Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.4% → 44.2%

Time: 24.4s

Alternatives: 15

Speedup: 151.0×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	return (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    t_0 = (c0 * (d_1 * d_1)) / ((w * h) * (d * d))
    code = (c0 / (2.0d0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (m * m))))
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	return (c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	return (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	return Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Initial Program: 24.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	return (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    t_0 = (c0 * (d_1 * d_1)) / ((w * h) * (d * d))
    code = (c0 / (2.0d0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (m * m))))
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	return (c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	return (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	return Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 44.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\frac{\frac{d}{D}}{w \cdot h}}{\frac{D}{c0 \cdot d}} \cdot \left(c0 \cdot 2\right)}{2 \cdot w}\\ \mathbf{if}\;D \cdot D \leq 10^{-293}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;D \cdot D \leq 10^{-204}:\\ \;\;\;\;0\\ \mathbf{elif}\;D \cdot D \leq 5 \cdot 10^{-60}:\\ \;\;\;\;\frac{\frac{c0}{D \cdot D}}{w \cdot \left(\frac{w}{c0 \cdot d} \cdot \frac{h}{d}\right)}\\ \mathbf{elif}\;D \cdot D \leq 10^{-7}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0
         (/ (* (/ (/ (/ d D) (* w h)) (/ D (* c0 d))) (* c0 2.0)) (* 2.0 w))))
   (if (<= (* D D) 1e-293)
     t_0
     (if (<= (* D D) 1e-204)
       0.0
       (if (<= (* D D) 5e-60)
         (/ (/ c0 (* D D)) (* w (* (/ w (* c0 d)) (/ h d))))
         (if (<= (* D D) 1e-7) 0.0 t_0))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((((d / D) / (w * h)) / (D / (c0 * d))) * (c0 * 2.0)) / (2.0 * w);
	double tmp;
	if ((D * D) <= 1e-293) {
		tmp = t_0;
	} else if ((D * D) <= 1e-204) {
		tmp = 0.0;
	} else if ((D * D) <= 5e-60) {
		tmp = (c0 / (D * D)) / (w * ((w / (c0 * d)) * (h / d)));
	} else if ((D * D) <= 1e-7) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((((d_1 / d) / (w * h)) / (d / (c0 * d_1))) * (c0 * 2.0d0)) / (2.0d0 * w)
    if ((d * d) <= 1d-293) then
        tmp = t_0
    else if ((d * d) <= 1d-204) then
        tmp = 0.0d0
    else if ((d * d) <= 5d-60) then
        tmp = (c0 / (d * d)) / (w * ((w / (c0 * d_1)) * (h / d_1)))
    else if ((d * d) <= 1d-7) then
        tmp = 0.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((((d / D) / (w * h)) / (D / (c0 * d))) * (c0 * 2.0)) / (2.0 * w);
	double tmp;
	if ((D * D) <= 1e-293) {
		tmp = t_0;
	} else if ((D * D) <= 1e-204) {
		tmp = 0.0;
	} else if ((D * D) <= 5e-60) {
		tmp = (c0 / (D * D)) / (w * ((w / (c0 * d)) * (h / d)));
	} else if ((D * D) <= 1e-7) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = ((((d / D) / (w * h)) / (D / (c0 * d))) * (c0 * 2.0)) / (2.0 * w)
	tmp = 0
	if (D * D) <= 1e-293:
		tmp = t_0
	elif (D * D) <= 1e-204:
		tmp = 0.0
	elif (D * D) <= 5e-60:
		tmp = (c0 / (D * D)) / (w * ((w / (c0 * d)) * (h / d)))
	elif (D * D) <= 1e-7:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(Float64(Float64(Float64(d / D) / Float64(w * h)) / Float64(D / Float64(c0 * d))) * Float64(c0 * 2.0)) / Float64(2.0 * w))
	tmp = 0.0
	if (Float64(D * D) <= 1e-293)
		tmp = t_0;
	elseif (Float64(D * D) <= 1e-204)
		tmp = 0.0;
	elseif (Float64(D * D) <= 5e-60)
		tmp = Float64(Float64(c0 / Float64(D * D)) / Float64(w * Float64(Float64(w / Float64(c0 * d)) * Float64(h / d))));
	elseif (Float64(D * D) <= 1e-7)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = ((((d / D) / (w * h)) / (D / (c0 * d))) * (c0 * 2.0)) / (2.0 * w);
	tmp = 0.0;
	if ((D * D) <= 1e-293)
		tmp = t_0;
	elseif ((D * D) <= 1e-204)
		tmp = 0.0;
	elseif ((D * D) <= 5e-60)
		tmp = (c0 / (D * D)) / (w * ((w / (c0 * d)) * (h / d)));
	elseif ((D * D) <= 1e-7)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(N[(N[(N[(d / D), $MachinePrecision] / N[(w * h), $MachinePrecision]), $MachinePrecision] / N[(D / N[(c0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c0 * 2.0), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(D * D), $MachinePrecision], 1e-293], t$95$0, If[LessEqual[N[(D * D), $MachinePrecision], 1e-204], 0.0, If[LessEqual[N[(D * D), $MachinePrecision], 5e-60], N[(N[(c0 / N[(D * D), $MachinePrecision]), $MachinePrecision] / N[(w * N[(N[(w / N[(c0 * d), $MachinePrecision]), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(D * D), $MachinePrecision], 1e-7], 0.0, t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 D D) < 1.0000000000000001e-293 or 9.9999999999999995e-8 < (*.f64 D D)

   1. Initial program 24.2%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Simplified 23.6%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{d \cdot \frac{\frac{c0 \cdot d}{w}}{h}}{D \cdot D} + \sqrt{\frac{\frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot \left(d \cdot \frac{\frac{c0 \cdot d}{w}}{h}\right)\right)}{D \cdot D}}{w \cdot h}}{D \cdot D} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\left(2 \cdot \left(c0 \cdot {d}^{2}\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot {d}^{2}\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left({d}^{2}\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(d \cdot d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(w \cdot h\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. *-lowering-*.f64 31.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(w, h\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 31.4%

      \[\leadsto \frac{c0 \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}}{2 \cdot w} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. count-2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{\left(c0 \cdot d\right) \cdot d}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{\left(c0 \cdot d\right) \cdot d}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{\left(c0 \cdot d\right) \cdot d}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{\left(c0 \cdot d\right) \cdot d}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. times-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      12. distribute-rgt-out N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)}\right), \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   9. Applied egg-rr 56.2%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)}}{2 \cdot w} \]
   10. Step-by-step derivation

       1. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       2. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right) \cdot c0 + \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right) \cdot c0\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

       3. distribute-lft-out N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right) \cdot \left(c0 + c0\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot d}{D}\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       6. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \frac{1}{\frac{D}{c0 \cdot d}}\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       7. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{d}{D \cdot \left(w \cdot h\right)}}{\frac{D}{c0 \cdot d}}\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)}\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       9. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{d}{D}}{w \cdot h}\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{D}\right), \left(w \cdot h\right)\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \left(w \cdot h\right)\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(D, \left(c0 \cdot d\right)\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(c0, d\right)\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       15. count-2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(c0, d\right)\right)\right), \left(2 \cdot c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       16. *-lowering-*.f64 56.7%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(c0, d\right)\right)\right), \mathsf{*.f64}\left(2, c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   11. Applied egg-rr 56.7%

       \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{d}{D}}{w \cdot h}}{\frac{D}{c0 \cdot d}} \cdot \left(2 \cdot c0\right)}}{2 \cdot w} \]

   if 1.0000000000000001e-293 < (*.f64 D D) < 1e-204 or 5.0000000000000001e-60 < (*.f64 D D) < 9.9999999999999995e-8

   1. Initial program 14.7%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Simplified 9.4%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{d \cdot \frac{\frac{c0 \cdot d}{w}}{h}}{D \cdot D} + \sqrt{\frac{\frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot \left(d \cdot \frac{\frac{c0 \cdot d}{w}}{h}\right)\right)}{D \cdot D}}{w \cdot h}}{D \cdot D} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot c0\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. mul0-lft N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(c0 \cdot \left(-1 + 1\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(\mathsf{neg}\left(\left(-1 + 1\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(\mathsf{neg}\left(0\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      13. metadata-eval 62.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 62.0%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(c0 \cdot 0\right)}}{2 \cdot w} \]
   8. Step-by-step derivation

      1. mul0-rgt N/A

         \[\leadsto \frac{c0 \cdot 0}{2 \cdot w} \]

      2. mul0-rgt N/A

         \[\leadsto \frac{0}{\color{blue}{2} \cdot w} \]

      3. div0 62.0%

         \[\leadsto 0 \]

   9. Applied egg-rr 62.0%

      \[\leadsto \color{blue}{0} \]

   if 1e-204 < (*.f64 D D) < 5.0000000000000001e-60

   1. Initial program 26.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Simplified 27.0%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{d \cdot \frac{\frac{c0 \cdot d}{w}}{h}}{D \cdot D} + \sqrt{\frac{\frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot \left(d \cdot \frac{\frac{c0 \cdot d}{w}}{h}\right)\right)}{D \cdot D}}{w \cdot h}}{D \cdot D} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\left(2 \cdot \left(c0 \cdot {d}^{2}\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot {d}^{2}\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left({d}^{2}\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(d \cdot d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(w \cdot h\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. *-lowering-*.f64 34.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(w, h\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 34.6%

      \[\leadsto \frac{c0 \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}}{2 \cdot w} \]
   8. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(\frac{2}{D \cdot D} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{w \cdot h}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(c0 \cdot \frac{2}{D \cdot D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{w \cdot h}\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(c0 \cdot \frac{2}{D \cdot D}\right) \cdot \frac{\frac{c0 \cdot \left(d \cdot d\right)}{w}}{h}\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(c0 \cdot \frac{2}{D \cdot D}\right) \cdot \frac{c0 \cdot \frac{d \cdot d}{w}}{h}\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(c0 \cdot \frac{2}{D \cdot D}\right), \left(\frac{c0 \cdot \frac{d \cdot d}{w}}{h}\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(c0 \cdot \frac{1}{\frac{D \cdot D}{2}}\right), \left(\frac{c0 \cdot \frac{d \cdot d}{w}}{h}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{c0}{\frac{D \cdot D}{2}}\right), \left(\frac{c0 \cdot \frac{d \cdot d}{w}}{h}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \left(\frac{D \cdot D}{2}\right)\right), \left(\frac{c0 \cdot \frac{d \cdot d}{w}}{h}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\left(D \cdot D\right), 2\right)\right), \left(\frac{c0 \cdot \frac{d \cdot d}{w}}{h}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \left(\frac{c0 \cdot \frac{d \cdot d}{w}}{h}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \left(c0 \cdot \frac{\frac{d \cdot d}{w}}{h}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      12. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \left(c0 \cdot \frac{1}{\frac{h}{\frac{d \cdot d}{w}}}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      13. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \left(\frac{c0}{\frac{h}{\frac{d \cdot d}{w}}}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \mathsf{/.f64}\left(c0, \left(\frac{h}{\frac{d \cdot d}{w}}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(h, \left(\frac{d \cdot d}{w}\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      16. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(h, \left(d \cdot \frac{d}{w}\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      17. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(h, \left(d \cdot \frac{1}{\frac{w}{d}}\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      18. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(h, \left(\frac{d}{\frac{w}{d}}\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      19. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \left(\frac{w}{d}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      20. /-lowering-/.f64 44.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{/.f64}\left(w, d\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   9. Applied egg-rr 44.4%

      \[\leadsto \frac{\color{blue}{\frac{c0}{\frac{D \cdot D}{2}} \cdot \frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}}{2 \cdot w} \]
   10. Step-by-step derivation

       1. times-frac N/A

          \[\leadsto \frac{\frac{c0}{\frac{D \cdot D}{2}}}{2} \cdot \color{blue}{\frac{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}{w}} \]

       2. clear-num N/A

          \[\leadsto \frac{\frac{c0}{\frac{D \cdot D}{2}}}{2} \cdot \frac{1}{\color{blue}{\frac{w}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}}} \]

       3. un-div-inv N/A

          \[\leadsto \frac{\frac{\frac{c0}{\frac{D \cdot D}{2}}}{2}}{\color{blue}{\frac{w}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}}} \]

       4. associate-/r/ N/A

          \[\leadsto \frac{\frac{\frac{c0}{D \cdot D} \cdot 2}{2}}{\frac{w}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}} \]

       5. associate-/l* N/A

          \[\leadsto \frac{\frac{c0}{D \cdot D} \cdot \frac{2}{2}}{\frac{\color{blue}{w}}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}} \]

       6. metadata-eval N/A

          \[\leadsto \frac{\frac{c0}{D \cdot D} \cdot 1}{\frac{w}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}} \]

       7. *-rgt-identity N/A

          \[\leadsto \frac{\frac{c0}{D \cdot D}}{\frac{\color{blue}{w}}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}} \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{c0}{D \cdot D}\right), \color{blue}{\left(\frac{w}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}\right)}\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \left(D \cdot D\right)\right), \left(\frac{\color{blue}{w}}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{w}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \color{blue}{\left(\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}\right)}\right)\right) \]

       12. associate-/r/ N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \left(\frac{c0}{h} \cdot \color{blue}{\frac{d}{\frac{w}{d}}}\right)\right)\right) \]

       13. frac-times N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \left(\frac{c0 \cdot d}{\color{blue}{h \cdot \frac{w}{d}}}\right)\right)\right) \]

       14. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \mathsf{/.f64}\left(\left(c0 \cdot d\right), \color{blue}{\left(h \cdot \frac{w}{d}\right)}\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, d\right), \left(\color{blue}{h} \cdot \frac{w}{d}\right)\right)\right)\right) \]

       16. clear-num N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, d\right), \left(h \cdot \frac{1}{\color{blue}{\frac{d}{w}}}\right)\right)\right)\right) \]

       17. un-div-inv N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, d\right), \left(\frac{h}{\color{blue}{\frac{d}{w}}}\right)\right)\right)\right) \]

       18. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, d\right), \mathsf{/.f64}\left(h, \color{blue}{\left(\frac{d}{w}\right)}\right)\right)\right)\right) \]

       19. /-lowering-/.f64 57.9%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, d\right), \mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \color{blue}{w}\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 57.9%

       \[\leadsto \color{blue}{\frac{\frac{c0}{D \cdot D}}{\frac{w}{\frac{c0 \cdot d}{\frac{h}{\frac{d}{w}}}}}} \]
   12. Step-by-step derivation

       1. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{w}{c0 \cdot d} \cdot \color{blue}{\frac{h}{\frac{d}{w}}}\right)\right) \]

       2. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{w}{c0 \cdot d} \cdot \left(\frac{h}{d} \cdot \color{blue}{w}\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \left(\left(\frac{w}{c0 \cdot d} \cdot \frac{h}{d}\right) \cdot \color{blue}{w}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\left(\frac{w}{c0 \cdot d} \cdot \frac{h}{d}\right), \color{blue}{w}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{w}{c0 \cdot d}\right), \left(\frac{h}{d}\right)\right), w\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(w, \left(c0 \cdot d\right)\right), \left(\frac{h}{d}\right)\right), w\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(w, \left(d \cdot c0\right)\right), \left(\frac{h}{d}\right)\right), w\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(w, \mathsf{*.f64}\left(d, c0\right)\right), \left(\frac{h}{d}\right)\right), w\right)\right) \]

       9. /-lowering-/.f64 58.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(w, \mathsf{*.f64}\left(d, c0\right)\right), \mathsf{/.f64}\left(h, d\right)\right), w\right)\right) \]

