) - ((-0.5d0) * ((h / l) * ((d_1 * (m * (0.5d0 / d))) ** 2.0d0))))
    else if (l <= 8.5d+50) then
        tmp = (1.0d0 + ((-0.5d0) * ((h * ((d_1 * ((m * 0.5d0) / d)) ** 2.0d0)) / l))) * sqrt(((d / l) * (d / h)))
    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 (l <= -1.25e+200) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-l)) * Math.sqrt((d / h));
	} else if (l <= -2.6e-259) {
		tmp = (d * Math.sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 * ((h / l) * Math.pow((D * (M * (0.5 / d))), 2.0))));
	} else if (l <= 8.5e+50) {
		tmp = (1.0 + (-0.5 * ((h * Math.pow((D * ((M * 0.5) / d)), 2.0)) / l))) * Math.sqrt(((d / l) * (d / h)));
	} 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 l <= -1.25e+200:
		tmp = (math.sqrt(-d) / math.sqrt(-l)) * math.sqrt((d / h))
	elif l <= -2.6e-259:
		tmp = (d * math.sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 * ((h / l) * math.pow((D * (M * (0.5 / d))), 2.0))))
	elif l <= 8.5e+50:
		tmp = (1.0 + (-0.5 * ((h * math.pow((D * ((M * 0.5) / d)), 2.0)) / l))) * math.sqrt(((d / l) * (d / h)))
	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 (l <= -1.25e+200)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))) * sqrt(Float64(d / h)));
	elseif (l <= -2.6e-259)
		tmp = Float64(Float64(d * sqrt(Float64(Float64(1.0 / h) / l))) * Float64(-1.0 - Float64(-0.5 * Float64(Float64(h / l) * (Float64(D * Float64(M * Float64(0.5 / d))) ^ 2.0)))));
	elseif (l <= 8.5e+50)
		tmp = Float64(Float64(1.0 + Float64(-0.5 * Float64(Float64(h * (Float64(D * Float64(Float64(M * 0.5) / d)) ^ 2.0)) / l))) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	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 (l <= -1.25e+200)
		tmp = (sqrt(-d) / sqrt(-l)) * sqrt((d / h));
	elseif (l <= -2.6e-259)
		tmp = (d * sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 * ((h / l) * ((D * (M * (0.5 / d))) ^ 2.0))));
	elseif (l <= 8.5e+50)
		tmp = (1.0 + (-0.5 * ((h * ((D * ((M * 0.5) / d)) ^ 2.0)) / l))) * sqrt(((d / l) * (d / h)));
	else
		tmp = d * ((l ^ -0.5) * (h ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1.25e+200], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2.6e-259], N[(N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 8.5e+50], N[(N[(1.0 + N[(-0.5 * N[(N[(h * N[Power[N[(D * N[(N[(M * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.25000000000000005e200