   13. Applied egg-rr 58.0%

       \[\leadsto \frac{\frac{c0}{D \cdot D}}{\color{blue}{\left(\frac{w}{d \cdot c0} \cdot \frac{h}{d}\right) \cdot w}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \cdot D \leq 10^{-293}:\\ \;\;\;\;\frac{\frac{\frac{\frac{d}{D}}{w \cdot h}}{\frac{D}{c0 \cdot d}} \cdot \left(c0 \cdot 2\right)}{2 \cdot w}\\ \mathbf{elif}\;D \cdot D \leq 10^{-204}:\\ \;\;\;\;0\\ \mathbf{elif}\;D \cdot D \leq 5 \cdot 10^{-60}:\\ \;\;\;\;\frac{\frac{c0}{D \cdot D}}{w \cdot \left(\frac{w}{c0 \cdot d} \cdot \frac{h}{d}\right)}\\ \mathbf{elif}\;D \cdot D \leq 10^{-7}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{d}{D}}{w \cdot h}}{\frac{D}{c0 \cdot d}} \cdot \left(c0 \cdot 2\right)}{2 \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 55.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0 \cdot \left(\frac{d}{\left(w \cdot h\right) \cdot D} \cdot \left(\frac{c0 \cdot d}{D} + d \cdot \frac{c0}{D}\right)\right)}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
        INFINITY)
     (/
      (* c0 (* (/ d (* (* w h) D)) (+ (/ (* c0 d) D) (* d (/ c0 D)))))
      (* 2.0 w))
     0.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= ((double) INFINITY)) {
		tmp = (c0 * ((d / ((w * h) * D)) * (((c0 * d) / D) + (d * (c0 / D))))) / (2.0 * w);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = (c0 * ((d / ((w * h) * D)) * (((c0 * d) / D) + (d * (c0 / D))))) / (2.0 * w);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))) <= math.inf:
		tmp = (c0 * ((d / ((w * h) * D)) * (((c0 * d) / D) + (d * (c0 / D))))) / (2.0 * w)
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M))))) <= Inf)
		tmp = Float64(Float64(c0 * Float64(Float64(d / Float64(Float64(w * h) * D)) * Float64(Float64(Float64(c0 * d) / D) + Float64(d * Float64(c0 / D))))) / Float64(2.0 * w));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= Inf)
		tmp = (c0 * ((d / ((w * h) * D)) * (((c0 * d) / D) + (d * (c0 / D))))) / (2.0 * w);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(c0 * N[(N[(d / N[(N[(w * h), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(c0 * d), $MachinePrecision] / D), $MachinePrecision] + N[(d * N[(c0 / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 76.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Simplified 71.6%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{d \cdot \frac{\frac{c0 \cdot d}{w}}{h}}{D \cdot D} + \sqrt{\frac{\frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot \left(d \cdot \frac{\frac{c0 \cdot d}{w}}{h}\right)\right)}{D \cdot D}}{w \cdot h}}{D \cdot D} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\left(2 \cdot \left(c0 \cdot {d}^{2}\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot {d}^{2}\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left({d}^{2}\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(d \cdot d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(w \cdot h\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. *-lowering-*.f64 72.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(w, h\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 72.9%

      \[\leadsto \frac{c0 \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}}{2 \cdot w} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. count-2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{\left(c0 \cdot d\right) \cdot d}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{\left(c0 \cdot d\right) \cdot d}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{\left(c0 \cdot d\right) \cdot d}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{\left(c0 \cdot d\right) \cdot d}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. times-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      12. distribute-rgt-out N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)}\right), \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   9. Applied egg-rr 85.0%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)}}{2 \cdot w} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, h\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, d\right), D\right), \left(\frac{d \cdot c0}{D}\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       2. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, h\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, d\right), D\right), \left(d \cdot \frac{c0}{D}\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, h\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, d\right), D\right), \mathsf{*.f64}\left(d, \left(\frac{c0}{D}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       4. /-lowering-/.f64 85.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, h\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, d\right), D\right), \mathsf{*.f64}\left(d, \mathsf{/.f64}\left(c0, D\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   11. Applied egg-rr 85.0%

       \[\leadsto \frac{c0 \cdot \left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \left(\frac{c0 \cdot d}{D} + \color{blue}{d \cdot \frac{c0}{D}}\right)\right)}{2 \cdot w} \]

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 0.0%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Simplified 0.6%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{d \cdot \frac{\frac{c0 \cdot d}{w}}{h}}{D \cdot D} + \sqrt{\frac{\frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot \left(d \cdot \frac{\frac{c0 \cdot d}{w}}{h}\right)\right)}{D \cdot D}}{w \cdot h}}{D \cdot D} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot c0\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. mul0-lft N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(c0 \cdot \left(-1 + 1\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(\mathsf{neg}\left(\left(-1 + 1\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(\mathsf{neg}\left(0\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      13. metadata-eval 44.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 44.6%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(c0 \cdot 0\right)}}{2 \cdot w} \]
   8. Step-by-step derivation

      1. mul0-rgt N/A

         \[\leadsto \frac{c0 \cdot 0}{2 \cdot w} \]

      2. mul0-rgt N/A

         \[\leadsto \frac{0}{\color{blue}{2} \cdot w} \]

      3. div0 44.6%

         \[\leadsto 0 \]

   9. Applied egg-rr 44.6%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0 \cdot \left(\frac{d}{\left(w \cdot h\right) \cdot D} \cdot \left(\frac{c0 \cdot d}{D} + d \cdot \frac{c0}{D}\right)\right)}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 44.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{2}{\frac{D}{c0 \cdot d}}}{\frac{w \cdot h}{\frac{d}{D}}}\\ \mathbf{if}\;D \cdot D \leq 10^{-293}:\\ \;\;\;\;c0 \cdot \frac{t\_0}{2 \cdot w}\\ \mathbf{elif}\;D \cdot D \leq 10^{-204}:\\ \;\;\;\;0\\ \mathbf{elif}\;D \cdot D \leq 5 \cdot 10^{-60}:\\ \;\;\;\;\frac{\frac{c0}{D \cdot D}}{w \cdot \left(\frac{w}{c0 \cdot d} \cdot \frac{h}{d}\right)}\\ \mathbf{elif}\;D \cdot D \leq 10^{-7}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot t\_0}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (/ 2.0 (/ D (* c0 d))) (/ (* w h) (/ d D)))))
   (if (<= (* D D) 1e-293)
     (* c0 (/ t_0 (* 2.0 w)))
     (if (<= (* D D) 1e-204)
       0.0
       (if (<= (* D D) 5e-60)
         (/ (/ c0 (* D D)) (* w (* (/ w (* c0 d)) (/ h d))))
         (if (<= (* D D) 1e-7) 0.0 (/ (* c0 t_0) (* 2.0 w))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (2.0 / (D / (c0 * d))) / ((w * h) / (d / D));
	double tmp;
	if ((D * D) <= 1e-293) {
		tmp = c0 * (t_0 / (2.0 * w));
	} else if ((D * D) <= 1e-204) {
		tmp = 0.0;
	} else if ((D * D) <= 5e-60) {
		tmp = (c0 / (D * D)) / (w * ((w / (c0 * d)) * (h / d)));
	} else if ((D * D) <= 1e-7) {
		tmp = 0.0;
	} else {
		tmp = (c0 * t_0) / (2.0 * w);
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (2.0d0 / (d / (c0 * d_1))) / ((w * h) / (d_1 / d))
    if ((d * d) <= 1d-293) then
        tmp = c0 * (t_0 / (2.0d0 * w))
    else if ((d * d) <= 1d-204) then
        tmp = 0.0d0
    else if ((d * d) <= 5d-60) then
        tmp = (c0 / (d * d)) / (w * ((w / (c0 * d_1)) * (h / d_1)))
    else if ((d * d) <= 1d-7) then
        tmp = 0.0d0
    else
        tmp = (c0 * t_0) / (2.0d0 * w)
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (2.0 / (D / (c0 * d))) / ((w * h) / (d / D));
	double tmp;
	if ((D * D) <= 1e-293) {
		tmp = c0 * (t_0 / (2.0 * w));
	} else if ((D * D) <= 1e-204) {
		tmp = 0.0;
	} else if ((D * D) <= 5e-60) {
		tmp = (c0 / (D * D)) / (w * ((w / (c0 * d)) * (h / d)));
	} else if ((D * D) <= 1e-7) {
		tmp = 0.0;
	} else {
		tmp = (c0 * t_0) / (2.0 * w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (2.0 / (D / (c0 * d))) / ((w * h) / (d / D))
	tmp = 0
	if (D * D) <= 1e-293:
		tmp = c0 * (t_0 / (2.0 * w))
	elif (D * D) <= 1e-204:
		tmp = 0.0
	elif (D * D) <= 5e-60:
		tmp = (c0 / (D * D)) / (w * ((w / (c0 * d)) * (h / d)))
	elif (D * D) <= 1e-7:
		tmp = 0.0
	else:
		tmp = (c0 * t_0) / (2.0 * w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(2.0 / Float64(D / Float64(c0 * d))) / Float64(Float64(w * h) / Float64(d / D)))
	tmp = 0.0
	if (Float64(D * D) <= 1e-293)
		tmp = Float64(c0 * Float64(t_0 / Float64(2.0 * w)));
	elseif (Float64(D * D) <= 1e-204)
		tmp = 0.0;
	elseif (Float64(D * D) <= 5e-60)
		tmp = Float64(Float64(c0 / Float64(D * D)) / Float64(w * Float64(Float64(w / Float64(c0 * d)) * Float64(h / d))));
	elseif (Float64(D * D) <= 1e-7)
		tmp = 0.0;
	else
		tmp = Float64(Float64(c0 * t_0) / Float64(2.0 * w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (2.0 / (D / (c0 * d))) / ((w * h) / (d / D));
	tmp = 0.0;
	if ((D * D) <= 1e-293)
		tmp = c0 * (t_0 / (2.0 * w));
	elseif ((D * D) <= 1e-204)
		tmp = 0.0;
	elseif ((D * D) <= 5e-60)
		tmp = (c0 / (D * D)) / (w * ((w / (c0 * d)) * (h / d)));
	elseif ((D * D) <= 1e-7)
		tmp = 0.0;
	else
		tmp = (c0 * t_0) / (2.0 * w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(2.0 / N[(D / N[(c0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(D * D), $MachinePrecision], 1e-293], N[(c0 * N[(t$95$0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(D * D), $MachinePrecision], 1e-204], 0.0, If[LessEqual[N[(D * D), $MachinePrecision], 5e-60], N[(N[(c0 / N[(D * D), $MachinePrecision]), $MachinePrecision] / N[(w * N[(N[(w / N[(c0 * d), $MachinePrecision]), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(D * D), $MachinePrecision], 1e-7], 0.0, N[(N[(c0 * t$95$0), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 D D) < 1.0000000000000001e-293

   1. Initial program 23.7%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Simplified 22.7%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{d \cdot \frac{\frac{c0 \cdot d}{w}}{h}}{D \cdot D} + \sqrt{\frac{\frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot \left(d \cdot \frac{\frac{c0 \cdot d}{w}}{h}\right)\right)}{D \cdot D}}{w \cdot h}}{D \cdot D} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\left(2 \cdot \left(c0 \cdot {d}^{2}\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot {d}^{2}\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left({d}^{2}\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(d \cdot d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(w \cdot h\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. *-lowering-*.f64 31.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(w, h\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 31.0%

      \[\leadsto \frac{c0 \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}}{2 \cdot w} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. count-2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{\left(c0 \cdot d\right) \cdot d}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{\left(c0 \cdot d\right) \cdot d}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{\left(c0 \cdot d\right) \cdot d}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{\left(c0 \cdot d\right) \cdot d}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. times-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      12. distribute-rgt-out N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)}\right), \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   9. Applied egg-rr 50.3%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)}}{2 \cdot w} \]
   10. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto c0 \cdot \color{blue}{\frac{\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)}{2 \cdot w}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)}{2 \cdot w} \cdot \color{blue}{c0} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)}{2 \cdot w}\right), \color{blue}{c0}\right) \]

   11. Applied egg-rr 51.1%

       \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\frac{D}{c0 \cdot d}}}{\frac{w \cdot h}{\frac{d}{D}}}}{w \cdot 2} \cdot c0} \]

   if 1.0000000000000001e-293 < (*.f64 D D) < 1e-204 or 5.0000000000000001e-60 < (*.f64 D D) < 9.9999999999999995e-8

   1. Initial program 14.7%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Simplified 9.4%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{d \cdot \frac{\frac{c0 \cdot d}{w}}{h}}{D \cdot D} + \sqrt{\frac{\frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot \left(d \cdot \frac{\frac{c0 \cdot d}{w}}{h}\right)\right)}{D \cdot D}}{w \cdot h}}{D \cdot D} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot c0\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. mul0-lft N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(c0 \cdot \left(-1 + 1\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(\mathsf{neg}\left(\left(-1 + 1\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(\mathsf{neg}\left(0\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      13. metadata-eval 62.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 62.0%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(c0 \cdot 0\right)}}{2 \cdot w} \]
   8. Step-by-step derivation

      1. mul0-rgt N/A

         \[\leadsto \frac{c0 \cdot 0}{2 \cdot w} \]

      2. mul0-rgt N/A

         \[\leadsto \frac{0}{\color{blue}{2} \cdot w} \]

      3. div0 62.0%

         \[\leadsto 0 \]

   9. Applied egg-rr 62.0%

      \[\leadsto \color{blue}{0} \]

   if 1e-204 < (*.f64 D D) < 5.0000000000000001e-60

   1. Initial program 26.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Simplified 27.0%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{d \cdot \frac{\frac{c0 \cdot d}{w}}{h}}{D \cdot D} + \sqrt{\frac{\frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot \left(d \cdot \frac{\frac{c0 \cdot d}{w}}{h}\right)\right)}{D \cdot D}}{w \cdot h}}{D \cdot D} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\left(2 \cdot \left(c0 \cdot {d}^{2}\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot {d}^{2}\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left({d}^{2}\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(d \cdot d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(w \cdot h\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. *-lowering-*.f64 34.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(w, h\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 34.6%

      \[\leadsto \frac{c0 \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}}{2 \cdot w} \]
   8. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(\frac{2}{D \cdot D} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{w \cdot h}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(c0 \cdot \frac{2}{D \cdot D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{w \cdot h}\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(c0 \cdot \frac{2}{D \cdot D}\right) \cdot \frac{\frac{c0 \cdot \left(d \cdot d\right)}{w}}{h}\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(c0 \cdot \frac{2}{D \cdot D}\right) \cdot \frac{c0 \cdot \frac{d \cdot d}{w}}{h}\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(c0 \cdot \frac{2}{D \cdot D}\right), \left(\frac{c0 \cdot \frac{d \cdot d}{w}}{h}\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(c0 \cdot \frac{1}{\frac{D \cdot D}{2}}\right), \left(\frac{c0 \cdot \frac{d \cdot d}{w}}{h}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{c0}{\frac{D \cdot D}{2}}\right), \left(\frac{c0 \cdot \frac{d \cdot d}{w}}{h}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \left(\frac{D \cdot D}{2}\right)\right), \left(\frac{c0 \cdot \frac{d \cdot d}{w}}{h}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\left(D \cdot D\right), 2\right)\right), \left(\frac{c0 \cdot \frac{d \cdot d}{w}}{h}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \left(\frac{c0 \cdot \frac{d \cdot d}{w}}{h}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \left(c0 \cdot \frac{\frac{d \cdot d}{w}}{h}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      12. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \left(c0 \cdot \frac{1}{\frac{h}{\frac{d \cdot d}{w}}}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      13. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \left(\frac{c0}{\frac{h}{\frac{d \cdot d}{w}}}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \mathsf{/.f64}\left(c0, \left(\frac{h}{\frac{d \cdot d}{w}}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(h, \left(\frac{d \cdot d}{w}\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      16. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(h, \left(d \cdot \frac{d}{w}\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      17. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(h, \left(d \cdot \frac{1}{\frac{w}{d}}\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      18. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(h, \left(\frac{d}{\frac{w}{d}}\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      19. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \left(\frac{w}{d}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      20. /-lowering-/.f64 44.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{/.f64}\left(w, d\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   9. Applied egg-rr 44.4%

      \[\leadsto \frac{\color{blue}{\frac{c0}{\frac{D \cdot D}{2}} \cdot \frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}}{2 \cdot w} \]
   10. Step-by-step derivation

       1. times-frac N/A

          \[\leadsto \frac{\frac{c0}{\frac{D \cdot D}{2}}}{2} \cdot \color{blue}{\frac{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}{w}} \]

       2. clear-num N/A

          \[\leadsto \frac{\frac{c0}{\frac{D \cdot D}{2}}}{2} \cdot \frac{1}{\color{blue}{\frac{w}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}}} \]

       3. un-div-inv N/A

          \[\leadsto \frac{\frac{\frac{c0}{\frac{D \cdot D}{2}}}{2}}{\color{blue}{\frac{w}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}}} \]

       4. associate-/r/ N/A

          \[\leadsto \frac{\frac{\frac{c0}{D \cdot D} \cdot 2}{2}}{\frac{w}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}} \]

       5. associate-/l* N/A

          \[\leadsto \frac{\frac{c0}{D \cdot D} \cdot \frac{2}{2}}{\frac{\color{blue}{w}}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}} \]

       6. metadata-eval N/A

          \[\leadsto \frac{\frac{c0}{D \cdot D} \cdot 1}{\frac{w}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}} \]

       7. *-rgt-identity N/A

          \[\leadsto \frac{\frac{c0}{D \cdot D}}{\frac{\color{blue}{w}}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}} \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{c0}{D \cdot D}\right), \color{blue}{\left(\frac{w}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}\right)}\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \left(D \cdot D\right)\right), \left(\frac{\color{blue}{w}}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{w}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \color{blue}{\left(\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}\right)}\right)\right) \]

       12. associate-/r/ N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \left(\frac{c0}{h} \cdot \color{blue}{\frac{d}{\frac{w}{d}}}\right)\right)\right) \]

       13. frac-times N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \left(\frac{c0 \cdot d}{\color{blue}{h \cdot \frac{w}{d}}}\right)\right)\right) \]

       14. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \mathsf{/.f64}\left(\left(c0 \cdot d\right), \color{blue}{\left(h \cdot \frac{w}{d}\right)}\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, d\right), \left(\color{blue}{h} \cdot \frac{w}{d}\right)\right)\right)\right) \]

       16. clear-num N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, d\right), \left(h \cdot \frac{1}{\color{blue}{\frac{d}{w}}}\right)\right)\right)\right) \]