   1. Initial program 53.8%

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

   3. Simplified 57.9%

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

   5. Taylor expanded in h around 0 41.8%

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

      1. frac-2neg 53.9%

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

      2. sqrt-div 65.5%

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

   7. Applied egg-rr 52.8%

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

   if -1.25000000000000005e200 < l < -2.60000000000000001e-259

   1. Initial program 73.6%

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

   3. Simplified 74.5%

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

      1. add-sqr-sqrt 74.5%

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

      2. pow2 74.5%

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

      3. sqrt-prod 74.5%

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

      4. unpow2 74.5%

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

      5. sqrt-prod 45.7%

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

      6. add-sqr-sqrt 74.5%

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

      7. div-inv 74.5%

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

      8. metadata-eval 74.5%

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

   6. Applied egg-rr 74.5%

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

      1. *-commutative 74.5%

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

      2. /-rgt-identity 74.5%

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

      3. associate-/l* 74.5%

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

      4. metadata-eval 74.5%

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

      5. times-frac 74.6%

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

      6. associate-*r/ 75.4%

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

      7. associate-*l* 75.4%

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

      8. *-lft-identity 75.4%

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

      9. associate-/l* 75.4%

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

      10. associate-/r/ 75.4%

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

      11. *-commutative 75.4%

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

      12. associate-/r* 75.4%

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

      13. metadata-eval 75.4%

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

   8. Simplified 75.4%

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

      1. expm1-log1p-u 43.7%

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

      2. expm1-udef 30.2%

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

   10. Applied egg-rr 27.7%

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

       1. expm1-def 39.4%

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

       2. expm1-log1p 66.7%

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

       3. rem-log-exp 60.0%

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

       4. rem-log-exp 66.7%

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

       5. *-commutative 66.7%

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

   12. Simplified 66.7%

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

   13. Taylor expanded in d around -inf 82.5%

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

       1. mul-1-neg 82.5%

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

       2. *-commutative 82.5%

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

       3. distribute-rgt-neg-in 82.5%

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

       4. associate-/r* 82.9%

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

   15. Simplified 82.9%

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

   if -2.60000000000000001e-259 < l < 8.49999999999999961e50

   1. Initial program 79.2%

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

   3. Simplified 78.2%

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

      1. add-sqr-sqrt 78.2%

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

      2. pow2 78.2%

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

      3. sqrt-prod 78.2%

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

      4. unpow2 78.2%

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

      5. sqrt-prod 45.4%

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

      6. add-sqr-sqrt 78.3%

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

      7. div-inv 78.3%

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

      8. metadata-eval 78.3%

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

   6. Applied egg-rr 78.3%

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

      1. *-commutative 78.3%

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

      2. /-rgt-identity 78.3%

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

      3. associate-/l* 78.3%

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

      4. metadata-eval 78.3%

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

      5. times-frac 79.3%

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

      6. associate-*r/ 78.2%

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

      7. associate-*l* 78.1%

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

      8. *-lft-identity 78.1%

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

      9. associate-/l* 78.2%

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

      10. associate-/r/ 78.2%

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

      11. *-commutative 78.2%

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

      12. associate-/r* 78.2%

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

      13. metadata-eval 78.2%

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

   8. Simplified 78.2%

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

      1. expm1-log1p-u 29.5%

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

      2. expm1-udef 25.3%

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

   10. Applied egg-rr 21.6%

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

       1. expm1-def 25.8%

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

       2. expm1-log1p 71.9%

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

       3. rem-log-exp 65.6%

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

       4. rem-log-exp 71.9%

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

       5. *-commutative 71.9%

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

   12. Simplified 71.9%

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

       1. associate-*l/ 73.2%

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

       2. *-commutative 73.2%

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

       3. associate-*r* 73.2%

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

       4. associate-*r* 73.2%

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

       5. *-commutative 73.2%

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

       6. associate-*r/ 73.2%

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

   14. Applied egg-rr 73.2%

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

   if 8.49999999999999961e50 < l

   1. Initial program 49.4%

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

   3. Simplified 49.4%

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

      1. add-sqr-sqrt 49.4%

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

      2. pow2 49.4%

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

      3. sqrt-prod 49.4%

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

      4. unpow2 49.4%

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

      5. sqrt-prod 32.5%

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

      6. add-sqr-sqrt 49.8%

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

      7. div-inv 49.8%

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

      8. metadata-eval 49.8%

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

   6. Applied egg-rr 49.8%

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

      1. *-commutative 49.8%

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

      2. /-rgt-identity 49.8%

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

      3. associate-/l* 49.8%

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

      4. metadata-eval 49.8%

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

      5. times-frac 52.2%

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

      6. associate-*r/ 52.0%

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

      7. associate-*l* 52.0%

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

      8. *-lft-identity 52.0%

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

      9. associate-/l* 52.0%

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

      10. associate-/r/ 52.0%

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

      11. *-commutative 52.0%

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

      12. associate-/r* 52.0%

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

      13. metadata-eval 52.0%

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

   8. Simplified 52.0%

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

   9. Taylor expanded in d around inf 55.1%

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

       1. unpow-1 55.1%

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

       2. metadata-eval 55.1%

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

       3. pow-sqr 55.1%

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

       4. rem-sqrt-square 55.1%

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

       5. rem-square-sqrt 54.9%

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

       6. fabs-sqr 54.9%

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

       7. rem-square-sqrt 55.1%

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

   11. Simplified 55.1%

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

       1. *-commutative 55.1%

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

       2. unpow-prod-down 61.7%

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

   13. Applied egg-rr 61.7%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.25 \cdot 10^{+200}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -2.6 \cdot 10^{-259}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 8.5 \cdot 10^{+50}:\\ \;\;\;\;\left(1 + -0.5 \cdot \frac{h \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}}{\ell}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + -0.5 \cdot \frac{h \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}}{\ell}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{if}\;\ell \leq -6.2 \cdot 10^{+155}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -2.55 \cdot 10^{-259}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 2.15 \cdot 10^{+51}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (+ 1.0 (* -0.5 (/ (* h (pow (* D (/ (* M 0.5) d)) 2.0)) l)))
          (sqrt (* (/ d l) (/ d h))))))
   (if (<= l -6.2e+155)
     t_0
     (if (<= l -2.55e-259)
       (*
        (* d (pow (* l h) -0.5))
        (- -1.0 (* -0.5 (* (/ h l) (pow (* D (* M (/ 0.5 d))) 2.0)))))
       (if (<= l 2.15e+51) t_0 (* d (* (pow l -0.5) (pow h -0.5))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (1.0 + (-0.5 * ((h * pow((D * ((M * 0.5) / d)), 2.0)) / l))) * sqrt(((d / l) * (d / h)));
	double tmp;
	if (l <= -6.2e+155) {
		tmp = t_0;
	} else if (l <= -2.55e-259) {
		tmp = (d * pow((l * h), -0.5)) * (-1.0 - (-0.5 * ((h / l) * pow((D * (M * (0.5 / d))), 2.0))));
	} else if (l <= 2.15e+51) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + ((-0.5d0) * ((h * ((d_1 * ((m * 0.5d0) / d)) ** 2.0d0)) / l))) * sqrt(((d / l) * (d / h)))
    if (l <= (-6.2d+155)) then
        tmp = t_0
    else if (l <= (-2.55d-259)) then
        tmp = (d * ((l * h) ** (-0.5d0))) * ((-1.0d0) - ((-0.5d0) * ((h / l) * ((d_1 * (m * (0.5d0 / d))) ** 2.0d0))))
    else if (l <= 2.15d+51) then
        tmp = t_0
    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 t_0 = (1.0 + (-0.5 * ((h * Math.pow((D * ((M * 0.5) / d)), 2.0)) / l))) * Math.sqrt(((d / l) * (d / h)));
	double tmp;
	if (l <= -6.2e+155) {
		tmp = t_0;
	} else if (l <= -2.55e-259) {
		tmp = (d * Math.pow((l * h), -0.5)) * (-1.0 - (-0.5 * ((h / l) * Math.pow((D * (M * (0.5 / d))), 2.0))));
	} else if (l <= 2.15e+51) {
		tmp = t_0;
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (1.0 + (-0.5 * ((h * math.pow((D * ((M * 0.5) / d)), 2.0)) / l))) * math.sqrt(((d / l) * (d / h)))
	tmp = 0
	if l <= -6.2e+155:
		tmp = t_0
	elif l <= -2.55e-259:
		tmp = (d * math.pow((l * h), -0.5)) * (-1.0 - (-0.5 * ((h / l) * math.pow((D * (M * (0.5 / d))), 2.0))))
	elif l <= 2.15e+51:
		tmp = t_0
	else:
		tmp = d * (math.pow(l, -0.5) * math.pow(h, -0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(1.0 + Float64(-0.5 * Float64(Float64(h * (Float64(D * Float64(Float64(M * 0.5) / d)) ^ 2.0)) / l))) * sqrt(Float64(Float64(d / l) * Float64(d / h))))
	tmp = 0.0
	if (l <= -6.2e+155)
		tmp = t_0;
	elseif (l <= -2.55e-259)
		tmp = Float64(Float64(d * (Float64(l * h) ^ -0.5)) * Float64(-1.0 - Float64(-0.5 * Float64(Float64(h / l) * (Float64(D * Float64(M * Float64(0.5 / d))) ^ 2.0)))));
	elseif (l <= 2.15e+51)
		tmp = t_0;
	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)
	t_0 = (1.0 + (-0.5 * ((h * ((D * ((M * 0.5) / d)) ^ 2.0)) / l))) * sqrt(((d / l) * (d / h)));
	tmp = 0.0;
	if (l <= -6.2e+155)
		tmp = t_0;
	elseif (l <= -2.55e-259)
		tmp = (d * ((l * h) ^ -0.5)) * (-1.0 - (-0.5 * ((h / l) * ((D * (M * (0.5 / d))) ^ 2.0))));
	elseif (l <= 2.15e+51)
		tmp = t_0;
	else
		tmp = d * ((l ^ -0.5) * (h ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(1.0 + N[(-0.5 * N[(N[(h * N[Power[N[(D * N[(N[(M * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -6.2e+155], t$95$0, If[LessEqual[l, -2.55e-259], N[(N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.15e+51], t$95$0, 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 l < -6.19999999999999978e155 or -2.5499999999999999e-259 < l < 2.1499999999999999e51

   1. Initial program 74.7%

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

   3. Simplified 74.8%

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

      1. add-sqr-sqrt 74.8%

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

      2. pow2 74.8%

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

      3. sqrt-prod 74.8%

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

      4. unpow2 74.8%

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

      5. sqrt-prod 44.6%

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

      6. add-sqr-sqrt 74.9%

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

      7. div-inv 74.9%

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

      8. metadata-eval 74.9%

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

   6. Applied egg-rr 74.9%

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

      1. *-commutative 74.9%

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

      2. /-rgt-identity 74.9%

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

      3. associate-/l* 74.9%

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

      4. metadata-eval 74.9%

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

      5. times-frac 74.9%

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

      6. associate-*r/ 74.8%

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

      7. associate-*l* 74.8%

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

      8. *-lft-identity 74.8%

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

      9. associate-/l* 74.8%

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

      10. associate-/r/ 74.8%

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

      11. *-commutative 74.8%

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

      12. associate-/r* 74.8%

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

      13. metadata-eval 74.8%

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

   8. Simplified 74.8%

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

      1. expm1-log1p-u 35.0%

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

      2. expm1-udef 23.6%

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

   10. Applied egg-rr 20.9%

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

       1. expm1-def 30.9%

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

       2. expm1-log1p 67.2%

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

       3. rem-log-exp 61.8%

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

       4. rem-log-exp 67.2%

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

       5. *-commutative 67.2%

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

   12. Simplified 67.2%

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

       1. associate-*l/ 68.2%

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

       2. *-commutative 68.2%

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

       3. associate-*r* 68.2%

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

       4. associate-*r* 68.2%

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

       5. *-commutative 68.2%

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

       6. associate-*r/ 68.2%

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

   14. Applied egg-rr 68.2%

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

   if -6.19999999999999978e155 < l < -2.5499999999999999e-259

   1. Initial program 72.6%

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

   3. Simplified 73.6%

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

      1. add-sqr-sqrt 73.6%

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

      2. pow2 73.6%

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

      3. sqrt-prod 73.6%

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

      4. unpow2 73.6%

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

      5. sqrt-prod 43.5%

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

      6. add-sqr-sqrt 73.6%

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

      7. div-inv 73.6%

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

      8. metadata-eval 73.6%

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

   6. Applied egg-rr 73.6%

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

      1. *-commutative 73.6%

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

      2. /-rgt-identity 73.6%

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

      3. associate-/l* 73.6%

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

      4. metadata-eval 73.6%

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

      5. times-frac 73.7%

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

      6. associate-*r/ 74.6%

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

      7. associate-*l* 74.6%

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

      8. *-lft-identity 74.6%

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

      9. associate-/l* 74.6%

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

      10. associate-/r/ 74.6%

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

      11. *-commutative 74.6%

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

      12. associate-/r* 74.6%

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

      13. metadata-eval 74.6%

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

   8. Simplified 74.6%

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

      1. expm1-log1p-u 41.4%

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

      2. expm1-udef 31.6%

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

   10. Applied egg-rr 28.8%

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

       1. expm1-def 36.6%

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

       2. expm1-log1p 64.9%

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

       3. rem-log-exp 57.4%

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

       4. rem-log-exp 64.9%

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

       5. *-commutative 64.9%

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

   12. Simplified 64.9%

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

   13. Taylor expanded in d around -inf 84.7%

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

       1. mul-1-neg 84.7%

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

       2. distribute-rgt-neg-in 84.7%

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

       3. unpow-1 84.7%

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

       4. metadata-eval 84.7%

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

       5. pow-sqr 84.7%

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

       6. rem-sqrt-square 84.7%

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

       7. rem-square-sqrt 84.5%

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

       8. fabs-sqr 84.5%

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

       9. rem-square-sqrt 84.7%

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

   15. Simplified 84.7%

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

   if 2.1499999999999999e51 < l

   1. Initial program 49.4%

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

   3. Simplified 49.4%

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

      1. add-sqr-sqrt 49.4%

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

      2. pow2 49.4%

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

      3. sqrt-prod 49.4%

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

      4. unpow2 49.4%

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

      5. sqrt-prod 32.5%

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

      6. add-sqr-sqrt 49.8%

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

      7. div-inv 49.8%

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

      8. metadata-eval 49.8%

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

   6. Applied egg-rr 49.8%

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

      1. *-commutative 49.8%

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

      2. /-rgt-identity 49.8%

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

      3. associate-/l* 49.8%

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

      4. metadata-eval 49.8%

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

      5. times-frac 52.2%

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

      6. associate-*r/ 52.0%

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

      7. associate-*l* 52.0%

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

      8. *-lft-identity 52.0%

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

      9. associate-/l* 52.0%

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

      10. associate-/r/ 52.0%

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

      11. *-commutative 52.0%

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

      12. associate-/r* 52.0%

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

      13. metadata-eval 52.0%

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

   8. Simplified 52.0%

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

   9. Taylor expanded in d around inf 55.1%

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

       1. unpow-1 55.1%

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

       2. metadata-eval 55.1%

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

       3. pow-sqr 55.1%

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

       4. rem-sqrt-square 55.1%

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

       5. rem-square-sqrt 54.9%

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

       6. fabs-sqr 54.9%

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

       7. rem-square-sqrt 55.1%

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

   11. Simplified 55.1%

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

       1. *-commutative 55.1%

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

       2. unpow-prod-down 61.7%

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

   13. Applied egg-rr 61.7%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.2 \cdot 10^{+155}:\\ \;\;\;\;\left(1 + -0.5 \cdot \frac{h \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}}{\ell}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -2.55 \cdot 10^{-259}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 2.15 \cdot 10^{+51}:\\ \;\;\;\;\left(1 + -0.5 \cdot \frac{h \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}}{\ell}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 66.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + -0.5 \cdot \frac{h \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}}{\ell}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{if}\;\ell \leq -2.9 \cdot 10^{+166}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -2.5 \cdot 10^{-259}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 3.1 \cdot 10^{+50}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (+ 1.0 (* -0.5 (/ (* h (pow (* D (/ (* M 0.5) d)) 2.0)) l)))
          (sqrt (* (/ d l) (/ d h))))))
   (if (<= l -2.9e+166)
     t_0
     (if (<= l -2.5e-259)
       (*
        (* d (sqrt (/ (/ 1.0 h) l)))
        (- -1.0 (* -0.5 (* (/ h l) (pow (* D (* M (/ 0.5 d))) 2.0)))))
       (if (<= l 3.1e+50) t_0 (* d (* (pow l -0.5) (pow h -0.5))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (1.0 + (-0.5 * ((h * pow((D * ((M * 0.5) / d)), 2.0)) / l))) * sqrt(((d / l) * (d / h)));
	double tmp;
	if (l <= -2.9e+166) {
		tmp = t_0;
	} else if (l <= -2.5e-259) {
		tmp = (d * sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 * ((h / l) * pow((D * (M * (0.5 / d))), 2.0))));
	} else if (l <= 3.1e+50) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + ((-0.5d0) * ((h * ((d_1 * ((m * 0.5d0) / d)) ** 2.0d0)) / l))) * sqrt(((d / l) * (d / h)))
    if (l <= (-2.9d+166)) then
        tmp = t_0
    else if (l <= (-2.5d-259)) then
        tmp = (d * sqrt(((1.0d0 / h) / l))) * ((-1.0d0) - ((-0.5d0) * ((h / l) * ((d_1 * (m * (0.5d0 / d))) ** 2.0d0))))
    else if (l <= 3.1d+50) then
        tmp = t_0
    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 t_0 = (1.0 + (-0.5 * ((h * Math.pow((D * ((M * 0.5) / d)), 2.0)) / l))) * Math.sqrt(((d / l) * (d / h)));
	double tmp;
	if (l <= -2.9e+166) {
		tmp = t_0;
	} else if (l <= -2.5e-259) {
		tmp = (d * Math.sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 * ((h / l) * Math.pow((D * (M * (0.5 / d))), 2.0))));
	} else if (l <= 3.1e+50) {
		tmp = t_0;
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (1.0 + (-0.5 * ((h * math.pow((D * ((M * 0.5) / d)), 2.0)) / l))) * math.sqrt(((d / l) * (d / h)))
	tmp = 0
	if l <= -2.9e+166:
		tmp = t_0
	elif l <= -2.5e-259:
		tmp = (d * math.sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 * ((h / l) * math.pow((D * (M * (0.5 / d))), 2.0))))
	elif l <= 3.1e+50:
		tmp = t_0
	else:
		tmp = d * (math.pow(l, -0.5) * math.pow(h, -0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(1.0 + Float64(-0.5 * Float64(Float64(h * (Float64(D * Float64(Float64(M * 0.5) / d)) ^ 2.0)) / l))) * sqrt(Float64(Float64(d / l) * Float64(d / h))))
	tmp = 0.0
	if (l <= -2.9e+166)
		tmp = t_0;
	elseif (l <= -2.5e-259)
		tmp = Float64(Float64(d * sqrt(Float64(Float64(1.0 / h) / l))) * Float64(-1.0 - Float64(-0.5 * Float64(Float64(h / l) * (Float64(D * Float64(M * Float64(0.5 / d))) ^ 2.0)))));
	elseif (l <= 3.1e+50)
		tmp = t_0;
	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)
	t_0 = (1.0 + (-0.5 * ((h * ((D * ((M * 0.5) / d)) ^ 2.0)) / l))) * sqrt(((d / l) * (d / h)));
	tmp = 0.0;
	if (l <= -2.9e+166)
		tmp = t_0;
	elseif (l <= -2.5e-259)
		tmp = (d * sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 * ((h / l) * ((D * (M * (0.5 / d))) ^ 2.0))));
	elseif (l <= 3.1e+50)
		tmp = t_0;
	else
		tmp = d * ((l ^ -0.5) * (h ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(1.0 + N[(-0.5 * N[(N[(h * N[Power[N[(D * N[(N[(M * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.9e+166], t$95$0, If[LessEqual[l, -2.5e-259], N[(N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.1e+50], t$95$0, 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 l < -2.9000000000000001e166 or -2.49999999999999989e-259 < l < 3.10000000000000003e50

   1. Initial program 74.3%

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

   3. Simplified 74.4%

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

      1. add-sqr-sqrt 74.4%

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

      2. pow2 74.4%

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

      3. sqrt-prod 74.4%

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

      4. unpow2 74.4%

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

      5. sqrt-prod 44.5%

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

      6. add-sqr-sqrt 74.5%

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

      7. div-inv 74.5%

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

      8. metadata-eval 74.5%

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

   6. Applied egg-rr 74.5%

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

      1. *-commutative 74.5%

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

      2. /-rgt-identity 74.5%

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

      3. associate-/l* 74.5%

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

      4. metadata-eval 74.5%

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

      5. times-frac 74.5%

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

      6. associate-*r/ 74.4%

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

      7. associate-*l* 74.3%

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

      8. *-lft-identity 74.3%

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

      9. associate-/l* 74.4%

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

      10. associate-/r/ 74.4%

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

      11. *-commutative 74.4%

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

      12. associate-/r* 74.4%

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

      13. metadata-eval 74.4%

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

   8. Simplified 74.4%

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

      1. expm1-log1p-u 33.9%

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

      2. expm1-udef 24.0%

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

   10. Applied egg-rr 21.2%

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

       1. expm1-def 29.7%

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

       2. expm1-log1p 66.6%

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

       3. rem-log-exp 61.2%

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

       4. rem-log-exp 66.6%

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

       5. *-commutative 66.6%

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

   12. Simplified 66.6%

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

       1. associate-*l/ 67.7%

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

       2. *-commutative 67.7%

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

       3. associate-*r* 67.7%

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

       4. associate-*r* 67.7%

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

       5. *-commutative 67.7%

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

       6. associate-*r/ 67.7%

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

   14. Applied egg-rr 67.7%

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

   if -2.9000000000000001e166 < l < -2.49999999999999989e-259

   1. Initial program 73.2%

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

   3. Simplified 74.2%

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

      1. add-sqr-sqrt 74.2%

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

      2. pow2 74.2%

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

      3. sqrt-prod 74.2%

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

      4. unpow2 74.2%

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

      5. sqrt-prod 43.6%

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

      6. add-sqr-sqrt 74.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. div-inv 74.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. metadata-eval 74.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 74.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   7. Step-by-step derivation

      1. *-commutative 74.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. /-rgt-identity 74.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \color{blue}{\frac{M \cdot 0.5}{1}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      3. associate-/l* 74.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \color{blue}{\frac{M}{\frac{1}{0.5}}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      4. metadata-eval 74.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \frac{M}{\color{blue}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. times-frac 74.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{D \cdot M}{d \cdot 2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. associate-*r/ 75.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{M}{d \cdot 2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. associate-*l* 75.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(D \cdot \left(\frac{M}{d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}}^{2}\right) \]

      8. *-lft-identity 75.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{\color{blue}{1 \cdot M}}{d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      9. associate-/l* 75.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\frac{1}{\frac{d \cdot 2}{M}}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      10. associate-/r/ 75.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\left(\frac{1}{d \cdot 2} \cdot M\right)} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      11. *-commutative 75.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\frac{1}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      12. associate-/r* 75.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      13. metadata-eval 75.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\frac{\color{blue}{0.5}}{d} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   8. Simplified 75.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(D \cdot \left(\left(\frac{0.5}{d} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}}\right) \]
   9. Step-by-step derivation

      1. expm1-log1p-u 42.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\frac{0.5}{d} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right)\right)\right)} \]

      2. expm1-udef 31.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\frac{0.5}{d} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right)\right)} - 1} \]

   10. Applied egg-rr 28.3%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left({\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 38.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left({\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

       2. expm1-log1p 65.7%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left({\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

       3. rem-log-exp 58.3%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\log \left(e^{1 + -0.5 \cdot \left({\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)}\right)} \]

       4. rem-log-exp 65.7%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(1 + -0.5 \cdot \left({\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

       5. *-commutative 65.7%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)}\right) \]

   12. Simplified 65.7%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)} \]

   13. Taylor expanded in d around -inf 85.0%

       \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right) \]
   14. Step-by-step derivation

       1. mul-1-neg 85.0%

          \[\leadsto \color{blue}{\left(-d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right) \]

       2. *-commutative 85.0%

          \[\leadsto \left(-\color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right) \]

       3. distribute-rgt-neg-in 85.0%

          \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\right)} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right) \]

       4. associate-/r* 85.1%

          \[\leadsto \left(\sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \cdot \left(-d\right)\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right) \]

   15. Simplified 85.1%

       \[\leadsto \color{blue}{\left(\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-d\right)\right)} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right) \]

   if 3.10000000000000003e50 < l

   1. Initial program 49.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 49.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 49.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 49.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 49.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. unpow2 49.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\sqrt{\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. sqrt-prod 32.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\sqrt{\frac{M}{2} \cdot \frac{D}{d}} \cdot \sqrt{\frac{M}{2} \cdot \frac{D}{d}}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. add-sqr-sqrt 49.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. div-inv 49.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. metadata-eval 49.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 49.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   7. Step-by-step derivation

      1. *-commutative 49.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. /-rgt-identity 49.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \color{blue}{\frac{M \cdot 0.5}{1}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      3. associate-/l* 49.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \color{blue}{\frac{M}{\frac{1}{0.5}}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      4. metadata-eval 49.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \frac{M}{\color{blue}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. times-frac 52.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{D \cdot M}{d \cdot 2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. associate-*r/ 52.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{M}{d \cdot 2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. associate-*l* 52.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(D \cdot \left(\frac{M}{d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}}^{2}\right) \]