       17. un-div-inv N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, d\right), \left(\frac{h}{\color{blue}{\frac{d}{w}}}\right)\right)\right)\right) \]

       18. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, d\right), \mathsf{/.f64}\left(h, \color{blue}{\left(\frac{d}{w}\right)}\right)\right)\right)\right) \]

       19. /-lowering-/.f64 57.9%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, d\right), \mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \color{blue}{w}\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 57.9%

       \[\leadsto \color{blue}{\frac{\frac{c0}{D \cdot D}}{\frac{w}{\frac{c0 \cdot d}{\frac{h}{\frac{d}{w}}}}}} \]
   12. Step-by-step derivation

       1. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{w}{c0 \cdot d} \cdot \color{blue}{\frac{h}{\frac{d}{w}}}\right)\right) \]

       2. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{w}{c0 \cdot d} \cdot \left(\frac{h}{d} \cdot \color{blue}{w}\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \left(\left(\frac{w}{c0 \cdot d} \cdot \frac{h}{d}\right) \cdot \color{blue}{w}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\left(\frac{w}{c0 \cdot d} \cdot \frac{h}{d}\right), \color{blue}{w}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{w}{c0 \cdot d}\right), \left(\frac{h}{d}\right)\right), w\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(w, \left(c0 \cdot d\right)\right), \left(\frac{h}{d}\right)\right), w\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(w, \left(d \cdot c0\right)\right), \left(\frac{h}{d}\right)\right), w\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(w, \mathsf{*.f64}\left(d, c0\right)\right), \left(\frac{h}{d}\right)\right), w\right)\right) \]

       9. /-lowering-/.f64 58.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(w, \mathsf{*.f64}\left(d, c0\right)\right), \mathsf{/.f64}\left(h, d\right)\right), w\right)\right) \]

   13. Applied egg-rr 58.0%

       \[\leadsto \frac{\frac{c0}{D \cdot D}}{\color{blue}{\left(\frac{w}{d \cdot c0} \cdot \frac{h}{d}\right) \cdot w}} \]

   if 9.9999999999999995e-8 < (*.f64 D D)

   1. Initial program 24.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Simplified 24.9%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{d \cdot \frac{\frac{c0 \cdot d}{w}}{h}}{D \cdot D} + \sqrt{\frac{\frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot \left(d \cdot \frac{\frac{c0 \cdot d}{w}}{h}\right)\right)}{D \cdot D}}{w \cdot h}}{D \cdot D} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\left(2 \cdot \left(c0 \cdot {d}^{2}\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot {d}^{2}\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left({d}^{2}\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(d \cdot d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(w \cdot h\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. *-lowering-*.f64 32.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(w, h\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 32.0%

      \[\leadsto \frac{c0 \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}}{2 \cdot w} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. count-2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{\left(c0 \cdot d\right) \cdot d}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{\left(c0 \cdot d\right) \cdot d}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{\left(c0 \cdot d\right) \cdot d}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{\left(c0 \cdot d\right) \cdot d}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. times-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      12. distribute-rgt-out N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)}\right), \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   9. Applied egg-rr 64.5%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)}}{2 \cdot w} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right) \cdot c0\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right), c0\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

   11. Applied egg-rr 64.7%

       \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\frac{D}{c0 \cdot d}}}{\frac{w \cdot h}{\frac{d}{D}}} \cdot c0}}{2 \cdot w} \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \cdot D \leq 10^{-293}:\\ \;\;\;\;c0 \cdot \frac{\frac{\frac{2}{\frac{D}{c0 \cdot d}}}{\frac{w \cdot h}{\frac{d}{D}}}}{2 \cdot w}\\ \mathbf{elif}\;D \cdot D \leq 10^{-204}:\\ \;\;\;\;0\\ \mathbf{elif}\;D \cdot D \leq 5 \cdot 10^{-60}:\\ \;\;\;\;\frac{\frac{c0}{D \cdot D}}{w \cdot \left(\frac{w}{c0 \cdot d} \cdot \frac{h}{d}\right)}\\ \mathbf{elif}\;D \cdot D \leq 10^{-7}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \frac{\frac{2}{\frac{D}{c0 \cdot d}}}{\frac{w \cdot h}{\frac{d}{D}}}}{2 \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 43.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \cdot D \leq 10^{-293}:\\ \;\;\;\;c0 \cdot \frac{\frac{\frac{2}{\frac{D}{c0 \cdot d}}}{\frac{w \cdot h}{\frac{d}{D}}}}{2 \cdot w}\\ \mathbf{elif}\;D \cdot D \leq 10^{-204}:\\ \;\;\;\;0\\ \mathbf{elif}\;D \cdot D \leq 5 \cdot 10^{-60}:\\ \;\;\;\;\frac{\frac{c0}{D \cdot D}}{w \cdot \left(\frac{w}{c0 \cdot d} \cdot \frac{h}{d}\right)}\\ \mathbf{elif}\;D \cdot D \leq 10^{-7}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c0 \cdot 2\right) \cdot \left(\left(c0 \cdot d\right) \cdot \frac{\frac{d}{D}}{\left(w \cdot h\right) \cdot D}\right)}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= (* D D) 1e-293)
   (* c0 (/ (/ (/ 2.0 (/ D (* c0 d))) (/ (* w h) (/ d D))) (* 2.0 w)))
   (if (<= (* D D) 1e-204)
     0.0
     (if (<= (* D D) 5e-60)
       (/ (/ c0 (* D D)) (* w (* (/ w (* c0 d)) (/ h d))))
       (if (<= (* D D) 1e-7)
         0.0
         (/
          (* (* c0 2.0) (* (* c0 d) (/ (/ d D) (* (* w h) D))))
          (* 2.0 w)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((D * D) <= 1e-293) {
		tmp = c0 * (((2.0 / (D / (c0 * d))) / ((w * h) / (d / D))) / (2.0 * w));
	} else if ((D * D) <= 1e-204) {
		tmp = 0.0;
	} else if ((D * D) <= 5e-60) {
		tmp = (c0 / (D * D)) / (w * ((w / (c0 * d)) * (h / d)));
	} else if ((D * D) <= 1e-7) {
		tmp = 0.0;
	} else {
		tmp = ((c0 * 2.0) * ((c0 * d) * ((d / D) / ((w * h) * D)))) / (2.0 * w);
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((d * d) <= 1d-293) then
        tmp = c0 * (((2.0d0 / (d / (c0 * d_1))) / ((w * h) / (d_1 / d))) / (2.0d0 * w))
    else if ((d * d) <= 1d-204) then
        tmp = 0.0d0
    else if ((d * d) <= 5d-60) then
        tmp = (c0 / (d * d)) / (w * ((w / (c0 * d_1)) * (h / d_1)))
    else if ((d * d) <= 1d-7) then
        tmp = 0.0d0
    else
        tmp = ((c0 * 2.0d0) * ((c0 * d_1) * ((d_1 / d) / ((w * h) * d)))) / (2.0d0 * w)
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((D * D) <= 1e-293) {
		tmp = c0 * (((2.0 / (D / (c0 * d))) / ((w * h) / (d / D))) / (2.0 * w));
	} else if ((D * D) <= 1e-204) {
		tmp = 0.0;
	} else if ((D * D) <= 5e-60) {
		tmp = (c0 / (D * D)) / (w * ((w / (c0 * d)) * (h / d)));
	} else if ((D * D) <= 1e-7) {
		tmp = 0.0;
	} else {
		tmp = ((c0 * 2.0) * ((c0 * d) * ((d / D) / ((w * h) * D)))) / (2.0 * w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (D * D) <= 1e-293:
		tmp = c0 * (((2.0 / (D / (c0 * d))) / ((w * h) / (d / D))) / (2.0 * w))
	elif (D * D) <= 1e-204:
		tmp = 0.0
	elif (D * D) <= 5e-60:
		tmp = (c0 / (D * D)) / (w * ((w / (c0 * d)) * (h / d)))
	elif (D * D) <= 1e-7:
		tmp = 0.0
	else:
		tmp = ((c0 * 2.0) * ((c0 * d) * ((d / D) / ((w * h) * D)))) / (2.0 * w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (Float64(D * D) <= 1e-293)
		tmp = Float64(c0 * Float64(Float64(Float64(2.0 / Float64(D / Float64(c0 * d))) / Float64(Float64(w * h) / Float64(d / D))) / Float64(2.0 * w)));
	elseif (Float64(D * D) <= 1e-204)
		tmp = 0.0;
	elseif (Float64(D * D) <= 5e-60)
		tmp = Float64(Float64(c0 / Float64(D * D)) / Float64(w * Float64(Float64(w / Float64(c0 * d)) * Float64(h / d))));
	elseif (Float64(D * D) <= 1e-7)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(c0 * 2.0) * Float64(Float64(c0 * d) * Float64(Float64(d / D) / Float64(Float64(w * h) * D)))) / Float64(2.0 * w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((D * D) <= 1e-293)
		tmp = c0 * (((2.0 / (D / (c0 * d))) / ((w * h) / (d / D))) / (2.0 * w));
	elseif ((D * D) <= 1e-204)
		tmp = 0.0;
	elseif ((D * D) <= 5e-60)
		tmp = (c0 / (D * D)) / (w * ((w / (c0 * d)) * (h / d)));
	elseif ((D * D) <= 1e-7)
		tmp = 0.0;
	else
		tmp = ((c0 * 2.0) * ((c0 * d) * ((d / D) / ((w * h) * D)))) / (2.0 * w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[N[(D * D), $MachinePrecision], 1e-293], N[(c0 * N[(N[(N[(2.0 / N[(D / N[(c0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(D * D), $MachinePrecision], 1e-204], 0.0, If[LessEqual[N[(D * D), $MachinePrecision], 5e-60], N[(N[(c0 / N[(D * D), $MachinePrecision]), $MachinePrecision] / N[(w * N[(N[(w / N[(c0 * d), $MachinePrecision]), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(D * D), $MachinePrecision], 1e-7], 0.0, N[(N[(N[(c0 * 2.0), $MachinePrecision] * N[(N[(c0 * d), $MachinePrecision] * N[(N[(d / D), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 D D) < 1.0000000000000001e-293

   1. Initial program 23.7%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Simplified 22.7%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{d \cdot \frac{\frac{c0 \cdot d}{w}}{h}}{D \cdot D} + \sqrt{\frac{\frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot \left(d \cdot \frac{\frac{c0 \cdot d}{w}}{h}\right)\right)}{D \cdot D}}{w \cdot h}}{D \cdot D} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\left(2 \cdot \left(c0 \cdot {d}^{2}\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot {d}^{2}\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left({d}^{2}\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(d \cdot d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(w \cdot h\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. *-lowering-*.f64 31.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(w, h\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 31.0%

      \[\leadsto \frac{c0 \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}}{2 \cdot w} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. count-2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{\left(c0 \cdot d\right) \cdot d}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{\left(c0 \cdot d\right) \cdot d}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{\left(c0 \cdot d\right) \cdot d}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{\left(c0 \cdot d\right) \cdot d}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. times-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      12. distribute-rgt-out N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)}\right), \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   9. Applied egg-rr 50.3%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)}}{2 \cdot w} \]
   10. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto c0 \cdot \color{blue}{\frac{\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)}{2 \cdot w}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)}{2 \cdot w} \cdot \color{blue}{c0} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)}{2 \cdot w}\right), \color{blue}{c0}\right) \]

   11. Applied egg-rr 51.1%

       \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\frac{D}{c0 \cdot d}}}{\frac{w \cdot h}{\frac{d}{D}}}}{w \cdot 2} \cdot c0} \]

   if 1.0000000000000001e-293 < (*.f64 D D) < 1e-204 or 5.0000000000000001e-60 < (*.f64 D D) < 9.9999999999999995e-8

   1. Initial program 14.7%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Simplified 9.4%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{d \cdot \frac{\frac{c0 \cdot d}{w}}{h}}{D \cdot D} + \sqrt{\frac{\frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot \left(d \cdot \frac{\frac{c0 \cdot d}{w}}{h}\right)\right)}{D \cdot D}}{w \cdot h}}{D \cdot D} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot c0\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. mul0-lft N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(c0 \cdot \left(-1 + 1\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(\mathsf{neg}\left(\left(-1 + 1\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(\mathsf{neg}\left(0\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      13. metadata-eval 62.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 62.0%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(c0 \cdot 0\right)}}{2 \cdot w} \]
   8. Step-by-step derivation

      1. mul0-rgt N/A

         \[\leadsto \frac{c0 \cdot 0}{2 \cdot w} \]

      2. mul0-rgt N/A

         \[\leadsto \frac{0}{\color{blue}{2} \cdot w} \]

      3. div0 62.0%

         \[\leadsto 0 \]

   9. Applied egg-rr 62.0%

      \[\leadsto \color{blue}{0} \]

   if 1e-204 < (*.f64 D D) < 5.0000000000000001e-60

   1. Initial program 26.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Simplified 27.0%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{d \cdot \frac{\frac{c0 \cdot d}{w}}{h}}{D \cdot D} + \sqrt{\frac{\frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot \left(d \cdot \frac{\frac{c0 \cdot d}{w}}{h}\right)\right)}{D \cdot D}}{w \cdot h}}{D \cdot D} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\left(2 \cdot \left(c0 \cdot {d}^{2}\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot {d}^{2}\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left({d}^{2}\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(d \cdot d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(w \cdot h\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. *-lowering-*.f64 34.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(w, h\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 34.6%

      \[\leadsto \frac{c0 \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}}{2 \cdot w} \]
   8. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(\frac{2}{D \cdot D} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{w \cdot h}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(c0 \cdot \frac{2}{D \cdot D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{w \cdot h}\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(c0 \cdot \frac{2}{D \cdot D}\right) \cdot \frac{\frac{c0 \cdot \left(d \cdot d\right)}{w}}{h}\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(c0 \cdot \frac{2}{D \cdot D}\right) \cdot \frac{c0 \cdot \frac{d \cdot d}{w}}{h}\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(c0 \cdot \frac{2}{D \cdot D}\right), \left(\frac{c0 \cdot \frac{d \cdot d}{w}}{h}\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(c0 \cdot \frac{1}{\frac{D \cdot D}{2}}\right), \left(\frac{c0 \cdot \frac{d \cdot d}{w}}{h}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{c0}{\frac{D \cdot D}{2}}\right), \left(\frac{c0 \cdot \frac{d \cdot d}{w}}{h}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \left(\frac{D \cdot D}{2}\right)\right), \left(\frac{c0 \cdot \frac{d \cdot d}{w}}{h}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\left(D \cdot D\right), 2\right)\right), \left(\frac{c0 \cdot \frac{d \cdot d}{w}}{h}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \left(\frac{c0 \cdot \frac{d \cdot d}{w}}{h}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \left(c0 \cdot \frac{\frac{d \cdot d}{w}}{h}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      12. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \left(c0 \cdot \frac{1}{\frac{h}{\frac{d \cdot d}{w}}}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      13. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \left(\frac{c0}{\frac{h}{\frac{d \cdot d}{w}}}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \mathsf{/.f64}\left(c0, \left(\frac{h}{\frac{d \cdot d}{w}}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(h, \left(\frac{d \cdot d}{w}\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      16. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(h, \left(d \cdot \frac{d}{w}\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      17. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(h, \left(d \cdot \frac{1}{\frac{w}{d}}\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      18. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(h, \left(\frac{d}{\frac{w}{d}}\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      19. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \left(\frac{w}{d}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      20. /-lowering-/.f64 44.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{/.f64}\left(w, d\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   9. Applied egg-rr 44.4%

      \[\leadsto \frac{\color{blue}{\frac{c0}{\frac{D \cdot D}{2}} \cdot \frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}}{2 \cdot w} \]
   10. Step-by-step derivation

       1. times-frac N/A

          \[\leadsto \frac{\frac{c0}{\frac{D \cdot D}{2}}}{2} \cdot \color{blue}{\frac{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}{w}} \]

       2. clear-num N/A

          \[\leadsto \frac{\frac{c0}{\frac{D \cdot D}{2}}}{2} \cdot \frac{1}{\color{blue}{\frac{w}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}}} \]

       3. un-div-inv N/A

          \[\leadsto \frac{\frac{\frac{c0}{\frac{D \cdot D}{2}}}{2}}{\color{blue}{\frac{w}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}}} \]

       4. associate-/r/ N/A

          \[\leadsto \frac{\frac{\frac{c0}{D \cdot D} \cdot 2}{2}}{\frac{w}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}} \]

       5. associate-/l* N/A

          \[\leadsto \frac{\frac{c0}{D \cdot D} \cdot \frac{2}{2}}{\frac{\color{blue}{w}}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}} \]

       6. metadata-eval N/A

          \[\leadsto \frac{\frac{c0}{D \cdot D} \cdot 1}{\frac{w}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}} \]

       7. *-rgt-identity N/A

          \[\leadsto \frac{\frac{c0}{D \cdot D}}{\frac{\color{blue}{w}}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}} \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{c0}{D \cdot D}\right), \color{blue}{\left(\frac{w}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}\right)}\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \left(D \cdot D\right)\right), \left(\frac{\color{blue}{w}}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{w}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \color{blue}{\left(\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}\right)}\right)\right) \]