      8. *-lft-identity 52.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{\color{blue}{1 \cdot M}}{d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      9. associate-/l* 52.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\frac{1}{\frac{d \cdot 2}{M}}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      10. associate-/r/ 52.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\left(\frac{1}{d \cdot 2} \cdot M\right)} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      11. *-commutative 52.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\frac{1}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      12. associate-/r* 52.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      13. metadata-eval 52.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\frac{\color{blue}{0.5}}{d} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   8. Simplified 52.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(D \cdot \left(\left(\frac{0.5}{d} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}}\right) \]

   9. Taylor expanded in d around inf 55.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. unpow-1 55.1%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

       2. metadata-eval 55.1%

          \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

       3. pow-sqr 55.1%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

       4. rem-sqrt-square 55.1%

          \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

       5. rem-square-sqrt 54.9%

          \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

       6. fabs-sqr 54.9%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

       7. rem-square-sqrt 55.1%

          \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   11. Simplified 55.1%

       \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
   12. Step-by-step derivation

       1. *-commutative 55.1%

          \[\leadsto d \cdot {\color{blue}{\left(\ell \cdot h\right)}}^{-0.5} \]

       2. unpow-prod-down 61.7%

          \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]

   13. Applied egg-rr 61.7%

       \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.9 \cdot 10^{+166}:\\ \;\;\;\;\left(1 + -0.5 \cdot \frac{h \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}}{\ell}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -2.5 \cdot 10^{-259}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 3.1 \cdot 10^{+50}:\\ \;\;\;\;\left(1 + -0.5 \cdot \frac{h \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}}{\ell}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 48.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 16000000000:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \frac{1}{\sqrt{\frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= D 16000000000.0)
   (* (sqrt (/ d l)) (/ 1.0 (sqrt (/ h d))))
   (*
    (+ 1.0 (* -0.5 (* (/ h l) (pow (* D (* M (/ 0.5 d))) 2.0))))
    (sqrt (* (/ d l) (/ d h))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (D <= 16000000000.0) {
		tmp = sqrt((d / l)) * (1.0 / sqrt((h / d)));
	} else {
		tmp = (1.0 + (-0.5 * ((h / l) * pow((D * (M * (0.5 / d))), 2.0)))) * sqrt(((d / l) * (d / 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 <= 16000000000.0d0) then
        tmp = sqrt((d / l)) * (1.0d0 / sqrt((h / d)))
    else
        tmp = (1.0d0 + ((-0.5d0) * ((h / l) * ((d_1 * (m * (0.5d0 / d))) ** 2.0d0)))) * sqrt(((d / l) * (d / 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 <= 16000000000.0) {
		tmp = Math.sqrt((d / l)) * (1.0 / Math.sqrt((h / d)));
	} else {
		tmp = (1.0 + (-0.5 * ((h / l) * Math.pow((D * (M * (0.5 / d))), 2.0)))) * Math.sqrt(((d / l) * (d / h)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if D <= 16000000000.0:
		tmp = math.sqrt((d / l)) * (1.0 / math.sqrt((h / d)))
	else:
		tmp = (1.0 + (-0.5 * ((h / l) * math.pow((D * (M * (0.5 / d))), 2.0)))) * math.sqrt(((d / l) * (d / h)))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (D <= 16000000000.0)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(1.0 / sqrt(Float64(h / d))));
	else
		tmp = Float64(Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(D * Float64(M * Float64(0.5 / d))) ^ 2.0)))) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (D <= 16000000000.0)
		tmp = sqrt((d / l)) * (1.0 / sqrt((h / d)));
	else
		tmp = (1.0 + (-0.5 * ((h / l) * ((D * (M * (0.5 / d))) ^ 2.0)))) * sqrt(((d / l) * (d / h)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[D, 16000000000.0], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 1.6e10

   1. Initial program 72.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 72.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(\frac{D}{\frac{2}{M} \cdot d}\right)}^{2} \cdot -0.5, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0 47.4%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]
   6. Step-by-step derivation

      1. clear-num 47.0%

         \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot 1\right) \]

      2. sqrt-div 47.7%

         \[\leadsto \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot 1\right) \]

      3. metadata-eval 47.7%

         \[\leadsto \frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot 1\right) \]

   7. Applied egg-rr 47.7%

      \[\leadsto \color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot 1\right) \]

   if 1.6e10 < D

   1. Initial program 63.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 62.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 62.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 62.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 62.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. unpow2 62.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\sqrt{\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. sqrt-prod 34.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\sqrt{\frac{M}{2} \cdot \frac{D}{d}} \cdot \sqrt{\frac{M}{2} \cdot \frac{D}{d}}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. add-sqr-sqrt 62.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. div-inv 62.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. metadata-eval 62.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 62.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   7. Step-by-step derivation

      1. *-commutative 62.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. /-rgt-identity 62.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \color{blue}{\frac{M \cdot 0.5}{1}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      3. associate-/l* 62.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \color{blue}{\frac{M}{\frac{1}{0.5}}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      4. metadata-eval 62.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \frac{M}{\color{blue}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. times-frac 63.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{D \cdot M}{d \cdot 2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. associate-*r/ 63.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{M}{d \cdot 2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. associate-*l* 63.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(D \cdot \left(\frac{M}{d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}}^{2}\right) \]

      8. *-lft-identity 63.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{\color{blue}{1 \cdot M}}{d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      9. associate-/l* 63.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\frac{1}{\frac{d \cdot 2}{M}}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      10. associate-/r/ 63.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\left(\frac{1}{d \cdot 2} \cdot M\right)} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      11. *-commutative 63.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\frac{1}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      12. associate-/r* 63.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      13. metadata-eval 63.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\frac{\color{blue}{0.5}}{d} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   8. Simplified 63.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(D \cdot \left(\left(\frac{0.5}{d} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}}\right) \]
   9. Step-by-step derivation

      1. expm1-log1p-u 21.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\frac{0.5}{d} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right)\right)\right)} \]

      2. expm1-udef 13.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\frac{0.5}{d} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right)\right)} - 1} \]

   10. Applied egg-rr 13.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left({\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 21.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left({\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

       2. expm1-log1p 60.3%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left({\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

       3. rem-log-exp 57.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\log \left(e^{1 + -0.5 \cdot \left({\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)}\right)} \]

       4. rem-log-exp 60.3%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(1 + -0.5 \cdot \left({\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

       5. *-commutative 60.3%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)}\right) \]

   12. Simplified 60.3%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 16000000000:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \frac{1}{\sqrt{\frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 48.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 5.5 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \frac{1}{\sqrt{\frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \frac{h \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}}{\ell}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= D 5.5e-10)
   (* (sqrt (/ d l)) (/ 1.0 (sqrt (/ h d))))
   (*
    (+ 1.0 (* -0.5 (/ (* h (pow (* D (/ (* M 0.5) d)) 2.0)) l)))
    (sqrt (* (/ d l) (/ d h))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (D <= 5.5e-10) {
		tmp = sqrt((d / l)) * (1.0 / sqrt((h / d)));
	} else {
		tmp = (1.0 + (-0.5 * ((h * pow((D * ((M * 0.5) / d)), 2.0)) / l))) * sqrt(((d / l) * (d / 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 <= 5.5d-10) then
        tmp = sqrt((d / l)) * (1.0d0 / sqrt((h / d)))
    else
        tmp = (1.0d0 + ((-0.5d0) * ((h * ((d_1 * ((m * 0.5d0) / d)) ** 2.0d0)) / l))) * sqrt(((d / l) * (d / 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 <= 5.5e-10) {
		tmp = Math.sqrt((d / l)) * (1.0 / Math.sqrt((h / d)));
	} else {
		tmp = (1.0 + (-0.5 * ((h * Math.pow((D * ((M * 0.5) / d)), 2.0)) / l))) * Math.sqrt(((d / l) * (d / h)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if D <= 5.5e-10:
		tmp = math.sqrt((d / l)) * (1.0 / math.sqrt((h / d)))
	else:
		tmp = (1.0 + (-0.5 * ((h * math.pow((D * ((M * 0.5) / d)), 2.0)) / l))) * math.sqrt(((d / l) * (d / h)))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (D <= 5.5e-10)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(1.0 / sqrt(Float64(h / d))));
	else
		tmp = Float64(Float64(1.0 + Float64(-0.5 * Float64(Float64(h * (Float64(D * Float64(Float64(M * 0.5) / d)) ^ 2.0)) / l))) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (D <= 5.5e-10)
		tmp = sqrt((d / l)) * (1.0 / sqrt((h / d)));
	else
		tmp = (1.0 + (-0.5 * ((h * ((D * ((M * 0.5) / d)) ^ 2.0)) / l))) * sqrt(((d / l) * (d / h)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[D, 5.5e-10], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(-0.5 * N[(N[(h * N[Power[N[(D * N[(N[(M * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 5.4999999999999996e-10

   1. Initial program 72.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 72.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(\frac{D}{\frac{2}{M} \cdot d}\right)}^{2} \cdot -0.5, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0 47.3%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]
   6. Step-by-step derivation

      1. clear-num 47.0%

         \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot 1\right) \]

      2. sqrt-div 47.6%

         \[\leadsto \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot 1\right) \]

      3. metadata-eval 47.6%

         \[\leadsto \frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot 1\right) \]

   7. Applied egg-rr 47.6%

      \[\leadsto \color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot 1\right) \]

   if 5.4999999999999996e-10 < D

   1. Initial program 64.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 62.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 62.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 62.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 62.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. unpow2 62.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\sqrt{\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. sqrt-prod 35.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\sqrt{\frac{M}{2} \cdot \frac{D}{d}} \cdot \sqrt{\frac{M}{2} \cdot \frac{D}{d}}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. add-sqr-sqrt 63.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. div-inv 63.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. metadata-eval 63.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 63.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   7. Step-by-step derivation

      1. *-commutative 63.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. /-rgt-identity 63.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \color{blue}{\frac{M \cdot 0.5}{1}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      3. associate-/l* 63.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \color{blue}{\frac{M}{\frac{1}{0.5}}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      4. metadata-eval 63.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \frac{M}{\color{blue}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. times-frac 64.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{D \cdot M}{d \cdot 2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. associate-*r/ 64.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{M}{d \cdot 2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. associate-*l* 64.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(D \cdot \left(\frac{M}{d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}}^{2}\right) \]

      8. *-lft-identity 64.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{\color{blue}{1 \cdot M}}{d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      9. associate-/l* 64.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\frac{1}{\frac{d \cdot 2}{M}}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      10. associate-/r/ 64.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\left(\frac{1}{d \cdot 2} \cdot M\right)} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      11. *-commutative 64.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\frac{1}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      12. associate-/r* 64.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      13. metadata-eval 64.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\frac{\color{blue}{0.5}}{d} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   8. Simplified 64.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(D \cdot \left(\left(\frac{0.5}{d} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}}\right) \]
   9. Step-by-step derivation

      1. expm1-log1p-u 23.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\frac{0.5}{d} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right)\right)\right)} \]

      2. expm1-udef 15.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\frac{0.5}{d} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right)\right)} - 1} \]

   10. Applied egg-rr 15.3%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left({\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 22.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left({\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

       2. expm1-log1p 59.7%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left({\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

       3. rem-log-exp 56.9%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\log \left(e^{1 + -0.5 \cdot \left({\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)}\right)} \]

       4. rem-log-exp 59.7%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(1 + -0.5 \cdot \left({\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

       5. *-commutative 59.7%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)}\right) \]

   12. Simplified 59.7%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*l/ 61.5%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \color{blue}{\frac{h \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}\right) \]

       2. *-commutative 61.5%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \frac{h \cdot {\color{blue}{\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D\right)}}^{2}}{\ell}\right) \]

       3. associate-*r* 60.3%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \frac{h \cdot {\color{blue}{\left(M \cdot \left(\frac{0.5}{d} \cdot D\right)\right)}}^{2}}{\ell}\right) \]

       4. associate-*r* 61.5%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \frac{h \cdot {\color{blue}{\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D\right)}}^{2}}{\ell}\right) \]

       5. *-commutative 61.5%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \frac{h \cdot {\color{blue}{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}}^{2}}{\ell}\right) \]

       6. associate-*r/ 61.5%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \frac{h \cdot {\left(D \cdot \color{blue}{\frac{M \cdot 0.5}{d}}\right)}^{2}}{\ell}\right) \]

   14. Applied egg-rr 61.5%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \color{blue}{\frac{h \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}}{\ell}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 5.5 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \frac{1}{\sqrt{\frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \frac{h \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}}{\ell}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 42.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -5.2 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;d \cdot {\left({\left(\ell \cdot h\right)}^{2}\right)}^{-0.25}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -5.2e-11)
   (sqrt (* (/ d l) (/ d h)))
   (if (<= d -1e-309)
     (* d (pow (pow (* l h) 2.0) -0.25))
     (* d (/ (pow h -0.5) (sqrt l))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -5.2e-11) {
		tmp = sqrt(((d / l) * (d / h)));
	} else if (d <= -1e-309) {
		tmp = d * pow(pow((l * h), 2.0), -0.25);
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= (-5.2d-11)) then
        tmp = sqrt(((d / l) * (d / h)))
    else if (d <= (-1d-309)) then
        tmp = d * (((l * h) ** 2.0d0) ** (-0.25d0))
    else
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -5.2e-11) {
		tmp = Math.sqrt(((d / l) * (d / h)));
	} else if (d <= -1e-309) {
		tmp = d * Math.pow(Math.pow((l * h), 2.0), -0.25);
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= -5.2e-11:
		tmp = math.sqrt(((d / l) * (d / h)))
	elif d <= -1e-309:
		tmp = d * math.pow(math.pow((l * h), 2.0), -0.25)
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -5.2e-11)
		tmp = sqrt(Float64(Float64(d / l) * Float64(d / h)));
	elseif (d <= -1e-309)
		tmp = Float64(d * ((Float64(l * h) ^ 2.0) ^ -0.25));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -5.2e-11)
		tmp = sqrt(((d / l) * (d / h)));
	elseif (d <= -1e-309)
		tmp = d * (((l * h) ^ 2.0) ^ -0.25);
	else
		tmp = d * ((h ^ -0.5) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -5.2e-11], N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[d, -1e-309], N[(d * N[Power[N[Power[N[(l * h), $MachinePrecision], 2.0], $MachinePrecision], -0.25], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.2000000000000001e-11

   1. Initial program 85.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 88.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(\frac{D}{\frac{2}{M} \cdot d}\right)}^{2} \cdot -0.5, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0 70.9%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]
   6. Step-by-step derivation

      1. *-rgt-identity 70.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. *-commutative 70.9%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}} \]

      3. sqrt-unprod 66.9%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

   7. Applied egg-rr 66.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

   if -5.2000000000000001e-11 < d < -1.000000000000002e-309

   1. Initial program 58.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 58.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 58.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 58.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 58.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. unpow2 58.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\sqrt{\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. sqrt-prod 35.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\sqrt{\frac{M}{2} \cdot \frac{D}{d}} \cdot \sqrt{\frac{M}{2} \cdot \frac{D}{d}}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. add-sqr-sqrt 58.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. div-inv 58.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. metadata-eval 58.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 58.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   7. Step-by-step derivation

      1. *-commutative 58.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. /-rgt-identity 58.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \color{blue}{\frac{M \cdot 0.5}{1}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      3. associate-/l* 58.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \color{blue}{\frac{M}{\frac{1}{0.5}}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      4. metadata-eval 58.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \frac{M}{\color{blue}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. times-frac 58.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{D \cdot M}{d \cdot 2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. associate-*r/ 58.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{M}{d \cdot 2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. associate-*l* 58.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(D \cdot \left(\frac{M}{d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}}^{2}\right) \]

      8. *-lft-identity 58.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{\color{blue}{1 \cdot M}}{d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      9. associate-/l* 58.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\frac{1}{\frac{d \cdot 2}{M}}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      10. associate-/r/ 58.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\left(\frac{1}{d \cdot 2} \cdot M\right)} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      11. *-commutative 58.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\frac{1}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      12. associate-/r* 58.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      13. metadata-eval 58.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\frac{\color{blue}{0.5}}{d} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   8. Simplified 58.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(D \cdot \left(\left(\frac{0.5}{d} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}}\right) \]

   9. Taylor expanded in d around inf 15.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. unpow-1 15.2%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

       2. metadata-eval 15.2%

          \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

       3. pow-sqr 15.2%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

       4. rem-sqrt-square 15.2%

          \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

       5. rem-square-sqrt 15.2%

          \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

       6. fabs-sqr 15.2%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

       7. rem-square-sqrt 15.2%

          \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   11. Simplified 15.2%

       \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   12. Taylor expanded in d around 0 15.2%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   13. Step-by-step derivation

       1. associate-/r* 15.2%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   14. Simplified 15.2%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   15. Step-by-step derivation

       1. associate-/l/ 15.2%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{\ell \cdot h}}} \]

       2. *-commutative 15.2%

          \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

       3. inv-pow 15.2%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

       4. metadata-eval 15.2%

          \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(-0.5 + -0.5\right)}}} \]

       5. pow-prod-up 15.2%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

       6. sqrt-unprod 15.2%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

       7. add-sqr-sqrt 15.2%

          \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

       8. sqr-pow 15.2%

          \[\leadsto d \cdot \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)}\right)} \]

       9. pow-prod-down 25.4%

          \[\leadsto d \cdot \color{blue}{{\left(\left(h \cdot \ell\right) \cdot \left(h \cdot \ell\right)\right)}^{\left(\frac{-0.5}{2}\right)}} \]

       10. pow2 25.4%

           \[\leadsto d \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{2}\right)}}^{\left(\frac{-0.5}{2}\right)} \]

       11. metadata-eval 25.4%

           \[\leadsto d \cdot {\left({\left(h \cdot \ell\right)}^{2}\right)}^{\color{blue}{-0.25}} \]

   16. Applied egg-rr 25.4%

       \[\leadsto d \cdot \color{blue}{{\left({\left(h \cdot \ell\right)}^{2}\right)}^{-0.25}} \]

   if -1.000000000000002e-309 < d

   1. Initial program 67.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 66.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 66.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 66.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 66.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. unpow2 66.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\sqrt{\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. sqrt-prod 39.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\sqrt{\frac{M}{2} \cdot \frac{D}{d}} \cdot \sqrt{\frac{M}{2} \cdot \frac{D}{d}}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. add-sqr-sqrt 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. div-inv 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. metadata-eval 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 66.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   7. Step-by-step derivation