       12. associate-/r/ N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \left(\frac{c0}{h} \cdot \color{blue}{\frac{d}{\frac{w}{d}}}\right)\right)\right) \]

       13. frac-times N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \left(\frac{c0 \cdot d}{\color{blue}{h \cdot \frac{w}{d}}}\right)\right)\right) \]

       14. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \mathsf{/.f64}\left(\left(c0 \cdot d\right), \color{blue}{\left(h \cdot \frac{w}{d}\right)}\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, d\right), \left(\color{blue}{h} \cdot \frac{w}{d}\right)\right)\right)\right) \]

       16. clear-num N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, d\right), \left(h \cdot \frac{1}{\color{blue}{\frac{d}{w}}}\right)\right)\right)\right) \]

       17. un-div-inv N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, d\right), \left(\frac{h}{\color{blue}{\frac{d}{w}}}\right)\right)\right)\right) \]

       18. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, d\right), \mathsf{/.f64}\left(h, \color{blue}{\left(\frac{d}{w}\right)}\right)\right)\right)\right) \]

       19. /-lowering-/.f64 57.9%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, d\right), \mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \color{blue}{w}\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 57.9%

       \[\leadsto \color{blue}{\frac{\frac{c0}{D \cdot D}}{\frac{w}{\frac{c0 \cdot d}{\frac{h}{\frac{d}{w}}}}}} \]
   12. Step-by-step derivation

       1. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{w}{c0 \cdot d} \cdot \color{blue}{\frac{h}{\frac{d}{w}}}\right)\right) \]

       2. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{w}{c0 \cdot d} \cdot \left(\frac{h}{d} \cdot \color{blue}{w}\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \left(\left(\frac{w}{c0 \cdot d} \cdot \frac{h}{d}\right) \cdot \color{blue}{w}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\left(\frac{w}{c0 \cdot d} \cdot \frac{h}{d}\right), \color{blue}{w}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{w}{c0 \cdot d}\right), \left(\frac{h}{d}\right)\right), w\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(w, \left(c0 \cdot d\right)\right), \left(\frac{h}{d}\right)\right), w\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(w, \left(d \cdot c0\right)\right), \left(\frac{h}{d}\right)\right), w\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(w, \mathsf{*.f64}\left(d, c0\right)\right), \left(\frac{h}{d}\right)\right), w\right)\right) \]

       9. /-lowering-/.f64 58.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(w, \mathsf{*.f64}\left(d, c0\right)\right), \mathsf{/.f64}\left(h, d\right)\right), w\right)\right) \]

   13. Applied egg-rr 58.0%

       \[\leadsto \frac{\frac{c0}{D \cdot D}}{\color{blue}{\left(\frac{w}{d \cdot c0} \cdot \frac{h}{d}\right) \cdot w}} \]

   if 9.9999999999999995e-8 < (*.f64 D D)

   1. Initial program 24.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Simplified 24.9%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{d \cdot \frac{\frac{c0 \cdot d}{w}}{h}}{D \cdot D} + \sqrt{\frac{\frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot \left(d \cdot \frac{\frac{c0 \cdot d}{w}}{h}\right)\right)}{D \cdot D}}{w \cdot h}}{D \cdot D} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\left(2 \cdot \left(c0 \cdot {d}^{2}\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot {d}^{2}\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left({d}^{2}\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(d \cdot d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(w \cdot h\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. *-lowering-*.f64 32.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(w, h\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 32.0%

      \[\leadsto \frac{c0 \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}}{2 \cdot w} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. count-2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{\left(c0 \cdot d\right) \cdot d}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{\left(c0 \cdot d\right) \cdot d}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{\left(c0 \cdot d\right) \cdot d}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{\left(c0 \cdot d\right) \cdot d}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. times-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      12. distribute-rgt-out N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)}\right), \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   9. Applied egg-rr 64.5%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)}}{2 \cdot w} \]
   10. Step-by-step derivation

       1. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       2. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right) \cdot c0 + \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right) \cdot c0\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

       3. distribute-lft-out N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right) \cdot \left(c0 + c0\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot d}{D}\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       6. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \frac{1}{\frac{D}{c0 \cdot d}}\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       7. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{d}{D \cdot \left(w \cdot h\right)}}{\frac{D}{c0 \cdot d}}\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)}\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       9. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{d}{D}}{w \cdot h}\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{D}\right), \left(w \cdot h\right)\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \left(w \cdot h\right)\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(D, \left(c0 \cdot d\right)\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(c0, d\right)\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       15. count-2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(c0, d\right)\right)\right), \left(2 \cdot c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       16. *-lowering-*.f64 64.7%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(c0, d\right)\right)\right), \mathsf{*.f64}\left(2, c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   11. Applied egg-rr 64.7%

       \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{d}{D}}{w \cdot h}}{\frac{D}{c0 \cdot d}} \cdot \left(2 \cdot c0\right)}}{2 \cdot w} \]
   12. Step-by-step derivation

       1. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{\frac{d}{D}}{w \cdot h}}{D} \cdot \left(c0 \cdot d\right)\right), \mathsf{*.f64}\left(2, c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{\frac{d}{D}}{w \cdot h}}{D}\right), \left(c0 \cdot d\right)\right), \mathsf{*.f64}\left(2, c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       3. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{d}{D}}{D \cdot \left(w \cdot h\right)}\right), \left(c0 \cdot d\right)\right), \mathsf{*.f64}\left(2, c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{d}{D}}{\left(w \cdot h\right) \cdot D}\right), \left(c0 \cdot d\right)\right), \mathsf{*.f64}\left(2, c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{D}\right), \left(\left(w \cdot h\right) \cdot D\right)\right), \left(c0 \cdot d\right)\right), \mathsf{*.f64}\left(2, c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \left(\left(w \cdot h\right) \cdot D\right)\right), \left(c0 \cdot d\right)\right), \mathsf{*.f64}\left(2, c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \left(D \cdot \left(w \cdot h\right)\right)\right), \left(c0 \cdot d\right)\right), \mathsf{*.f64}\left(2, c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(D, \left(w \cdot h\right)\right)\right), \left(c0 \cdot d\right)\right), \mathsf{*.f64}\left(2, c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, h\right)\right)\right), \left(c0 \cdot d\right)\right), \mathsf{*.f64}\left(2, c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, h\right)\right)\right), \left(d \cdot c0\right)\right), \mathsf{*.f64}\left(2, c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       11. *-lowering-*.f64 64.4%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, h\right)\right)\right), \mathsf{*.f64}\left(d, c0\right)\right), \mathsf{*.f64}\left(2, c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   13. Applied egg-rr 64.4%

       \[\leadsto \frac{\color{blue}{\left(\frac{\frac{d}{D}}{D \cdot \left(w \cdot h\right)} \cdot \left(d \cdot c0\right)\right)} \cdot \left(2 \cdot c0\right)}{2 \cdot w} \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \cdot D \leq 10^{-293}:\\ \;\;\;\;c0 \cdot \frac{\frac{\frac{2}{\frac{D}{c0 \cdot d}}}{\frac{w \cdot h}{\frac{d}{D}}}}{2 \cdot w}\\ \mathbf{elif}\;D \cdot D \leq 10^{-204}:\\ \;\;\;\;0\\ \mathbf{elif}\;D \cdot D \leq 5 \cdot 10^{-60}:\\ \;\;\;\;\frac{\frac{c0}{D \cdot D}}{w \cdot \left(\frac{w}{c0 \cdot d} \cdot \frac{h}{d}\right)}\\ \mathbf{elif}\;D \cdot D \leq 10^{-7}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c0 \cdot 2\right) \cdot \left(\left(c0 \cdot d\right) \cdot \frac{\frac{d}{D}}{\left(w \cdot h\right) \cdot D}\right)}{2 \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 44.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \cdot D \leq 10^{-293}:\\ \;\;\;\;c0 \cdot \frac{\frac{\frac{2}{\frac{D}{c0 \cdot d}}}{\frac{w \cdot h}{\frac{d}{D}}}}{2 \cdot w}\\ \mathbf{elif}\;D \cdot D \leq 10^{-204}:\\ \;\;\;\;0\\ \mathbf{elif}\;D \cdot D \leq 5 \cdot 10^{-60}:\\ \;\;\;\;\frac{\frac{c0}{D \cdot D}}{w \cdot \left(\frac{w}{c0 \cdot d} \cdot \frac{h}{d}\right)}\\ \mathbf{elif}\;D \cdot D \leq 1:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\frac{D}{c0}}}{\frac{h}{\frac{d}{w \cdot D}}} \cdot \frac{c0}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= (* D D) 1e-293)
   (* c0 (/ (/ (/ 2.0 (/ D (* c0 d))) (/ (* w h) (/ d D))) (* 2.0 w)))
   (if (<= (* D D) 1e-204)
     0.0
     (if (<= (* D D) 5e-60)
       (/ (/ c0 (* D D)) (* w (* (/ w (* c0 d)) (/ h d))))
       (if (<= (* D D) 1.0)
         0.0
         (* (/ (/ d (/ D c0)) (/ h (/ d (* w D)))) (/ c0 w)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((D * D) <= 1e-293) {
		tmp = c0 * (((2.0 / (D / (c0 * d))) / ((w * h) / (d / D))) / (2.0 * w));
	} else if ((D * D) <= 1e-204) {
		tmp = 0.0;
	} else if ((D * D) <= 5e-60) {
		tmp = (c0 / (D * D)) / (w * ((w / (c0 * d)) * (h / d)));
	} else if ((D * D) <= 1.0) {
		tmp = 0.0;
	} else {
		tmp = ((d / (D / c0)) / (h / (d / (w * D)))) * (c0 / w);
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((d * d) <= 1d-293) then
        tmp = c0 * (((2.0d0 / (d / (c0 * d_1))) / ((w * h) / (d_1 / d))) / (2.0d0 * w))
    else if ((d * d) <= 1d-204) then
        tmp = 0.0d0
    else if ((d * d) <= 5d-60) then
        tmp = (c0 / (d * d)) / (w * ((w / (c0 * d_1)) * (h / d_1)))
    else if ((d * d) <= 1.0d0) then
        tmp = 0.0d0
    else
        tmp = ((d_1 / (d / c0)) / (h / (d_1 / (w * d)))) * (c0 / w)
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((D * D) <= 1e-293) {
		tmp = c0 * (((2.0 / (D / (c0 * d))) / ((w * h) / (d / D))) / (2.0 * w));
	} else if ((D * D) <= 1e-204) {
		tmp = 0.0;
	} else if ((D * D) <= 5e-60) {
		tmp = (c0 / (D * D)) / (w * ((w / (c0 * d)) * (h / d)));
	} else if ((D * D) <= 1.0) {
		tmp = 0.0;
	} else {
		tmp = ((d / (D / c0)) / (h / (d / (w * D)))) * (c0 / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (D * D) <= 1e-293:
		tmp = c0 * (((2.0 / (D / (c0 * d))) / ((w * h) / (d / D))) / (2.0 * w))
	elif (D * D) <= 1e-204:
		tmp = 0.0
	elif (D * D) <= 5e-60:
		tmp = (c0 / (D * D)) / (w * ((w / (c0 * d)) * (h / d)))
	elif (D * D) <= 1.0:
		tmp = 0.0
	else:
		tmp = ((d / (D / c0)) / (h / (d / (w * D)))) * (c0 / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (Float64(D * D) <= 1e-293)
		tmp = Float64(c0 * Float64(Float64(Float64(2.0 / Float64(D / Float64(c0 * d))) / Float64(Float64(w * h) / Float64(d / D))) / Float64(2.0 * w)));
	elseif (Float64(D * D) <= 1e-204)
		tmp = 0.0;
	elseif (Float64(D * D) <= 5e-60)
		tmp = Float64(Float64(c0 / Float64(D * D)) / Float64(w * Float64(Float64(w / Float64(c0 * d)) * Float64(h / d))));
	elseif (Float64(D * D) <= 1.0)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(d / Float64(D / c0)) / Float64(h / Float64(d / Float64(w * D)))) * Float64(c0 / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((D * D) <= 1e-293)
		tmp = c0 * (((2.0 / (D / (c0 * d))) / ((w * h) / (d / D))) / (2.0 * w));
	elseif ((D * D) <= 1e-204)
		tmp = 0.0;
	elseif ((D * D) <= 5e-60)
		tmp = (c0 / (D * D)) / (w * ((w / (c0 * d)) * (h / d)));
	elseif ((D * D) <= 1.0)
		tmp = 0.0;
	else
		tmp = ((d / (D / c0)) / (h / (d / (w * D)))) * (c0 / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[N[(D * D), $MachinePrecision], 1e-293], N[(c0 * N[(N[(N[(2.0 / N[(D / N[(c0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(D * D), $MachinePrecision], 1e-204], 0.0, If[LessEqual[N[(D * D), $MachinePrecision], 5e-60], N[(N[(c0 / N[(D * D), $MachinePrecision]), $MachinePrecision] / N[(w * N[(N[(w / N[(c0 * d), $MachinePrecision]), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(D * D), $MachinePrecision], 1.0], 0.0, N[(N[(N[(d / N[(D / c0), $MachinePrecision]), $MachinePrecision] / N[(h / N[(d / N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c0 / w), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 D D) < 1.0000000000000001e-293

   1. Initial program 23.7%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Simplified 22.7%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{d \cdot \frac{\frac{c0 \cdot d}{w}}{h}}{D \cdot D} + \sqrt{\frac{\frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot \left(d \cdot \frac{\frac{c0 \cdot d}{w}}{h}\right)\right)}{D \cdot D}}{w \cdot h}}{D \cdot D} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\left(2 \cdot \left(c0 \cdot {d}^{2}\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot {d}^{2}\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left({d}^{2}\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(d \cdot d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(w \cdot h\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. *-lowering-*.f64 31.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(w, h\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 31.0%

      \[\leadsto \frac{c0 \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}}{2 \cdot w} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. count-2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{\left(c0 \cdot d\right) \cdot d}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{\left(c0 \cdot d\right) \cdot d}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{\left(c0 \cdot d\right) \cdot d}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{\left(c0 \cdot d\right) \cdot d}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. times-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      12. distribute-rgt-out N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)}\right), \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   9. Applied egg-rr 50.3%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)}}{2 \cdot w} \]
   10. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto c0 \cdot \color{blue}{\frac{\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)}{2 \cdot w}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)}{2 \cdot w} \cdot \color{blue}{c0} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)}{2 \cdot w}\right), \color{blue}{c0}\right) \]

   11. Applied egg-rr 51.1%

       \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\frac{D}{c0 \cdot d}}}{\frac{w \cdot h}{\frac{d}{D}}}}{w \cdot 2} \cdot c0} \]

   if 1.0000000000000001e-293 < (*.f64 D D) < 1e-204 or 5.0000000000000001e-60 < (*.f64 D D) < 1

   1. Initial program 14.2%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Simplified 9.2%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{d \cdot \frac{\frac{c0 \cdot d}{w}}{h}}{D \cdot D} + \sqrt{\frac{\frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot \left(d \cdot \frac{\frac{c0 \cdot d}{w}}{h}\right)\right)}{D \cdot D}}{w \cdot h}}{D \cdot D} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot c0\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. mul0-lft N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(c0 \cdot \left(-1 + 1\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(\mathsf{neg}\left(\left(-1 + 1\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(\mathsf{neg}\left(0\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      13. metadata-eval 60.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 60.2%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(c0 \cdot 0\right)}}{2 \cdot w} \]
   8. Step-by-step derivation

      1. mul0-rgt N/A

         \[\leadsto \frac{c0 \cdot 0}{2 \cdot w} \]

      2. mul0-rgt N/A

         \[\leadsto \frac{0}{\color{blue}{2} \cdot w} \]

      3. div0 60.2%

         \[\leadsto 0 \]

   9. Applied egg-rr 60.2%

      \[\leadsto \color{blue}{0} \]

   if 1e-204 < (*.f64 D D) < 5.0000000000000001e-60

   1. Initial program 26.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Simplified 27.0%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{d \cdot \frac{\frac{c0 \cdot d}{w}}{h}}{D \cdot D} + \sqrt{\frac{\frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot \left(d \cdot \frac{\frac{c0 \cdot d}{w}}{h}\right)\right)}{D \cdot D}}{w \cdot h}}{D \cdot D} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\left(2 \cdot \left(c0 \cdot {d}^{2}\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot {d}^{2}\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left({d}^{2}\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(d \cdot d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(w \cdot h\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. *-lowering-*.f64 34.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(w, h\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 34.6%

      \[\leadsto \frac{c0 \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}}{2 \cdot w} \]
   8. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(\frac{2}{D \cdot D} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{w \cdot h}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(c0 \cdot \frac{2}{D \cdot D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{w \cdot h}\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(c0 \cdot \frac{2}{D \cdot D}\right) \cdot \frac{\frac{c0 \cdot \left(d \cdot d\right)}{w}}{h}\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(c0 \cdot \frac{2}{D \cdot D}\right) \cdot \frac{c0 \cdot \frac{d \cdot d}{w}}{h}\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(c0 \cdot \frac{2}{D \cdot D}\right), \left(\frac{c0 \cdot \frac{d \cdot d}{w}}{h}\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(c0 \cdot \frac{1}{\frac{D \cdot D}{2}}\right), \left(\frac{c0 \cdot \frac{d \cdot d}{w}}{h}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{c0}{\frac{D \cdot D}{2}}\right), \left(\frac{c0 \cdot \frac{d \cdot d}{w}}{h}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \left(\frac{D \cdot D}{2}\right)\right), \left(\frac{c0 \cdot \frac{d \cdot d}{w}}{h}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\left(D \cdot D\right), 2\right)\right), \left(\frac{c0 \cdot \frac{d \cdot d}{w}}{h}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \left(\frac{c0 \cdot \frac{d \cdot d}{w}}{h}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \left(c0 \cdot \frac{\frac{d \cdot d}{w}}{h}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      12. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \left(c0 \cdot \frac{1}{\frac{h}{\frac{d \cdot d}{w}}}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      13. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \left(\frac{c0}{\frac{h}{\frac{d \cdot d}{w}}}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \mathsf{/.f64}\left(c0, \left(\frac{h}{\frac{d \cdot d}{w}}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(h, \left(\frac{d \cdot d}{w}\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      16. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(h, \left(d \cdot \frac{d}{w}\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      17. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(h, \left(d \cdot \frac{1}{\frac{w}{d}}\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      18. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(h, \left(\frac{d}{\frac{w}{d}}\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      19. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \left(\frac{w}{d}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      20. /-lowering-/.f64 44.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{/.f64}\left(w, d\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   9. Applied egg-rr 44.4%

      \[\leadsto \frac{\color{blue}{\frac{c0}{\frac{D \cdot D}{2}} \cdot \frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}}{2 \cdot w} \]
   10. Step-by-step derivation

       1. times-frac N/A

          \[\leadsto \frac{\frac{c0}{\frac{D \cdot D}{2}}}{2} \cdot \color{blue}{\frac{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}{w}} \]

       2. clear-num N/A

          \[\leadsto \frac{\frac{c0}{\frac{D \cdot D}{2}}}{2} \cdot \frac{1}{\color{blue}{\frac{w}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}}} \]

       3. un-div-inv N/A

          \[\leadsto \frac{\frac{\frac{c0}{\frac{D \cdot D}{2}}}{2}}{\color{blue}{\frac{w}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}}} \]

       4. associate-/r/ N/A

          \[\leadsto \frac{\frac{\frac{c0}{D \cdot D} \cdot 2}{2}}{\frac{w}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}} \]

       5. associate-/l* N/A

          \[\leadsto \frac{\frac{c0}{D \cdot D} \cdot \frac{2}{2}}{\frac{\color{blue}{w}}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}} \]

       6. metadata-eval N/A

          \[\leadsto \frac{\frac{c0}{D \cdot D} \cdot 1}{\frac{w}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}} \]

       7. *-rgt-identity N/A

          \[\leadsto \frac{\frac{c0}{D \cdot D}}{\frac{\color{blue}{w}}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}} \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{c0}{D \cdot D}\right), \color{blue}{\left(\frac{w}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}\right)}\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \left(D \cdot D\right)\right), \left(\frac{\color{blue}{w}}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{w}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \color{blue}{\left(\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}\right)}\right)\right) \]

       12. associate-/r/ N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \left(\frac{c0}{h} \cdot \color{blue}{\frac{d}{\frac{w}{d}}}\right)\right)\right) \]

       13. frac-times N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \left(\frac{c0 \cdot d}{\color{blue}{h \cdot \frac{w}{d}}}\right)\right)\right) \]

       14. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \mathsf{/.f64}\left(\left(c0 \cdot d\right), \color{blue}{\left(h \cdot \frac{w}{d}\right)}\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, d\right), \left(\color{blue}{h} \cdot \frac{w}{d}\right)\right)\right)\right) \]

       16. clear-num N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, d\right), \left(h \cdot \frac{1}{\color{blue}{\frac{d}{w}}}\right)\right)\right)\right) \]

       17. un-div-inv N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, d\right), \left(\frac{h}{\color{blue}{\frac{d}{w}}}\right)\right)\right)\right) \]

       18. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, d\right), \mathsf{/.f64}\left(h, \color{blue}{\left(\frac{d}{w}\right)}\right)\right)\right)\right) \]

       19. /-lowering-/.f64 57.9%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, d\right), \mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \color{blue}{w}\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 57.9%

       \[\leadsto \color{blue}{\frac{\frac{c0}{D \cdot D}}{\frac{w}{\frac{c0 \cdot d}{\frac{h}{\frac{d}{w}}}}}} \]
   12. Step-by-step derivation

       1. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{w}{c0 \cdot d} \cdot \color{blue}{\frac{h}{\frac{d}{w}}}\right)\right) \]

       2. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{w}{c0 \cdot d} \cdot \left(\frac{h}{d} \cdot \color{blue}{w}\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \left(\left(\frac{w}{c0 \cdot d} \cdot \frac{h}{d}\right) \cdot \color{blue}{w}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\left(\frac{w}{c0 \cdot d} \cdot \frac{h}{d}\right), \color{blue}{w}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{w}{c0 \cdot d}\right), \left(\frac{h}{d}\right)\right), w\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(w, \left(c0 \cdot d\right)\right), \left(\frac{h}{d}\right)\right), w\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(w, \left(d \cdot c0\right)\right), \left(\frac{h}{d}\right)\right), w\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(w, \mathsf{*.f64}\left(d, c0\right)\right), \left(\frac{h}{d}\right)\right), w\right)\right) \]

       9. /-lowering-/.f64 58.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(w, \mathsf{*.f64}\left(d, c0\right)\right), \mathsf{/.f64}\left(h, d\right)\right), w\right)\right) \]

   13. Applied egg-rr 58.0%

       \[\leadsto \frac{\frac{c0}{D \cdot D}}{\color{blue}{\left(\frac{w}{d \cdot c0} \cdot \frac{h}{d}\right) \cdot w}} \]

   if 1 < (*.f64 D D)

   1. Initial program 25.4%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Simplified 25.4%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{d \cdot \frac{\frac{c0 \cdot d}{w}}{h}}{D \cdot D} + \sqrt{\frac{\frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot \left(d \cdot \frac{\frac{c0 \cdot d}{w}}{h}\right)\right)}{D \cdot D}}{w \cdot h}}{D \cdot D} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\left(2 \cdot \left(c0 \cdot {d}^{2}\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot {d}^{2}\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left({d}^{2}\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(d \cdot d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(w \cdot h\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. *-lowering-*.f64 32.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(w, h\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 32.8%

      \[\leadsto \frac{c0 \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}}{2 \cdot w} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. count-2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{\left(c0 \cdot d\right) \cdot d}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{\left(c0 \cdot d\right) \cdot d}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{\left(c0 \cdot d\right) \cdot d}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{\left(c0 \cdot d\right) \cdot d}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. times-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      12. distribute-rgt-out N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)}\right), \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   9. Applied egg-rr 64.9%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)}}{2 \cdot w} \]
   10. Step-by-step derivation

       1. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       2. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right) \cdot c0 + \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right) \cdot c0\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

       3. distribute-lft-out N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right) \cdot \left(c0 + c0\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot d}{D}\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       6. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \frac{1}{\frac{D}{c0 \cdot d}}\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       7. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{d}{D \cdot \left(w \cdot h\right)}}{\frac{D}{c0 \cdot d}}\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)}\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       9. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{d}{D}}{w \cdot h}\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{D}\right), \left(w \cdot h\right)\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \left(w \cdot h\right)\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(D, \left(c0 \cdot d\right)\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(c0, d\right)\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       15. count-2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(c0, d\right)\right)\right), \left(2 \cdot c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       16. *-lowering-*.f64 65.1%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(c0, d\right)\right)\right), \mathsf{*.f64}\left(2, c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   11. Applied egg-rr 65.1%

       \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{d}{D}}{w \cdot h}}{\frac{D}{c0 \cdot d}} \cdot \left(2 \cdot c0\right)}}{2 \cdot w} \]
   12. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \frac{\frac{\frac{d}{D}}{w \cdot h}}{\frac{D}{c0 \cdot d}} \cdot \color{blue}{\frac{2 \cdot c0}{2 \cdot w}} \]

       2. times-frac N/A

          \[\leadsto \frac{\frac{\frac{d}{D}}{w \cdot h}}{\frac{D}{c0 \cdot d}} \cdot \left(\frac{2}{2} \cdot \color{blue}{\frac{c0}{w}}\right) \]

       3. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{d}{D}}{w \cdot h}}{\frac{D}{c0 \cdot d}} \cdot \left(1 \cdot \frac{\color{blue}{c0}}{w}\right) \]

       4. *-lft-identity N/A

          \[\leadsto \frac{\frac{\frac{d}{D}}{w \cdot h}}{\frac{D}{c0 \cdot d}} \cdot \frac{c0}{\color{blue}{w}} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{d}{D}}{w \cdot h}}{\frac{D}{c0 \cdot d}}\right), \color{blue}{\left(\frac{c0}{w}\right)}\right) \]

   13. Applied egg-rr 65.0%

       \[\leadsto \color{blue}{\frac{\frac{d}{\frac{D}{c0}}}{\frac{h}{\frac{d}{w \cdot D}}} \cdot \frac{c0}{w}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \cdot D \leq 10^{-293}:\\ \;\;\;\;c0 \cdot \frac{\frac{\frac{2}{\frac{D}{c0 \cdot d}}}{\frac{w \cdot h}{\frac{d}{D}}}}{2 \cdot w}\\ \mathbf{elif}\;D \cdot D \leq 10^{-204}:\\ \;\;\;\;0\\ \mathbf{elif}\;D \cdot D \leq 5 \cdot 10^{-60}:\\ \;\;\;\;\frac{\frac{c0}{D \cdot D}}{w \cdot \left(\frac{w}{c0 \cdot d} \cdot \frac{h}{d}\right)}\\ \mathbf{elif}\;D \cdot D \leq 1:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\frac{D}{c0}}}{\frac{h}{\frac{d}{w \cdot D}}} \cdot \frac{c0}{w}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 46.7% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 1.2 \cdot 10^{-143}:\\ \;\;\;\;\frac{1}{D} \cdot \left(\frac{d}{w} \cdot \frac{\frac{\frac{c0 \cdot \left(c0 \cdot d\right)}{w}}{h}}{D}\right)\\ \mathbf{elif}\;D \leq 2 \cdot 10^{-102}:\\ \;\;\;\;0\\ \mathbf{elif}\;D \leq 1.6 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{c0}{D \cdot D}}{w \cdot \left(\frac{w}{c0 \cdot d} \cdot \frac{h}{d}\right)}\\ \mathbf{elif}\;D \leq 0.78:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\frac{D}{c0}}}{\frac{h}{\frac{d}{w \cdot D}}} \cdot \frac{c0}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= D 1.2e-143)
   (* (/ 1.0 D) (* (/ d w) (/ (/ (/ (* c0 (* c0 d)) w) h) D)))
   (if (<= D 2e-102)
     0.0
     (if (<= D 1.6e-29)
       (/ (/ c0 (* D D)) (* w (* (/ w (* c0 d)) (/ h d))))
       (if (<= D 0.78)
         0.0
         (* (/ (/ d (/ D c0)) (/ h (/ d (* w D)))) (/ c0 w)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (D <= 1.2e-143) {
		tmp = (1.0 / D) * ((d / w) * ((((c0 * (c0 * d)) / w) / h) / D));
	} else if (D <= 2e-102) {
		tmp = 0.0;
	} else if (D <= 1.6e-29) {
		tmp = (c0 / (D * D)) / (w * ((w / (c0 * d)) * (h / d)));
	} else if (D <= 0.78) {
		tmp = 0.0;
	} else {
		tmp = ((d / (D / c0)) / (h / (d / (w * D)))) * (c0 / w);
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (d <= 1.2d-143) then
        tmp = (1.0d0 / d) * ((d_1 / w) * ((((c0 * (c0 * d_1)) / w) / h) / d))
    else if (d <= 2d-102) then
        tmp = 0.0d0
    else if (d <= 1.6d-29) then
        tmp = (c0 / (d * d)) / (w * ((w / (c0 * d_1)) * (h / d_1)))
    else if (d <= 0.78d0) then
        tmp = 0.0d0
    else
        tmp = ((d_1 / (d / c0)) / (h / (d_1 / (w * d)))) * (c0 / w)
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (D <= 1.2e-143) {
		tmp = (1.0 / D) * ((d / w) * ((((c0 * (c0 * d)) / w) / h) / D));
	} else if (D <= 2e-102) {
		tmp = 0.0;
	} else if (D <= 1.6e-29) {
		tmp = (c0 / (D * D)) / (w * ((w / (c0 * d)) * (h / d)));
	} else if (D <= 0.78) {
		tmp = 0.0;
	} else {
		tmp = ((d / (D / c0)) / (h / (d / (w * D)))) * (c0 / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if D <= 1.2e-143:
		tmp = (1.0 / D) * ((d / w) * ((((c0 * (c0 * d)) / w) / h) / D))
	elif D <= 2e-102:
		tmp = 0.0
	elif D <= 1.6e-29:
		tmp = (c0 / (D * D)) / (w * ((w / (c0 * d)) * (h / d)))
	elif D <= 0.78:
		tmp = 0.0
	else:
		tmp = ((d / (D / c0)) / (h / (d / (w * D)))) * (c0 / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (D <= 1.2e-143)
		tmp = Float64(Float64(1.0 / D) * Float64(Float64(d / w) * Float64(Float64(Float64(Float64(c0 * Float64(c0 * d)) / w) / h) / D)));
	elseif (D <= 2e-102)
		tmp = 0.0;
	elseif (D <= 1.6e-29)
		tmp = Float64(Float64(c0 / Float64(D * D)) / Float64(w * Float64(Float64(w / Float64(c0 * d)) * Float64(h / d))));
	elseif (D <= 0.78)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(d / Float64(D / c0)) / Float64(h / Float64(d / Float64(w * D)))) * Float64(c0 / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (D <= 1.2e-143)
		tmp = (1.0 / D) * ((d / w) * ((((c0 * (c0 * d)) / w) / h) / D));
	elseif (D <= 2e-102)
		tmp = 0.0;
	elseif (D <= 1.6e-29)
		tmp = (c0 / (D * D)) / (w * ((w / (c0 * d)) * (h / d)));
	elseif (D <= 0.78)
		tmp = 0.0;
	else
		tmp = ((d / (D / c0)) / (h / (d / (w * D)))) * (c0 / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[D, 1.2e-143], N[(N[(1.0 / D), $MachinePrecision] * N[(N[(d / w), $MachinePrecision] * N[(N[(N[(N[(c0 * N[(c0 * d), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision] / h), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[D, 2e-102], 0.0, If[LessEqual[D, 1.6e-29], N[(N[(c0 / N[(D * D), $MachinePrecision]), $MachinePrecision] / N[(w * N[(N[(w / N[(c0 * d), $MachinePrecision]), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[D, 0.78], 0.0, N[(N[(N[(d / N[(D / c0), $MachinePrecision]), $MachinePrecision] / N[(h / N[(d / N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c0 / w), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if D < 1.1999999999999999e-143

   1. Initial program 20.8%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({c0}^{2} \cdot {d}^{2}\right), \color{blue}{\left({D}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left({d}^{2} \cdot {c0}^{2}\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({d}^{2}\right), \left({c0}^{2}\right)\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(d \cdot d\right), \left({c0}^{2}\right)\right), \left({\color{blue}{D}}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \left({c0}^{2}\right)\right), \left({\color{blue}{D}}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \left(c0 \cdot c0\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(h \cdot {w}^{2}\right)}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{h} \cdot {w}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{h} \cdot {w}^{2}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \color{blue}{\left({w}^{2}\right)}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(w \cdot \color{blue}{w}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 20.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w, \color{blue}{w}\right)\right)\right)\right) \]

   5. Simplified 20.7%

      \[\leadsto \color{blue}{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{\left(h \cdot \left(w \cdot w\right)\right) \cdot \color{blue}{\left(D \cdot D\right)}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{h \cdot \left(w \cdot w\right)}}{\color{blue}{D \cdot D}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{h \cdot \left(w \cdot w\right)}\right), \color{blue}{\left(D \cdot D\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)\right), \left(h \cdot \left(w \cdot w\right)\right)\right), \left(\color{blue}{D} \cdot D\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)\right), \left(h \cdot \left(w \cdot w\right)\right)\right), \left(D \cdot D\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \left(d \cdot \left(c0 \cdot c0\right)\right)\right), \left(h \cdot \left(w \cdot w\right)\right)\right), \left(D \cdot D\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \left(c0 \cdot c0\right)\right)\right), \left(h \cdot \left(w \cdot w\right)\right)\right), \left(D \cdot D\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(c0, c0\right)\right)\right), \left(h \cdot \left(w \cdot w\right)\right)\right), \left(D \cdot D\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(c0, c0\right)\right)\right), \left(\left(w \cdot w\right) \cdot h\right)\right), \left(D \cdot D\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(c0, c0\right)\right)\right), \left(w \cdot \left(w \cdot h\right)\right)\right), \left(D \cdot D\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(c0, c0\right)\right)\right), \mathsf{*.f64}\left(w, \left(w \cdot h\right)\right)\right), \left(D \cdot D\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(c0, c0\right)\right)\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, h\right)\right)\right), \left(D \cdot D\right)\right) \]