      1. *-commutative 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. /-rgt-identity 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \color{blue}{\frac{M \cdot 0.5}{1}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      3. associate-/l* 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \color{blue}{\frac{M}{\frac{1}{0.5}}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      4. metadata-eval 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \frac{M}{\color{blue}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. times-frac 68.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{D \cdot M}{d \cdot 2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. associate-*r/ 67.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{M}{d \cdot 2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. associate-*l* 67.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(D \cdot \left(\frac{M}{d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}}^{2}\right) \]

      8. *-lft-identity 67.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{\color{blue}{1 \cdot M}}{d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      9. associate-/l* 67.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\frac{1}{\frac{d \cdot 2}{M}}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      10. associate-/r/ 67.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\left(\frac{1}{d \cdot 2} \cdot M\right)} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      11. *-commutative 67.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\frac{1}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      12. associate-/r* 67.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      13. metadata-eval 67.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\frac{\color{blue}{0.5}}{d} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   8. Simplified 67.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(D \cdot \left(\left(\frac{0.5}{d} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}}\right) \]

   9. Taylor expanded in d around inf 43.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. unpow-1 43.2%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

       2. metadata-eval 43.2%

          \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

       3. pow-sqr 43.2%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

       4. rem-sqrt-square 43.6%

          \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

       5. rem-square-sqrt 43.5%

          \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

       6. fabs-sqr 43.5%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

       7. rem-square-sqrt 43.6%

          \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   11. Simplified 43.6%

       \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   12. Taylor expanded in d around 0 43.2%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   13. Step-by-step derivation

       1. associate-/r* 43.2%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   14. Simplified 43.2%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   15. Step-by-step derivation

       1. sqrt-div 47.8%

          \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]

       2. inv-pow 47.8%

          \[\leadsto d \cdot \frac{\sqrt{\color{blue}{{h}^{-1}}}}{\sqrt{\ell}} \]

       3. sqrt-pow1 47.8%

          \[\leadsto d \cdot \frac{\color{blue}{{h}^{\left(\frac{-1}{2}\right)}}}{\sqrt{\ell}} \]

       4. metadata-eval 47.8%

          \[\leadsto d \cdot \frac{{h}^{\color{blue}{-0.5}}}{\sqrt{\ell}} \]

   16. Applied egg-rr 47.8%

       \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.2 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;d \cdot {\left({\left(\ell \cdot h\right)}^{2}\right)}^{-0.25}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 46.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -5.4 \cdot 10^{-139}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;d \cdot {\left({\left(\ell \cdot h\right)}^{2}\right)}^{-0.25}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -5.4e-139)
   (* (sqrt (/ d l)) (sqrt (/ d h)))
   (if (<= d -1e-309)
     (* d (pow (pow (* l h) 2.0) -0.25))
     (* d (/ (pow h -0.5) (sqrt l))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -5.4e-139) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (d <= -1e-309) {
		tmp = d * pow(pow((l * h), 2.0), -0.25);
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= (-5.4d-139)) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else if (d <= (-1d-309)) then
        tmp = d * (((l * h) ** 2.0d0) ** (-0.25d0))
    else
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -5.4e-139) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else if (d <= -1e-309) {
		tmp = d * Math.pow(Math.pow((l * h), 2.0), -0.25);
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= -5.4e-139:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	elif d <= -1e-309:
		tmp = d * math.pow(math.pow((l * h), 2.0), -0.25)
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -5.4e-139)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (d <= -1e-309)
		tmp = Float64(d * ((Float64(l * h) ^ 2.0) ^ -0.25));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -5.4e-139)
		tmp = sqrt((d / l)) * sqrt((d / h));
	elseif (d <= -1e-309)
		tmp = d * (((l * h) ^ 2.0) ^ -0.25);
	else
		tmp = d * ((h ^ -0.5) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -5.4e-139], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1e-309], N[(d * N[Power[N[Power[N[(l * h), $MachinePrecision], 2.0], $MachinePrecision], -0.25], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.3999999999999997e-139

   1. Initial program 85.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 87.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(\frac{D}{\frac{2}{M} \cdot d}\right)}^{2} \cdot -0.5, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0 59.9%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]

   if -5.3999999999999997e-139 < d < -1.000000000000002e-309

   1. Initial program 43.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 43.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 43.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 43.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 43.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. unpow2 43.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\sqrt{\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. sqrt-prod 19.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\sqrt{\frac{M}{2} \cdot \frac{D}{d}} \cdot \sqrt{\frac{M}{2} \cdot \frac{D}{d}}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. add-sqr-sqrt 43.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. div-inv 43.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. metadata-eval 43.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 43.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   7. Step-by-step derivation

      1. *-commutative 43.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. /-rgt-identity 43.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \color{blue}{\frac{M \cdot 0.5}{1}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      3. associate-/l* 43.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \color{blue}{\frac{M}{\frac{1}{0.5}}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      4. metadata-eval 43.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \frac{M}{\color{blue}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. times-frac 43.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{D \cdot M}{d \cdot 2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. associate-*r/ 43.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{M}{d \cdot 2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. associate-*l* 43.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(D \cdot \left(\frac{M}{d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}}^{2}\right) \]

      8. *-lft-identity 43.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{\color{blue}{1 \cdot M}}{d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      9. associate-/l* 43.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\frac{1}{\frac{d \cdot 2}{M}}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      10. associate-/r/ 43.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\left(\frac{1}{d \cdot 2} \cdot M\right)} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      11. *-commutative 43.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\frac{1}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      12. associate-/r* 43.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      13. metadata-eval 43.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\frac{\color{blue}{0.5}}{d} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   8. Simplified 43.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(D \cdot \left(\left(\frac{0.5}{d} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}}\right) \]

   9. Taylor expanded in d around inf 17.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. unpow-1 17.0%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

       2. metadata-eval 17.0%

          \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

       3. pow-sqr 17.0%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

       4. rem-sqrt-square 17.0%

          \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

       5. rem-square-sqrt 17.0%

          \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

       6. fabs-sqr 17.0%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

       7. rem-square-sqrt 17.0%

          \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   11. Simplified 17.0%

       \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   12. Taylor expanded in d around 0 17.0%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   13. Step-by-step derivation

       1. associate-/r* 17.0%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   14. Simplified 17.0%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   15. Step-by-step derivation

       1. associate-/l/ 17.0%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{\ell \cdot h}}} \]

       2. *-commutative 17.0%

          \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

       3. inv-pow 17.0%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

       4. metadata-eval 17.0%

          \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(-0.5 + -0.5\right)}}} \]

       5. pow-prod-up 17.0%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

       6. sqrt-unprod 17.0%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

       7. add-sqr-sqrt 17.0%

          \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

       8. sqr-pow 17.0%

          \[\leadsto d \cdot \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)}\right)} \]

       9. pow-prod-down 26.1%

          \[\leadsto d \cdot \color{blue}{{\left(\left(h \cdot \ell\right) \cdot \left(h \cdot \ell\right)\right)}^{\left(\frac{-0.5}{2}\right)}} \]

       10. pow2 26.1%

           \[\leadsto d \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{2}\right)}}^{\left(\frac{-0.5}{2}\right)} \]

       11. metadata-eval 26.1%

           \[\leadsto d \cdot {\left({\left(h \cdot \ell\right)}^{2}\right)}^{\color{blue}{-0.25}} \]

   16. Applied egg-rr 26.1%

       \[\leadsto d \cdot \color{blue}{{\left({\left(h \cdot \ell\right)}^{2}\right)}^{-0.25}} \]

   if -1.000000000000002e-309 < d

   1. Initial program 67.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 66.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 66.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 66.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 66.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. unpow2 66.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\sqrt{\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. sqrt-prod 39.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\sqrt{\frac{M}{2} \cdot \frac{D}{d}} \cdot \sqrt{\frac{M}{2} \cdot \frac{D}{d}}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. add-sqr-sqrt 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. div-inv 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. metadata-eval 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 66.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   7. Step-by-step derivation

      1. *-commutative 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. /-rgt-identity 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \color{blue}{\frac{M \cdot 0.5}{1}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      3. associate-/l* 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \color{blue}{\frac{M}{\frac{1}{0.5}}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      4. metadata-eval 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \frac{M}{\color{blue}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. times-frac 68.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{D \cdot M}{d \cdot 2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. associate-*r/ 67.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{M}{d \cdot 2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. associate-*l* 67.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(D \cdot \left(\frac{M}{d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}}^{2}\right) \]

      8. *-lft-identity 67.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{\color{blue}{1 \cdot M}}{d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      9. associate-/l* 67.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\frac{1}{\frac{d \cdot 2}{M}}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      10. associate-/r/ 67.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\left(\frac{1}{d \cdot 2} \cdot M\right)} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      11. *-commutative 67.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\frac{1}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      12. associate-/r* 67.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      13. metadata-eval 67.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\frac{\color{blue}{0.5}}{d} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   8. Simplified 67.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(D \cdot \left(\left(\frac{0.5}{d} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}}\right) \]

   9. Taylor expanded in d around inf 43.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. unpow-1 43.2%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

       2. metadata-eval 43.2%

          \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

       3. pow-sqr 43.2%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

       4. rem-sqrt-square 43.6%

          \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

       5. rem-square-sqrt 43.5%

          \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

       6. fabs-sqr 43.5%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

       7. rem-square-sqrt 43.6%

          \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   11. Simplified 43.6%

       \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   12. Taylor expanded in d around 0 43.2%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   13. Step-by-step derivation

       1. associate-/r* 43.2%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   14. Simplified 43.2%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   15. Step-by-step derivation

       1. sqrt-div 47.8%

          \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]

       2. inv-pow 47.8%

          \[\leadsto d \cdot \frac{\sqrt{\color{blue}{{h}^{-1}}}}{\sqrt{\ell}} \]