      13. *-lowering-*.f64 31.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(c0, c0\right)\right)\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, h\right)\right)\right), \mathsf{*.f64}\left(D, \color{blue}{D}\right)\right) \]

   7. Applied egg-rr 31.8%

      \[\leadsto \color{blue}{\frac{\frac{d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)}{w \cdot \left(w \cdot h\right)}}{D \cdot D}} \]
   8. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{D \cdot D}{\frac{d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)}{w \cdot \left(w \cdot h\right)}}}} \]

      2. inv-pow N/A

         \[\leadsto {\left(\frac{D \cdot D}{\frac{d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)}{w \cdot \left(w \cdot h\right)}}\right)}^{\color{blue}{-1}} \]

      3. associate-/l* N/A

         \[\leadsto {\left(D \cdot \frac{D}{\frac{d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)}{w \cdot \left(w \cdot h\right)}}\right)}^{-1} \]

      4. unpow-prod-down N/A

         \[\leadsto {D}^{-1} \cdot \color{blue}{{\left(\frac{D}{\frac{d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)}{w \cdot \left(w \cdot h\right)}}\right)}^{-1}} \]

      5. inv-pow N/A

         \[\leadsto \frac{1}{D} \cdot {\color{blue}{\left(\frac{D}{\frac{d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)}{w \cdot \left(w \cdot h\right)}}\right)}}^{-1} \]

      6. inv-pow N/A

         \[\leadsto \frac{1}{D} \cdot \frac{1}{\color{blue}{\frac{D}{\frac{d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)}{w \cdot \left(w \cdot h\right)}}}} \]

      7. clear-num N/A

         \[\leadsto \frac{1}{D} \cdot \frac{\frac{d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)}{w \cdot \left(w \cdot h\right)}}{\color{blue}{D}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{D}\right), \color{blue}{\left(\frac{\frac{d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)}{w \cdot \left(w \cdot h\right)}}{D}\right)}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, D\right), \left(\frac{\color{blue}{\frac{d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)}{w \cdot \left(w \cdot h\right)}}}{D}\right)\right) \]

      10. times-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, D\right), \left(\frac{\frac{d}{w} \cdot \frac{d \cdot \left(c0 \cdot c0\right)}{w \cdot h}}{D}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, D\right), \left(\frac{d}{w} \cdot \color{blue}{\frac{\frac{d \cdot \left(c0 \cdot c0\right)}{w \cdot h}}{D}}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, D\right), \mathsf{*.f64}\left(\left(\frac{d}{w}\right), \color{blue}{\left(\frac{\frac{d \cdot \left(c0 \cdot c0\right)}{w \cdot h}}{D}\right)}\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, w\right), \left(\frac{\color{blue}{\frac{d \cdot \left(c0 \cdot c0\right)}{w \cdot h}}}{D}\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, w\right), \mathsf{/.f64}\left(\left(\frac{d \cdot \left(c0 \cdot c0\right)}{w \cdot h}\right), \color{blue}{D}\right)\right)\right) \]

   9. Applied egg-rr 46.6%

      \[\leadsto \color{blue}{\frac{1}{D} \cdot \left(\frac{d}{w} \cdot \frac{\frac{\frac{c0 \cdot \left(c0 \cdot d\right)}{w}}{h}}{D}\right)} \]

   if 1.1999999999999999e-143 < D < 1.99999999999999987e-102 or 1.6e-29 < D < 0.78000000000000003

   1. Initial program 24.0%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Simplified 16.5%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{d \cdot \frac{\frac{c0 \cdot d}{w}}{h}}{D \cdot D} + \sqrt{\frac{\frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot \left(d \cdot \frac{\frac{c0 \cdot d}{w}}{h}\right)\right)}{D \cdot D}}{w \cdot h}}{D \cdot D} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot c0\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. mul0-lft N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(c0 \cdot \left(-1 + 1\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(\mathsf{neg}\left(\left(-1 + 1\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(\mathsf{neg}\left(0\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      13. metadata-eval 55.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 55.4%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(c0 \cdot 0\right)}}{2 \cdot w} \]
   8. Step-by-step derivation

      1. mul0-rgt N/A

         \[\leadsto \frac{c0 \cdot 0}{2 \cdot w} \]

      2. mul0-rgt N/A

         \[\leadsto \frac{0}{\color{blue}{2} \cdot w} \]

      3. div0 55.4%

         \[\leadsto 0 \]

   9. Applied egg-rr 55.4%

      \[\leadsto \color{blue}{0} \]

   if 1.99999999999999987e-102 < D < 1.6e-29

   1. Initial program 39.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Simplified 39.6%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{d \cdot \frac{\frac{c0 \cdot d}{w}}{h}}{D \cdot D} + \sqrt{\frac{\frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot \left(d \cdot \frac{\frac{c0 \cdot d}{w}}{h}\right)\right)}{D \cdot D}}{w \cdot h}}{D \cdot D} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\left(2 \cdot \left(c0 \cdot {d}^{2}\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot {d}^{2}\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left({d}^{2}\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(d \cdot d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(w \cdot h\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. *-lowering-*.f64 44.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(w, h\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 44.7%

      \[\leadsto \frac{c0 \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}}{2 \cdot w} \]
   8. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(\frac{2}{D \cdot D} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{w \cdot h}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(c0 \cdot \frac{2}{D \cdot D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{w \cdot h}\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(c0 \cdot \frac{2}{D \cdot D}\right) \cdot \frac{\frac{c0 \cdot \left(d \cdot d\right)}{w}}{h}\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(c0 \cdot \frac{2}{D \cdot D}\right) \cdot \frac{c0 \cdot \frac{d \cdot d}{w}}{h}\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(c0 \cdot \frac{2}{D \cdot D}\right), \left(\frac{c0 \cdot \frac{d \cdot d}{w}}{h}\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(c0 \cdot \frac{1}{\frac{D \cdot D}{2}}\right), \left(\frac{c0 \cdot \frac{d \cdot d}{w}}{h}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{c0}{\frac{D \cdot D}{2}}\right), \left(\frac{c0 \cdot \frac{d \cdot d}{w}}{h}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \left(\frac{D \cdot D}{2}\right)\right), \left(\frac{c0 \cdot \frac{d \cdot d}{w}}{h}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\left(D \cdot D\right), 2\right)\right), \left(\frac{c0 \cdot \frac{d \cdot d}{w}}{h}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \left(\frac{c0 \cdot \frac{d \cdot d}{w}}{h}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \left(c0 \cdot \frac{\frac{d \cdot d}{w}}{h}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      12. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \left(c0 \cdot \frac{1}{\frac{h}{\frac{d \cdot d}{w}}}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      13. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \left(\frac{c0}{\frac{h}{\frac{d \cdot d}{w}}}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \mathsf{/.f64}\left(c0, \left(\frac{h}{\frac{d \cdot d}{w}}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(h, \left(\frac{d \cdot d}{w}\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      16. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(h, \left(d \cdot \frac{d}{w}\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      17. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(h, \left(d \cdot \frac{1}{\frac{w}{d}}\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      18. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(h, \left(\frac{d}{\frac{w}{d}}\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      19. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \left(\frac{w}{d}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      20. /-lowering-/.f64 49.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{/.f64}\left(w, d\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   9. Applied egg-rr 49.0%

      \[\leadsto \frac{\color{blue}{\frac{c0}{\frac{D \cdot D}{2}} \cdot \frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}}{2 \cdot w} \]
   10. Step-by-step derivation

       1. times-frac N/A

          \[\leadsto \frac{\frac{c0}{\frac{D \cdot D}{2}}}{2} \cdot \color{blue}{\frac{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}{w}} \]

       2. clear-num N/A

          \[\leadsto \frac{\frac{c0}{\frac{D \cdot D}{2}}}{2} \cdot \frac{1}{\color{blue}{\frac{w}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}}} \]

       3. un-div-inv N/A

          \[\leadsto \frac{\frac{\frac{c0}{\frac{D \cdot D}{2}}}{2}}{\color{blue}{\frac{w}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}}} \]

       4. associate-/r/ N/A

          \[\leadsto \frac{\frac{\frac{c0}{D \cdot D} \cdot 2}{2}}{\frac{w}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}} \]

       5. associate-/l* N/A

          \[\leadsto \frac{\frac{c0}{D \cdot D} \cdot \frac{2}{2}}{\frac{\color{blue}{w}}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}} \]

       6. metadata-eval N/A

          \[\leadsto \frac{\frac{c0}{D \cdot D} \cdot 1}{\frac{w}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}} \]

       7. *-rgt-identity N/A

          \[\leadsto \frac{\frac{c0}{D \cdot D}}{\frac{\color{blue}{w}}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}} \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{c0}{D \cdot D}\right), \color{blue}{\left(\frac{w}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}\right)}\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \left(D \cdot D\right)\right), \left(\frac{\color{blue}{w}}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{w}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \color{blue}{\left(\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}\right)}\right)\right) \]

       12. associate-/r/ N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \left(\frac{c0}{h} \cdot \color{blue}{\frac{d}{\frac{w}{d}}}\right)\right)\right) \]

       13. frac-times N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \left(\frac{c0 \cdot d}{\color{blue}{h \cdot \frac{w}{d}}}\right)\right)\right) \]

       14. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \mathsf{/.f64}\left(\left(c0 \cdot d\right), \color{blue}{\left(h \cdot \frac{w}{d}\right)}\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, d\right), \left(\color{blue}{h} \cdot \frac{w}{d}\right)\right)\right)\right) \]

       16. clear-num N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, d\right), \left(h \cdot \frac{1}{\color{blue}{\frac{d}{w}}}\right)\right)\right)\right) \]

       17. un-div-inv N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, d\right), \left(\frac{h}{\color{blue}{\frac{d}{w}}}\right)\right)\right)\right) \]

       18. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, d\right), \mathsf{/.f64}\left(h, \color{blue}{\left(\frac{d}{w}\right)}\right)\right)\right)\right) \]

       19. /-lowering-/.f64 57.1%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, d\right), \mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \color{blue}{w}\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 57.1%

       \[\leadsto \color{blue}{\frac{\frac{c0}{D \cdot D}}{\frac{w}{\frac{c0 \cdot d}{\frac{h}{\frac{d}{w}}}}}} \]
   12. Step-by-step derivation

       1. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{w}{c0 \cdot d} \cdot \color{blue}{\frac{h}{\frac{d}{w}}}\right)\right) \]

       2. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{w}{c0 \cdot d} \cdot \left(\frac{h}{d} \cdot \color{blue}{w}\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \left(\left(\frac{w}{c0 \cdot d} \cdot \frac{h}{d}\right) \cdot \color{blue}{w}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\left(\frac{w}{c0 \cdot d} \cdot \frac{h}{d}\right), \color{blue}{w}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{w}{c0 \cdot d}\right), \left(\frac{h}{d}\right)\right), w\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(w, \left(c0 \cdot d\right)\right), \left(\frac{h}{d}\right)\right), w\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(w, \left(d \cdot c0\right)\right), \left(\frac{h}{d}\right)\right), w\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(w, \mathsf{*.f64}\left(d, c0\right)\right), \left(\frac{h}{d}\right)\right), w\right)\right) \]

       9. /-lowering-/.f64 57.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(w, \mathsf{*.f64}\left(d, c0\right)\right), \mathsf{/.f64}\left(h, d\right)\right), w\right)\right) \]

   13. Applied egg-rr 57.3%

       \[\leadsto \frac{\frac{c0}{D \cdot D}}{\color{blue}{\left(\frac{w}{d \cdot c0} \cdot \frac{h}{d}\right) \cdot w}} \]

   if 0.78000000000000003 < D

   1. Initial program 25.4%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Simplified 22.9%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{d \cdot \frac{\frac{c0 \cdot d}{w}}{h}}{D \cdot D} + \sqrt{\frac{\frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot \left(d \cdot \frac{\frac{c0 \cdot d}{w}}{h}\right)\right)}{D \cdot D}}{w \cdot h}}{D \cdot D} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\left(2 \cdot \left(c0 \cdot {d}^{2}\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot {d}^{2}\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left({d}^{2}\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(d \cdot d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(w \cdot h\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. *-lowering-*.f64 37.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(w, h\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 37.1%

      \[\leadsto \frac{c0 \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}}{2 \cdot w} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. count-2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{\left(c0 \cdot d\right) \cdot d}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{\left(c0 \cdot d\right) \cdot d}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{\left(c0 \cdot d\right) \cdot d}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{\left(c0 \cdot d\right) \cdot d}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. times-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      12. distribute-rgt-out N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)}\right), \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   9. Applied egg-rr 62.6%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)}}{2 \cdot w} \]
   10. Step-by-step derivation

       1. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       2. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right) \cdot c0 + \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right) \cdot c0\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

       3. distribute-lft-out N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right) \cdot \left(c0 + c0\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot d}{D}\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       6. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \frac{1}{\frac{D}{c0 \cdot d}}\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       7. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{d}{D \cdot \left(w \cdot h\right)}}{\frac{D}{c0 \cdot d}}\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)}\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       9. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{d}{D}}{w \cdot h}\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{D}\right), \left(w \cdot h\right)\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \left(w \cdot h\right)\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(D, \left(c0 \cdot d\right)\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(c0, d\right)\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       15. count-2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(c0, d\right)\right)\right), \left(2 \cdot c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       16. *-lowering-*.f64 62.6%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(c0, d\right)\right)\right), \mathsf{*.f64}\left(2, c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   11. Applied egg-rr 62.6%

       \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{d}{D}}{w \cdot h}}{\frac{D}{c0 \cdot d}} \cdot \left(2 \cdot c0\right)}}{2 \cdot w} \]
   12. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \frac{\frac{\frac{d}{D}}{w \cdot h}}{\frac{D}{c0 \cdot d}} \cdot \color{blue}{\frac{2 \cdot c0}{2 \cdot w}} \]

       2. times-frac N/A

          \[\leadsto \frac{\frac{\frac{d}{D}}{w \cdot h}}{\frac{D}{c0 \cdot d}} \cdot \left(\frac{2}{2} \cdot \color{blue}{\frac{c0}{w}}\right) \]

       3. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{d}{D}}{w \cdot h}}{\frac{D}{c0 \cdot d}} \cdot \left(1 \cdot \frac{\color{blue}{c0}}{w}\right) \]

       4. *-lft-identity N/A

          \[\leadsto \frac{\frac{\frac{d}{D}}{w \cdot h}}{\frac{D}{c0 \cdot d}} \cdot \frac{c0}{\color{blue}{w}} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{d}{D}}{w \cdot h}}{\frac{D}{c0 \cdot d}}\right), \color{blue}{\left(\frac{c0}{w}\right)}\right) \]

   13. Applied egg-rr 64.9%

       \[\leadsto \color{blue}{\frac{\frac{d}{\frac{D}{c0}}}{\frac{h}{\frac{d}{w \cdot D}}} \cdot \frac{c0}{w}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 1.2 \cdot 10^{-143}:\\ \;\;\;\;\frac{1}{D} \cdot \left(\frac{d}{w} \cdot \frac{\frac{\frac{c0 \cdot \left(c0 \cdot d\right)}{w}}{h}}{D}\right)\\ \mathbf{elif}\;D \leq 2 \cdot 10^{-102}:\\ \;\;\;\;0\\ \mathbf{elif}\;D \leq 1.6 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{c0}{D \cdot D}}{w \cdot \left(\frac{w}{c0 \cdot d} \cdot \frac{h}{d}\right)}\\ \mathbf{elif}\;D \leq 0.78:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\frac{D}{c0}}}{\frac{h}{\frac{d}{w \cdot D}}} \cdot \frac{c0}{w}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 45.4% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 1.45 \cdot 10^{-144}:\\ \;\;\;\;\frac{\frac{\frac{c0 \cdot \left(c0 \cdot d\right)}{w}}{h}}{D} \cdot \frac{\frac{d}{w}}{D}\\ \mathbf{elif}\;D \leq 8.2 \cdot 10^{-103}:\\ \;\;\;\;0\\ \mathbf{elif}\;D \leq 1.8 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{c0}{D \cdot D}}{w \cdot \left(\frac{w}{c0 \cdot d} \cdot \frac{h}{d}\right)}\\ \mathbf{elif}\;D \leq 1.3:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\frac{D}{c0}}}{\frac{h}{\frac{d}{w \cdot D}}} \cdot \frac{c0}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= D 1.45e-144)
   (* (/ (/ (/ (* c0 (* c0 d)) w) h) D) (/ (/ d w) D))
   (if (<= D 8.2e-103)
     0.0
     (if (<= D 1.8e-28)
       (/ (/ c0 (* D D)) (* w (* (/ w (* c0 d)) (/ h d))))
       (if (<= D 1.3)
         0.0
         (* (/ (/ d (/ D c0)) (/ h (/ d (* w D)))) (/ c0 w)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (D <= 1.45e-144) {
		tmp = ((((c0 * (c0 * d)) / w) / h) / D) * ((d / w) / D);
	} else if (D <= 8.2e-103) {
		tmp = 0.0;
	} else if (D <= 1.8e-28) {
		tmp = (c0 / (D * D)) / (w * ((w / (c0 * d)) * (h / d)));
	} else if (D <= 1.3) {
		tmp = 0.0;
	} else {
		tmp = ((d / (D / c0)) / (h / (d / (w * D)))) * (c0 / w);
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (d <= 1.45d-144) then
        tmp = ((((c0 * (c0 * d_1)) / w) / h) / d) * ((d_1 / w) / d)
    else if (d <= 8.2d-103) then
        tmp = 0.0d0
    else if (d <= 1.8d-28) then
        tmp = (c0 / (d * d)) / (w * ((w / (c0 * d_1)) * (h / d_1)))
    else if (d <= 1.3d0) then
        tmp = 0.0d0
    else
        tmp = ((d_1 / (d / c0)) / (h / (d_1 / (w * d)))) * (c0 / w)
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (D <= 1.45e-144) {
		tmp = ((((c0 * (c0 * d)) / w) / h) / D) * ((d / w) / D);
	} else if (D <= 8.2e-103) {
		tmp = 0.0;
	} else if (D <= 1.8e-28) {
		tmp = (c0 / (D * D)) / (w * ((w / (c0 * d)) * (h / d)));
	} else if (D <= 1.3) {
		tmp = 0.0;
	} else {
		tmp = ((d / (D / c0)) / (h / (d / (w * D)))) * (c0 / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if D <= 1.45e-144:
		tmp = ((((c0 * (c0 * d)) / w) / h) / D) * ((d / w) / D)
	elif D <= 8.2e-103:
		tmp = 0.0
	elif D <= 1.8e-28:
		tmp = (c0 / (D * D)) / (w * ((w / (c0 * d)) * (h / d)))
	elif D <= 1.3:
		tmp = 0.0
	else:
		tmp = ((d / (D / c0)) / (h / (d / (w * D)))) * (c0 / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (D <= 1.45e-144)
		tmp = Float64(Float64(Float64(Float64(Float64(c0 * Float64(c0 * d)) / w) / h) / D) * Float64(Float64(d / w) / D));
	elseif (D <= 8.2e-103)
		tmp = 0.0;
	elseif (D <= 1.8e-28)
		tmp = Float64(Float64(c0 / Float64(D * D)) / Float64(w * Float64(Float64(w / Float64(c0 * d)) * Float64(h / d))));
	elseif (D <= 1.3)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(d / Float64(D / c0)) / Float64(h / Float64(d / Float64(w * D)))) * Float64(c0 / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (D <= 1.45e-144)
		tmp = ((((c0 * (c0 * d)) / w) / h) / D) * ((d / w) / D);
	elseif (D <= 8.2e-103)
		tmp = 0.0;
	elseif (D <= 1.8e-28)
		tmp = (c0 / (D * D)) / (w * ((w / (c0 * d)) * (h / d)));
	elseif (D <= 1.3)
		tmp = 0.0;
	else
		tmp = ((d / (D / c0)) / (h / (d / (w * D)))) * (c0 / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[D, 1.45e-144], N[(N[(N[(N[(N[(c0 * N[(c0 * d), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision] / h), $MachinePrecision] / D), $MachinePrecision] * N[(N[(d / w), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision], If[LessEqual[D, 8.2e-103], 0.0, If[LessEqual[D, 1.8e-28], N[(N[(c0 / N[(D * D), $MachinePrecision]), $MachinePrecision] / N[(w * N[(N[(w / N[(c0 * d), $MachinePrecision]), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[D, 1.3], 0.0, N[(N[(N[(d / N[(D / c0), $MachinePrecision]), $MachinePrecision] / N[(h / N[(d / N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c0 / w), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if D < 1.4500000000000001e-144

   1. Initial program 20.8%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({c0}^{2} \cdot {d}^{2}\right), \color{blue}{\left({D}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left({d}^{2} \cdot {c0}^{2}\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({d}^{2}\right), \left({c0}^{2}\right)\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(d \cdot d\right), \left({c0}^{2}\right)\right), \left({\color{blue}{D}}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \left({c0}^{2}\right)\right), \left({\color{blue}{D}}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \left(c0 \cdot c0\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(h \cdot {w}^{2}\right)}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{h} \cdot {w}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{h} \cdot {w}^{2}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \color{blue}{\left({w}^{2}\right)}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(w \cdot \color{blue}{w}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 20.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w, \color{blue}{w}\right)\right)\right)\right) \]

   5. Simplified 20.7%

      \[\leadsto \color{blue}{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{\left(h \cdot \left(w \cdot w\right)\right) \cdot \color{blue}{\left(D \cdot D\right)}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{h \cdot \left(w \cdot w\right)}}{\color{blue}{D \cdot D}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{h \cdot \left(w \cdot w\right)}\right), \color{blue}{\left(D \cdot D\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)\right), \left(h \cdot \left(w \cdot w\right)\right)\right), \left(\color{blue}{D} \cdot D\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)\right), \left(h \cdot \left(w \cdot w\right)\right)\right), \left(D \cdot D\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \left(d \cdot \left(c0 \cdot c0\right)\right)\right), \left(h \cdot \left(w \cdot w\right)\right)\right), \left(D \cdot D\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \left(c0 \cdot c0\right)\right)\right), \left(h \cdot \left(w \cdot w\right)\right)\right), \left(D \cdot D\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(c0, c0\right)\right)\right), \left(h \cdot \left(w \cdot w\right)\right)\right), \left(D \cdot D\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(c0, c0\right)\right)\right), \left(\left(w \cdot w\right) \cdot h\right)\right), \left(D \cdot D\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(c0, c0\right)\right)\right), \left(w \cdot \left(w \cdot h\right)\right)\right), \left(D \cdot D\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(c0, c0\right)\right)\right), \mathsf{*.f64}\left(w, \left(w \cdot h\right)\right)\right), \left(D \cdot D\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(c0, c0\right)\right)\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, h\right)\right)\right), \left(D \cdot D\right)\right) \]

      13. *-lowering-*.f64 31.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(c0, c0\right)\right)\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, h\right)\right)\right), \mathsf{*.f64}\left(D, \color{blue}{D}\right)\right) \]

   7. Applied egg-rr 31.8%

      \[\leadsto \color{blue}{\frac{\frac{d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)}{w \cdot \left(w \cdot h\right)}}{D \cdot D}} \]
   8. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \frac{\frac{d}{w} \cdot \frac{d \cdot \left(c0 \cdot c0\right)}{w \cdot h}}{\color{blue}{D} \cdot D} \]

      2. times-frac N/A

         \[\leadsto \frac{\frac{d}{w}}{D} \cdot \color{blue}{\frac{\frac{d \cdot \left(c0 \cdot c0\right)}{w \cdot h}}{D}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{d}{w}}{D}\right), \color{blue}{\left(\frac{\frac{d \cdot \left(c0 \cdot c0\right)}{w \cdot h}}{D}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{w}\right), D\right), \left(\frac{\color{blue}{\frac{d \cdot \left(c0 \cdot c0\right)}{w \cdot h}}}{D}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, w\right), D\right), \left(\frac{\frac{\color{blue}{d \cdot \left(c0 \cdot c0\right)}}{w \cdot h}}{D}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, w\right), D\right), \mathsf{/.f64}\left(\left(\frac{d \cdot \left(c0 \cdot c0\right)}{w \cdot h}\right), \color{blue}{D}\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, w\right), D\right), \mathsf{/.f64}\left(\left(\frac{\frac{d \cdot \left(c0 \cdot c0\right)}{w}}{h}\right), D\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, w\right), D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{d \cdot \left(c0 \cdot c0\right)}{w}\right), h\right), D\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, w\right), D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(d \cdot \left(c0 \cdot c0\right)\right), w\right), h\right), D\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, w\right), D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(c0 \cdot c0\right) \cdot d\right), w\right), h\right), D\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, w\right), D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(c0 \cdot \left(c0 \cdot d\right)\right), w\right), h\right), D\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, w\right), D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot d\right)\right), w\right), h\right), D\right)\right) \]

      13. *-lowering-*.f64 47.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, w\right), D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, d\right)\right), w\right), h\right), D\right)\right) \]

   9. Applied egg-rr 47.0%

      \[\leadsto \color{blue}{\frac{\frac{d}{w}}{D} \cdot \frac{\frac{\frac{c0 \cdot \left(c0 \cdot d\right)}{w}}{h}}{D}} \]

   if 1.4500000000000001e-144 < D < 8.19999999999999992e-103 or 1.7999999999999999e-28 < D < 1.30000000000000004

   1. Initial program 24.0%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Simplified 16.5%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{d \cdot \frac{\frac{c0 \cdot d}{w}}{h}}{D \cdot D} + \sqrt{\frac{\frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot \left(d \cdot \frac{\frac{c0 \cdot d}{w}}{h}\right)\right)}{D \cdot D}}{w \cdot h}}{D \cdot D} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot c0\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. mul0-lft N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(c0 \cdot \left(-1 + 1\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(\mathsf{neg}\left(\left(-1 + 1\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(\mathsf{neg}\left(0\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      13. metadata-eval 55.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 55.4%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(c0 \cdot 0\right)}}{2 \cdot w} \]
   8. Step-by-step derivation

      1. mul0-rgt N/A

         \[\leadsto \frac{c0 \cdot 0}{2 \cdot w} \]

      2. mul0-rgt N/A

         \[\leadsto \frac{0}{\color{blue}{2} \cdot w} \]

      3. div0 55.4%

         \[\leadsto 0 \]

   9. Applied egg-rr 55.4%

      \[\leadsto \color{blue}{0} \]

   if 8.19999999999999992e-103 < D < 1.7999999999999999e-28

   1. Initial program 39.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Simplified 39.6%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{d \cdot \frac{\frac{c0 \cdot d}{w}}{h}}{D \cdot D} + \sqrt{\frac{\frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot \left(d \cdot \frac{\frac{c0 \cdot d}{w}}{h}\right)\right)}{D \cdot D}}{w \cdot h}}{D \cdot D} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\left(2 \cdot \left(c0 \cdot {d}^{2}\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot {d}^{2}\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left({d}^{2}\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(d \cdot d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(w \cdot h\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. *-lowering-*.f64 44.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(w, h\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 44.7%

      \[\leadsto \frac{c0 \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}}{2 \cdot w} \]
   8. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(\frac{2}{D \cdot D} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{w \cdot h}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(c0 \cdot \frac{2}{D \cdot D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{w \cdot h}\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(c0 \cdot \frac{2}{D \cdot D}\right) \cdot \frac{\frac{c0 \cdot \left(d \cdot d\right)}{w}}{h}\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(c0 \cdot \frac{2}{D \cdot D}\right) \cdot \frac{c0 \cdot \frac{d \cdot d}{w}}{h}\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(c0 \cdot \frac{2}{D \cdot D}\right), \left(\frac{c0 \cdot \frac{d \cdot d}{w}}{h}\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(c0 \cdot \frac{1}{\frac{D \cdot D}{2}}\right), \left(\frac{c0 \cdot \frac{d \cdot d}{w}}{h}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{c0}{\frac{D \cdot D}{2}}\right), \left(\frac{c0 \cdot \frac{d \cdot d}{w}}{h}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \left(\frac{D \cdot D}{2}\right)\right), \left(\frac{c0 \cdot \frac{d \cdot d}{w}}{h}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\left(D \cdot D\right), 2\right)\right), \left(\frac{c0 \cdot \frac{d \cdot d}{w}}{h}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \left(\frac{c0 \cdot \frac{d \cdot d}{w}}{h}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \left(c0 \cdot \frac{\frac{d \cdot d}{w}}{h}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      12. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \left(c0 \cdot \frac{1}{\frac{h}{\frac{d \cdot d}{w}}}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      13. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \left(\frac{c0}{\frac{h}{\frac{d \cdot d}{w}}}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \mathsf{/.f64}\left(c0, \left(\frac{h}{\frac{d \cdot d}{w}}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(h, \left(\frac{d \cdot d}{w}\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      16. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(h, \left(d \cdot \frac{d}{w}\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      17. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(h, \left(d \cdot \frac{1}{\frac{w}{d}}\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      18. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(h, \left(\frac{d}{\frac{w}{d}}\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      19. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \left(\frac{w}{d}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      20. /-lowering-/.f64 49.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), 2\right)\right), \mathsf{/.f64}\left(c0, \mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{/.f64}\left(w, d\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   9. Applied egg-rr 49.0%

      \[\leadsto \frac{\color{blue}{\frac{c0}{\frac{D \cdot D}{2}} \cdot \frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}}{2 \cdot w} \]
   10. Step-by-step derivation

       1. times-frac N/A

          \[\leadsto \frac{\frac{c0}{\frac{D \cdot D}{2}}}{2} \cdot \color{blue}{\frac{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}{w}} \]

       2. clear-num N/A

          \[\leadsto \frac{\frac{c0}{\frac{D \cdot D}{2}}}{2} \cdot \frac{1}{\color{blue}{\frac{w}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}}} \]

       3. un-div-inv N/A

          \[\leadsto \frac{\frac{\frac{c0}{\frac{D \cdot D}{2}}}{2}}{\color{blue}{\frac{w}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}}} \]

       4. associate-/r/ N/A

          \[\leadsto \frac{\frac{\frac{c0}{D \cdot D} \cdot 2}{2}}{\frac{w}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}} \]

       5. associate-/l* N/A

          \[\leadsto \frac{\frac{c0}{D \cdot D} \cdot \frac{2}{2}}{\frac{\color{blue}{w}}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}} \]

       6. metadata-eval N/A

          \[\leadsto \frac{\frac{c0}{D \cdot D} \cdot 1}{\frac{w}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}} \]

       7. *-rgt-identity N/A

          \[\leadsto \frac{\frac{c0}{D \cdot D}}{\frac{\color{blue}{w}}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}} \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{c0}{D \cdot D}\right), \color{blue}{\left(\frac{w}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}\right)}\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \left(D \cdot D\right)\right), \left(\frac{\color{blue}{w}}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{w}{\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}}\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \color{blue}{\left(\frac{c0}{\frac{h}{\frac{d}{\frac{w}{d}}}}\right)}\right)\right) \]

       12. associate-/r/ N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \left(\frac{c0}{h} \cdot \color{blue}{\frac{d}{\frac{w}{d}}}\right)\right)\right) \]

       13. frac-times N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \left(\frac{c0 \cdot d}{\color{blue}{h \cdot \frac{w}{d}}}\right)\right)\right) \]

       14. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \mathsf{/.f64}\left(\left(c0 \cdot d\right), \color{blue}{\left(h \cdot \frac{w}{d}\right)}\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, d\right), \left(\color{blue}{h} \cdot \frac{w}{d}\right)\right)\right)\right) \]

       16. clear-num N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, d\right), \left(h \cdot \frac{1}{\color{blue}{\frac{d}{w}}}\right)\right)\right)\right) \]

       17. un-div-inv N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, d\right), \left(\frac{h}{\color{blue}{\frac{d}{w}}}\right)\right)\right)\right) \]

       18. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, d\right), \mathsf{/.f64}\left(h, \color{blue}{\left(\frac{d}{w}\right)}\right)\right)\right)\right) \]

       19. /-lowering-/.f64 57.1%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(w, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, d\right), \mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \color{blue}{w}\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 57.1%

       \[\leadsto \color{blue}{\frac{\frac{c0}{D \cdot D}}{\frac{w}{\frac{c0 \cdot d}{\frac{h}{\frac{d}{w}}}}}} \]
   12. Step-by-step derivation

       1. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{w}{c0 \cdot d} \cdot \color{blue}{\frac{h}{\frac{d}{w}}}\right)\right) \]

       2. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{w}{c0 \cdot d} \cdot \left(\frac{h}{d} \cdot \color{blue}{w}\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \left(\left(\frac{w}{c0 \cdot d} \cdot \frac{h}{d}\right) \cdot \color{blue}{w}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\left(\frac{w}{c0 \cdot d} \cdot \frac{h}{d}\right), \color{blue}{w}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{w}{c0 \cdot d}\right), \left(\frac{h}{d}\right)\right), w\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(w, \left(c0 \cdot d\right)\right), \left(\frac{h}{d}\right)\right), w\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(w, \left(d \cdot c0\right)\right), \left(\frac{h}{d}\right)\right), w\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(w, \mathsf{*.f64}\left(d, c0\right)\right), \left(\frac{h}{d}\right)\right), w\right)\right) \]

       9. /-lowering-/.f64 57.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(w, \mathsf{*.f64}\left(d, c0\right)\right), \mathsf{/.f64}\left(h, d\right)\right), w\right)\right) \]