       3. sqrt-pow1 47.8%

          \[\leadsto d \cdot \frac{\color{blue}{{h}^{\left(\frac{-1}{2}\right)}}}{\sqrt{\ell}} \]

       4. metadata-eval 47.8%

          \[\leadsto d \cdot \frac{{h}^{\color{blue}{-0.5}}}{\sqrt{\ell}} \]

   16. Applied egg-rr 47.8%

       \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.4 \cdot 10^{-139}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;d \cdot {\left({\left(\ell \cdot h\right)}^{2}\right)}^{-0.25}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 42.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -5.5 \cdot 10^{-189}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -5.5e-189)
   (sqrt (* (/ d l) (/ d h)))
   (if (<= d -1e-309)
     (* d (pow (* l h) -0.5))
     (* d (/ (pow h -0.5) (sqrt l))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -5.5e-189) {
		tmp = sqrt(((d / l) * (d / h)));
	} else if (d <= -1e-309) {
		tmp = d * pow((l * h), -0.5);
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= (-5.5d-189)) then
        tmp = sqrt(((d / l) * (d / h)))
    else if (d <= (-1d-309)) then
        tmp = d * ((l * h) ** (-0.5d0))
    else
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -5.5e-189) {
		tmp = Math.sqrt(((d / l) * (d / h)));
	} else if (d <= -1e-309) {
		tmp = d * Math.pow((l * h), -0.5);
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= -5.5e-189:
		tmp = math.sqrt(((d / l) * (d / h)))
	elif d <= -1e-309:
		tmp = d * math.pow((l * h), -0.5)
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -5.5e-189)
		tmp = sqrt(Float64(Float64(d / l) * Float64(d / h)));
	elseif (d <= -1e-309)
		tmp = Float64(d * (Float64(l * h) ^ -0.5));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -5.5e-189)
		tmp = sqrt(((d / l) * (d / h)));
	elseif (d <= -1e-309)
		tmp = d * ((l * h) ^ -0.5);
	else
		tmp = d * ((h ^ -0.5) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -5.5e-189], N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[d, -1e-309], N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.4999999999999999e-189

   1. Initial program 85.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 86.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(\frac{D}{\frac{2}{M} \cdot d}\right)}^{2} \cdot -0.5, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0 56.6%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]
   6. Step-by-step derivation

      1. *-rgt-identity 56.6%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. *-commutative 56.6%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}} \]

      3. sqrt-unprod 52.1%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

   7. Applied egg-rr 52.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

   if -5.4999999999999999e-189 < d < -1.000000000000002e-309

   1. Initial program 34.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 34.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 34.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 34.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 34.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. unpow2 34.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\sqrt{\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. sqrt-prod 16.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\sqrt{\frac{M}{2} \cdot \frac{D}{d}} \cdot \sqrt{\frac{M}{2} \cdot \frac{D}{d}}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. add-sqr-sqrt 34.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. div-inv 34.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. metadata-eval 34.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 34.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   7. Step-by-step derivation

      1. *-commutative 34.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. /-rgt-identity 34.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \color{blue}{\frac{M \cdot 0.5}{1}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      3. associate-/l* 34.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \color{blue}{\frac{M}{\frac{1}{0.5}}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      4. metadata-eval 34.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \frac{M}{\color{blue}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. times-frac 34.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{D \cdot M}{d \cdot 2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. associate-*r/ 34.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{M}{d \cdot 2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. associate-*l* 34.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(D \cdot \left(\frac{M}{d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}}^{2}\right) \]

      8. *-lft-identity 34.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{\color{blue}{1 \cdot M}}{d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      9. associate-/l* 34.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\frac{1}{\frac{d \cdot 2}{M}}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      10. associate-/r/ 34.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\left(\frac{1}{d \cdot 2} \cdot M\right)} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      11. *-commutative 34.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\frac{1}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      12. associate-/r* 34.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      13. metadata-eval 34.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\frac{\color{blue}{0.5}}{d} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   8. Simplified 34.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(D \cdot \left(\left(\frac{0.5}{d} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}}\right) \]

   9. Taylor expanded in d around inf 20.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. unpow-1 20.7%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

       2. metadata-eval 20.7%

          \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

       3. pow-sqr 20.7%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

       4. rem-sqrt-square 20.7%

          \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

       5. rem-square-sqrt 20.7%

          \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

       6. fabs-sqr 20.7%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

       7. rem-square-sqrt 20.7%

          \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   11. Simplified 20.7%

       \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   if -1.000000000000002e-309 < d

   1. Initial program 67.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 66.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 66.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 66.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 66.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. unpow2 66.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\sqrt{\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. sqrt-prod 39.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\sqrt{\frac{M}{2} \cdot \frac{D}{d}} \cdot \sqrt{\frac{M}{2} \cdot \frac{D}{d}}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. add-sqr-sqrt 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. div-inv 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. metadata-eval 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 66.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   7. Step-by-step derivation

      1. *-commutative 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. /-rgt-identity 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \color{blue}{\frac{M \cdot 0.5}{1}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      3. associate-/l* 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \color{blue}{\frac{M}{\frac{1}{0.5}}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      4. metadata-eval 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \frac{M}{\color{blue}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. times-frac 68.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{D \cdot M}{d \cdot 2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. associate-*r/ 67.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{M}{d \cdot 2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. associate-*l* 67.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(D \cdot \left(\frac{M}{d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}}^{2}\right) \]

      8. *-lft-identity 67.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{\color{blue}{1 \cdot M}}{d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      9. associate-/l* 67.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\frac{1}{\frac{d \cdot 2}{M}}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      10. associate-/r/ 67.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\left(\frac{1}{d \cdot 2} \cdot M\right)} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      11. *-commutative 67.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\frac{1}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      12. associate-/r* 67.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      13. metadata-eval 67.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\frac{\color{blue}{0.5}}{d} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   8. Simplified 67.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(D \cdot \left(\left(\frac{0.5}{d} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}}\right) \]

   9. Taylor expanded in d around inf 43.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. unpow-1 43.2%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

       2. metadata-eval 43.2%

          \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

       3. pow-sqr 43.2%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

       4. rem-sqrt-square 43.6%

          \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

       5. rem-square-sqrt 43.5%

          \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

       6. fabs-sqr 43.5%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

       7. rem-square-sqrt 43.6%

          \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   11. Simplified 43.6%

       \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   12. Taylor expanded in d around 0 43.2%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   13. Step-by-step derivation

       1. associate-/r* 43.2%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   14. Simplified 43.2%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   15. Step-by-step derivation

       1. sqrt-div 47.8%

          \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]

       2. inv-pow 47.8%

          \[\leadsto d \cdot \frac{\sqrt{\color{blue}{{h}^{-1}}}}{\sqrt{\ell}} \]

       3. sqrt-pow1 47.8%

          \[\leadsto d \cdot \frac{\color{blue}{{h}^{\left(\frac{-1}{2}\right)}}}{\sqrt{\ell}} \]

       4. metadata-eval 47.8%

          \[\leadsto d \cdot \frac{{h}^{\color{blue}{-0.5}}}{\sqrt{\ell}} \]

   16. Applied egg-rr 47.8%

       \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.5 \cdot 10^{-189}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 37.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.48 \cdot 10^{-227}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1}{\sqrt{\frac{\ell}{\frac{1}{h}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1.48e-227)
   (sqrt (* (/ d l) (/ d h)))
   (* d (/ 1.0 (sqrt (/ l (/ 1.0 h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.48e-227) {
		tmp = sqrt(((d / l) * (d / h)));
	} else {
		tmp = d * (1.0 / sqrt((l / (1.0 / h))));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-1.48d-227)) then
        tmp = sqrt(((d / l) * (d / h)))
    else
        tmp = d * (1.0d0 / sqrt((l / (1.0d0 / h))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.48e-227) {
		tmp = Math.sqrt(((d / l) * (d / h)));
	} else {
		tmp = d * (1.0 / Math.sqrt((l / (1.0 / h))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -1.48e-227:
		tmp = math.sqrt(((d / l) * (d / h)))
	else:
		tmp = d * (1.0 / math.sqrt((l / (1.0 / h))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1.48e-227)
		tmp = sqrt(Float64(Float64(d / l) * Float64(d / h)));
	else
		tmp = Float64(d * Float64(1.0 / sqrt(Float64(l / Float64(1.0 / h)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -1.48e-227)
		tmp = sqrt(((d / l) * (d / h)));
	else
		tmp = d * (1.0 / sqrt((l / (1.0 / h))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1.48e-227], N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(d * N[(1.0 / N[Sqrt[N[(l / N[(1.0 / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.48e-227

   1. Initial program 69.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 71.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(\frac{D}{\frac{2}{M} \cdot d}\right)}^{2} \cdot -0.5, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0 47.5%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]
   6. Step-by-step derivation

      1. *-rgt-identity 47.5%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. *-commutative 47.5%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}} \]

      3. sqrt-unprod 43.6%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

   7. Applied egg-rr 43.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

   if -1.48e-227 < l

   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 69.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 69.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 69.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 69.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. unpow2 69.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\sqrt{\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. sqrt-prod 41.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\sqrt{\frac{M}{2} \cdot \frac{D}{d}} \cdot \sqrt{\frac{M}{2} \cdot \frac{D}{d}}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. add-sqr-sqrt 69.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. div-inv 69.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. metadata-eval 69.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 69.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   7. Step-by-step derivation

      1. *-commutative 69.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. /-rgt-identity 69.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \color{blue}{\frac{M \cdot 0.5}{1}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      3. associate-/l* 69.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \color{blue}{\frac{M}{\frac{1}{0.5}}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      4. metadata-eval 69.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \frac{M}{\color{blue}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. times-frac 71.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{D \cdot M}{d \cdot 2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. associate-*r/ 70.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{M}{d \cdot 2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. associate-*l* 70.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(D \cdot \left(\frac{M}{d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}}^{2}\right) \]

      8. *-lft-identity 70.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{\color{blue}{1 \cdot M}}{d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      9. associate-/l* 70.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\frac{1}{\frac{d \cdot 2}{M}}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      10. associate-/r/ 70.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\left(\frac{1}{d \cdot 2} \cdot M\right)} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      11. *-commutative 70.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\frac{1}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      12. associate-/r* 70.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      13. metadata-eval 70.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\frac{\color{blue}{0.5}}{d} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   8. Simplified 70.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(D \cdot \left(\left(\frac{0.5}{d} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}}\right) \]

   9. Taylor expanded in d around inf 45.3%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. unpow-1 45.3%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

       2. metadata-eval 45.3%

          \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

       3. pow-sqr 45.3%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

       4. rem-sqrt-square 45.6%

          \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

       5. rem-square-sqrt 45.6%

          \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

       6. fabs-sqr 45.6%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

       7. rem-square-sqrt 45.6%

          \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   11. Simplified 45.6%

       \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   12. Taylor expanded in d around 0 45.3%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   13. Step-by-step derivation

       1. associate-/r* 45.3%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   14. Simplified 45.3%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   15. Step-by-step derivation

       1. clear-num 45.3%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{\frac{1}{h}}}}} \]

       2. sqrt-div 45.7%

          \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{\frac{1}{h}}}}} \]

       3. metadata-eval 45.7%

          \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{\frac{1}{h}}}} \]

   16. Applied egg-rr 45.7%

       \[\leadsto d \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{\frac{1}{h}}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.48 \cdot 10^{-227}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1}{\sqrt{\frac{\ell}{\frac{1}{h}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 38.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.3 \cdot 10^{-227}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -6.3e-227) (sqrt (* (/ d l) (/ d h))) (* d (pow (* l h) -0.5))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -6.3e-227) {
		tmp = sqrt(((d / l) * (d / h)));
	} else {
		tmp = d * pow((l * h), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-6.3d-227)) then
        tmp = sqrt(((d / l) * (d / h)))
    else
        tmp = d * ((l * h) ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -6.3e-227) {
		tmp = Math.sqrt(((d / l) * (d / h)));
	} else {
		tmp = d * Math.pow((l * h), -0.5);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -6.3e-227:
		tmp = math.sqrt(((d / l) * (d / h)))
	else:
		tmp = d * math.pow((l * h), -0.5)
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -6.3e-227)
		tmp = sqrt(Float64(Float64(d / l) * Float64(d / h)));
	else
		tmp = Float64(d * (Float64(l * h) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -6.3e-227)
		tmp = sqrt(((d / l) * (d / h)));
	else
		tmp = d * ((l * h) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -6.3e-227], N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -6.2999999999999999e-227

   1. Initial program 69.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 71.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(\frac{D}{\frac{2}{M} \cdot d}\right)}^{2} \cdot -0.5, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0 47.5%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]
   6. Step-by-step derivation

      1. *-rgt-identity 47.5%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. *-commutative 47.5%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}} \]

      3. sqrt-unprod 43.6%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

   7. Applied egg-rr 43.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

   if -6.2999999999999999e-227 < l

   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 69.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 69.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 69.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 69.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. unpow2 69.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\sqrt{\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. sqrt-prod 41.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\sqrt{\frac{M}{2} \cdot \frac{D}{d}} \cdot \sqrt{\frac{M}{2} \cdot \frac{D}{d}}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. add-sqr-sqrt 69.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. div-inv 69.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. metadata-eval 69.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 69.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   7. Step-by-step derivation

      1. *-commutative 69.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. /-rgt-identity 69.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \color{blue}{\frac{M \cdot 0.5}{1}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      3. associate-/l* 69.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \color{blue}{\frac{M}{\frac{1}{0.5}}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      4. metadata-eval 69.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \frac{M}{\color{blue}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. times-frac 71.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{D \cdot M}{d \cdot 2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. associate-*r/ 70.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{M}{d \cdot 2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. associate-*l* 70.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(D \cdot \left(\frac{M}{d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}}^{2}\right) \]

      8. *-lft-identity 70.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{\color{blue}{1 \cdot M}}{d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      9. associate-/l* 70.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\frac{1}{\frac{d \cdot 2}{M}}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      10. associate-/r/ 70.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\left(\frac{1}{d \cdot 2} \cdot M\right)} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      11. *-commutative 70.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\frac{1}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      12. associate-/r* 70.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      13. metadata-eval 70.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\frac{\color{blue}{0.5}}{d} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   8. Simplified 70.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(D \cdot \left(\left(\frac{0.5}{d} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}}\right) \]

   9. Taylor expanded in d around inf 45.3%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. unpow-1 45.3%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

       2. metadata-eval 45.3%

          \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

       3. pow-sqr 45.3%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

       4. rem-sqrt-square 45.6%

          \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

       5. rem-square-sqrt 45.6%

          \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

       6. fabs-sqr 45.6%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

       7. rem-square-sqrt 45.6%

          \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   11. Simplified 45.6%

       \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.3 \cdot 10^{-227}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 38.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.5 \cdot 10^{-226}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1.5e-226) (sqrt (* (/ d l) (/ d h))) (* d (/ 1.0 (sqrt (* l h))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.5e-226) {
		tmp = sqrt(((d / l) * (d / h)));
	} else {
		tmp = d * (1.0 / sqrt((l * h)));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-1.5d-226)) then
        tmp = sqrt(((d / l) * (d / h)))
    else
        tmp = d * (1.0d0 / sqrt((l * h)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.5e-226) {
		tmp = Math.sqrt(((d / l) * (d / h)));
	} else {
		tmp = d * (1.0 / Math.sqrt((l * h)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -1.5e-226:
		tmp = math.sqrt(((d / l) * (d / h)))
	else:
		tmp = d * (1.0 / math.sqrt((l * h)))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1.5e-226)
		tmp = sqrt(Float64(Float64(d / l) * Float64(d / h)));
	else
		tmp = Float64(d * Float64(1.0 / sqrt(Float64(l * h))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -1.5e-226)
		tmp = sqrt(((d / l) * (d / h)));
	else
		tmp = d * (1.0 / sqrt((l * h)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1.5e-226], N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(d * N[(1.0 / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.49999999999999998e-226

   1. Initial program 69.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 71.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(\frac{D}{\frac{2}{M} \cdot d}\right)}^{2} \cdot -0.5, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0 47.5%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]
   6. Step-by-step derivation

      1. *-rgt-identity 47.5%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. *-commutative 47.5%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}} \]

      3. sqrt-unprod 43.6%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

   7. Applied egg-rr 43.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

   if -1.49999999999999998e-226 < l

   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 69.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 45.3%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. sqrt-div 45.7%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      2. metadata-eval 45.7%

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

   7. Applied egg-rr 45.7%

      \[\leadsto d \cdot \color{blue}{\frac{1}{\sqrt{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.5 \cdot 10^{-226}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 26.7% 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 69.9%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 70.3%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 70.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

   2. pow2 70.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

   3. sqrt-prod 70.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

   4. unpow2 70.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\sqrt{\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   5. sqrt-prod 42.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\sqrt{\frac{M}{2} \cdot \frac{D}{d}} \cdot \sqrt{\frac{M}{2} \cdot \frac{D}{d}}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. add-sqr-sqrt 70.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   7. div-inv 70.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   8. metadata-eval 70.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

6. Applied egg-rr 70.4%

   \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
7. Step-by-step derivation

   1. *-commutative 70.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   2. /-rgt-identity 70.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \color{blue}{\frac{M \cdot 0.5}{1}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   3. associate-/l* 70.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \color{blue}{\frac{M}{\frac{1}{0.5}}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   4. metadata-eval 70.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \frac{M}{\color{blue}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   5. times-frac 70.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{D \cdot M}{d \cdot 2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. associate-*r/ 71.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{M}{d \cdot 2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   7. associate-*l* 71.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(D \cdot \left(\frac{M}{d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}}^{2}\right) \]

   8. *-lft-identity 71.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{\color{blue}{1 \cdot M}}{d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   9. associate-/l* 71.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\frac{1}{\frac{d \cdot 2}{M}}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   10. associate-/r/ 71.1%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\left(\frac{1}{d \cdot 2} \cdot M\right)} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   11. *-commutative 71.1%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\frac{1}{\color{blue}{2 \cdot d}} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   12. associate-/r* 71.1%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   13. metadata-eval 71.1%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\frac{\color{blue}{0.5}}{d} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

8. Simplified 71.1%

   \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(D \cdot \left(\left(\frac{0.5}{d} \cdot M\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}}\right) \]

9. Taylor expanded in d around inf 27.1%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
10. Step-by-step derivation

    1. unpow-1 27.1%

       \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

    2. metadata-eval 27.1%

       \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

    3. pow-sqr 27.1%

       \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

    4. rem-sqrt-square 27.3%

       \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

    5. rem-square-sqrt 27.3%

       \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

    6. fabs-sqr 27.3%

       \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

    7. rem-square-sqrt 27.3%

       \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

11. Simplified 27.3%

    \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

12. Final simplification 27.3%

    \[\leadsto d \cdot {\left(\ell \cdot h\right)}^{-0.5} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(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)))))