   13. Applied egg-rr 57.3%

       \[\leadsto \frac{\frac{c0}{D \cdot D}}{\color{blue}{\left(\frac{w}{d \cdot c0} \cdot \frac{h}{d}\right) \cdot w}} \]

   if 1.30000000000000004 < D

   1. Initial program 25.4%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Simplified 22.9%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{d \cdot \frac{\frac{c0 \cdot d}{w}}{h}}{D \cdot D} + \sqrt{\frac{\frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot \left(d \cdot \frac{\frac{c0 \cdot d}{w}}{h}\right)\right)}{D \cdot D}}{w \cdot h}}{D \cdot D} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\left(2 \cdot \left(c0 \cdot {d}^{2}\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot {d}^{2}\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left({d}^{2}\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(d \cdot d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(w \cdot h\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. *-lowering-*.f64 37.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(w, h\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 37.1%

      \[\leadsto \frac{c0 \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}}{2 \cdot w} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. count-2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{\left(c0 \cdot d\right) \cdot d}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{\left(c0 \cdot d\right) \cdot d}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{\left(c0 \cdot d\right) \cdot d}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{\left(c0 \cdot d\right) \cdot d}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. times-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      12. distribute-rgt-out N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)}\right), \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   9. Applied egg-rr 62.6%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)}}{2 \cdot w} \]
   10. Step-by-step derivation

       1. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       2. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right) \cdot c0 + \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right) \cdot c0\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

       3. distribute-lft-out N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right) \cdot \left(c0 + c0\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot d}{D}\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       6. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \frac{1}{\frac{D}{c0 \cdot d}}\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       7. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{d}{D \cdot \left(w \cdot h\right)}}{\frac{D}{c0 \cdot d}}\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)}\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       9. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{d}{D}}{w \cdot h}\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{D}\right), \left(w \cdot h\right)\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \left(w \cdot h\right)\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(D, \left(c0 \cdot d\right)\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(c0, d\right)\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       15. count-2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(c0, d\right)\right)\right), \left(2 \cdot c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       16. *-lowering-*.f64 62.6%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(c0, d\right)\right)\right), \mathsf{*.f64}\left(2, c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   11. Applied egg-rr 62.6%

       \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{d}{D}}{w \cdot h}}{\frac{D}{c0 \cdot d}} \cdot \left(2 \cdot c0\right)}}{2 \cdot w} \]
   12. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \frac{\frac{\frac{d}{D}}{w \cdot h}}{\frac{D}{c0 \cdot d}} \cdot \color{blue}{\frac{2 \cdot c0}{2 \cdot w}} \]

       2. times-frac N/A

          \[\leadsto \frac{\frac{\frac{d}{D}}{w \cdot h}}{\frac{D}{c0 \cdot d}} \cdot \left(\frac{2}{2} \cdot \color{blue}{\frac{c0}{w}}\right) \]

       3. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{d}{D}}{w \cdot h}}{\frac{D}{c0 \cdot d}} \cdot \left(1 \cdot \frac{\color{blue}{c0}}{w}\right) \]

       4. *-lft-identity N/A

          \[\leadsto \frac{\frac{\frac{d}{D}}{w \cdot h}}{\frac{D}{c0 \cdot d}} \cdot \frac{c0}{\color{blue}{w}} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{d}{D}}{w \cdot h}}{\frac{D}{c0 \cdot d}}\right), \color{blue}{\left(\frac{c0}{w}\right)}\right) \]

   13. Applied egg-rr 64.9%

       \[\leadsto \color{blue}{\frac{\frac{d}{\frac{D}{c0}}}{\frac{h}{\frac{d}{w \cdot D}}} \cdot \frac{c0}{w}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 1.45 \cdot 10^{-144}:\\ \;\;\;\;\frac{\frac{\frac{c0 \cdot \left(c0 \cdot d\right)}{w}}{h}}{D} \cdot \frac{\frac{d}{w}}{D}\\ \mathbf{elif}\;D \leq 8.2 \cdot 10^{-103}:\\ \;\;\;\;0\\ \mathbf{elif}\;D \leq 1.8 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{c0}{D \cdot D}}{w \cdot \left(\frac{w}{c0 \cdot d} \cdot \frac{h}{d}\right)}\\ \mathbf{elif}\;D \leq 1.3:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\frac{D}{c0}}}{\frac{h}{\frac{d}{w \cdot D}}} \cdot \frac{c0}{w}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 42.5% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d}{D} \cdot \frac{\frac{\frac{\frac{c0 \cdot \left(c0 \cdot d\right)}{w}}{h}}{w}}{D}\\ \mathbf{if}\;D \cdot D \leq 10^{-293}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;D \cdot D \leq 10^{-7}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ d D) (/ (/ (/ (/ (* c0 (* c0 d)) w) h) w) D))))
   (if (<= (* D D) 1e-293) t_0 (if (<= (* D D) 1e-7) 0.0 t_0))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (d / D) * (((((c0 * (c0 * d)) / w) / h) / w) / D);
	double tmp;
	if ((D * D) <= 1e-293) {
		tmp = t_0;
	} else if ((D * D) <= 1e-7) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (d_1 / d) * (((((c0 * (c0 * d_1)) / w) / h) / w) / d)
    if ((d * d) <= 1d-293) then
        tmp = t_0
    else if ((d * d) <= 1d-7) then
        tmp = 0.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (d / D) * (((((c0 * (c0 * d)) / w) / h) / w) / D);
	double tmp;
	if ((D * D) <= 1e-293) {
		tmp = t_0;
	} else if ((D * D) <= 1e-7) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (d / D) * (((((c0 * (c0 * d)) / w) / h) / w) / D)
	tmp = 0
	if (D * D) <= 1e-293:
		tmp = t_0
	elif (D * D) <= 1e-7:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(d / D) * Float64(Float64(Float64(Float64(Float64(c0 * Float64(c0 * d)) / w) / h) / w) / D))
	tmp = 0.0
	if (Float64(D * D) <= 1e-293)
		tmp = t_0;
	elseif (Float64(D * D) <= 1e-7)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (d / D) * (((((c0 * (c0 * d)) / w) / h) / w) / D);
	tmp = 0.0;
	if ((D * D) <= 1e-293)
		tmp = t_0;
	elseif ((D * D) <= 1e-7)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(d / D), $MachinePrecision] * N[(N[(N[(N[(N[(c0 * N[(c0 * d), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision] / h), $MachinePrecision] / w), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(D * D), $MachinePrecision], 1e-293], t$95$0, If[LessEqual[N[(D * D), $MachinePrecision], 1e-7], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 D D) < 1.0000000000000001e-293 or 9.9999999999999995e-8 < (*.f64 D D)

   1. Initial program 24.2%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({c0}^{2} \cdot {d}^{2}\right), \color{blue}{\left({D}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left({d}^{2} \cdot {c0}^{2}\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({d}^{2}\right), \left({c0}^{2}\right)\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(d \cdot d\right), \left({c0}^{2}\right)\right), \left({\color{blue}{D}}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \left({c0}^{2}\right)\right), \left({\color{blue}{D}}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \left(c0 \cdot c0\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(h \cdot {w}^{2}\right)}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{h} \cdot {w}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{h} \cdot {w}^{2}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \color{blue}{\left({w}^{2}\right)}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(w \cdot \color{blue}{w}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 26.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w, \color{blue}{w}\right)\right)\right)\right) \]

   5. Simplified 26.1%

      \[\leadsto \color{blue}{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{\left(h \cdot \left(w \cdot w\right)\right) \cdot \color{blue}{\left(D \cdot D\right)}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{h \cdot \left(w \cdot w\right)}}{\color{blue}{D \cdot D}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{h \cdot \left(w \cdot w\right)}\right), \color{blue}{\left(D \cdot D\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)\right), \left(h \cdot \left(w \cdot w\right)\right)\right), \left(\color{blue}{D} \cdot D\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)\right), \left(h \cdot \left(w \cdot w\right)\right)\right), \left(D \cdot D\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \left(d \cdot \left(c0 \cdot c0\right)\right)\right), \left(h \cdot \left(w \cdot w\right)\right)\right), \left(D \cdot D\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \left(c0 \cdot c0\right)\right)\right), \left(h \cdot \left(w \cdot w\right)\right)\right), \left(D \cdot D\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(c0, c0\right)\right)\right), \left(h \cdot \left(w \cdot w\right)\right)\right), \left(D \cdot D\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(c0, c0\right)\right)\right), \left(\left(w \cdot w\right) \cdot h\right)\right), \left(D \cdot D\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(c0, c0\right)\right)\right), \left(w \cdot \left(w \cdot h\right)\right)\right), \left(D \cdot D\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(c0, c0\right)\right)\right), \mathsf{*.f64}\left(w, \left(w \cdot h\right)\right)\right), \left(D \cdot D\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(c0, c0\right)\right)\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, h\right)\right)\right), \left(D \cdot D\right)\right) \]

      13. *-lowering-*.f64 37.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(c0, c0\right)\right)\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, h\right)\right)\right), \mathsf{*.f64}\left(D, \color{blue}{D}\right)\right) \]

   7. Applied egg-rr 37.0%

      \[\leadsto \color{blue}{\frac{\frac{d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)}{w \cdot \left(w \cdot h\right)}}{D \cdot D}} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{d \cdot \frac{d \cdot \left(c0 \cdot c0\right)}{w \cdot \left(w \cdot h\right)}}{\color{blue}{D} \cdot D} \]

      2. times-frac N/A

         \[\leadsto \frac{d}{D} \cdot \color{blue}{\frac{\frac{d \cdot \left(c0 \cdot c0\right)}{w \cdot \left(w \cdot h\right)}}{D}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{d}{D}\right), \color{blue}{\left(\frac{\frac{d \cdot \left(c0 \cdot c0\right)}{w \cdot \left(w \cdot h\right)}}{D}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, D\right), \left(\frac{\color{blue}{\frac{d \cdot \left(c0 \cdot c0\right)}{w \cdot \left(w \cdot h\right)}}}{D}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{/.f64}\left(\left(\frac{d \cdot \left(c0 \cdot c0\right)}{w \cdot \left(w \cdot h\right)}\right), \color{blue}{D}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{/.f64}\left(\left(\frac{d \cdot \left(c0 \cdot c0\right)}{\left(w \cdot h\right) \cdot w}\right), D\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{/.f64}\left(\left(\frac{\frac{d \cdot \left(c0 \cdot c0\right)}{w \cdot h}}{w}\right), D\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{d \cdot \left(c0 \cdot c0\right)}{w \cdot h}\right), w\right), D\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{d \cdot \left(c0 \cdot c0\right)}{w}}{h}\right), w\right), D\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{d \cdot \left(c0 \cdot c0\right)}{w}\right), h\right), w\right), D\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(d \cdot \left(c0 \cdot c0\right)\right), w\right), h\right), w\right), D\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(c0 \cdot c0\right) \cdot d\right), w\right), h\right), w\right), D\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(c0 \cdot \left(c0 \cdot d\right)\right), w\right), h\right), w\right), D\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot d\right)\right), w\right), h\right), w\right), D\right)\right) \]

      15. *-lowering-*.f64 52.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, d\right)\right), w\right), h\right), w\right), D\right)\right) \]

   9. Applied egg-rr 52.5%

      \[\leadsto \color{blue}{\frac{d}{D} \cdot \frac{\frac{\frac{\frac{c0 \cdot \left(c0 \cdot d\right)}{w}}{h}}{w}}{D}} \]

   if 1.0000000000000001e-293 < (*.f64 D D) < 9.9999999999999995e-8

   1. Initial program 21.3%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Simplified 19.1%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{d \cdot \frac{\frac{c0 \cdot d}{w}}{h}}{D \cdot D} + \sqrt{\frac{\frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot \left(d \cdot \frac{\frac{c0 \cdot d}{w}}{h}\right)\right)}{D \cdot D}}{w \cdot h}}{D \cdot D} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot c0\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. mul0-lft N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(c0 \cdot \left(-1 + 1\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(\mathsf{neg}\left(\left(-1 + 1\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(\mathsf{neg}\left(0\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      13. metadata-eval 47.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 47.1%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(c0 \cdot 0\right)}}{2 \cdot w} \]
   8. Step-by-step derivation

      1. mul0-rgt N/A

         \[\leadsto \frac{c0 \cdot 0}{2 \cdot w} \]

      2. mul0-rgt N/A

         \[\leadsto \frac{0}{\color{blue}{2} \cdot w} \]

      3. div0 47.1%

         \[\leadsto 0 \]

   9. Applied egg-rr 47.1%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 44.4% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 2.5 \cdot 10^{-143}:\\ \;\;\;\;\frac{\frac{\frac{c0 \cdot \left(c0 \cdot d\right)}{w}}{h}}{D} \cdot \frac{\frac{d}{w}}{D}\\ \mathbf{elif}\;D \leq 0.88:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\frac{D}{c0}}}{\frac{h}{\frac{d}{w \cdot D}}} \cdot \frac{c0}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= D 2.5e-143)
   (* (/ (/ (/ (* c0 (* c0 d)) w) h) D) (/ (/ d w) D))
   (if (<= D 0.88) 0.0 (* (/ (/ d (/ D c0)) (/ h (/ d (* w D)))) (/ c0 w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (D <= 2.5e-143) {
		tmp = ((((c0 * (c0 * d)) / w) / h) / D) * ((d / w) / D);
	} else if (D <= 0.88) {
		tmp = 0.0;
	} else {
		tmp = ((d / (D / c0)) / (h / (d / (w * D)))) * (c0 / w);
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (d <= 2.5d-143) then
        tmp = ((((c0 * (c0 * d_1)) / w) / h) / d) * ((d_1 / w) / d)
    else if (d <= 0.88d0) then
        tmp = 0.0d0
    else
        tmp = ((d_1 / (d / c0)) / (h / (d_1 / (w * d)))) * (c0 / w)
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (D <= 2.5e-143) {
		tmp = ((((c0 * (c0 * d)) / w) / h) / D) * ((d / w) / D);
	} else if (D <= 0.88) {
		tmp = 0.0;
	} else {
		tmp = ((d / (D / c0)) / (h / (d / (w * D)))) * (c0 / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if D <= 2.5e-143:
		tmp = ((((c0 * (c0 * d)) / w) / h) / D) * ((d / w) / D)
	elif D <= 0.88:
		tmp = 0.0
	else:
		tmp = ((d / (D / c0)) / (h / (d / (w * D)))) * (c0 / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (D <= 2.5e-143)
		tmp = Float64(Float64(Float64(Float64(Float64(c0 * Float64(c0 * d)) / w) / h) / D) * Float64(Float64(d / w) / D));
	elseif (D <= 0.88)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(d / Float64(D / c0)) / Float64(h / Float64(d / Float64(w * D)))) * Float64(c0 / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (D <= 2.5e-143)
		tmp = ((((c0 * (c0 * d)) / w) / h) / D) * ((d / w) / D);
	elseif (D <= 0.88)
		tmp = 0.0;
	else
		tmp = ((d / (D / c0)) / (h / (d / (w * D)))) * (c0 / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[D, 2.5e-143], N[(N[(N[(N[(N[(c0 * N[(c0 * d), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision] / h), $MachinePrecision] / D), $MachinePrecision] * N[(N[(d / w), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision], If[LessEqual[D, 0.88], 0.0, N[(N[(N[(d / N[(D / c0), $MachinePrecision]), $MachinePrecision] / N[(h / N[(d / N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c0 / w), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if D < 2.5000000000000001e-143

   1. Initial program 20.8%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({c0}^{2} \cdot {d}^{2}\right), \color{blue}{\left({D}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left({d}^{2} \cdot {c0}^{2}\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({d}^{2}\right), \left({c0}^{2}\right)\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(d \cdot d\right), \left({c0}^{2}\right)\right), \left({\color{blue}{D}}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \left({c0}^{2}\right)\right), \left({\color{blue}{D}}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \left(c0 \cdot c0\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(h \cdot {w}^{2}\right)}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{h} \cdot {w}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{h} \cdot {w}^{2}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \color{blue}{\left({w}^{2}\right)}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(w \cdot \color{blue}{w}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 20.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w, \color{blue}{w}\right)\right)\right)\right) \]

   5. Simplified 20.7%

      \[\leadsto \color{blue}{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{\left(h \cdot \left(w \cdot w\right)\right) \cdot \color{blue}{\left(D \cdot D\right)}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{h \cdot \left(w \cdot w\right)}}{\color{blue}{D \cdot D}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{h \cdot \left(w \cdot w\right)}\right), \color{blue}{\left(D \cdot D\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)\right), \left(h \cdot \left(w \cdot w\right)\right)\right), \left(\color{blue}{D} \cdot D\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)\right), \left(h \cdot \left(w \cdot w\right)\right)\right), \left(D \cdot D\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \left(d \cdot \left(c0 \cdot c0\right)\right)\right), \left(h \cdot \left(w \cdot w\right)\right)\right), \left(D \cdot D\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \left(c0 \cdot c0\right)\right)\right), \left(h \cdot \left(w \cdot w\right)\right)\right), \left(D \cdot D\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(c0, c0\right)\right)\right), \left(h \cdot \left(w \cdot w\right)\right)\right), \left(D \cdot D\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(c0, c0\right)\right)\right), \left(\left(w \cdot w\right) \cdot h\right)\right), \left(D \cdot D\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(c0, c0\right)\right)\right), \left(w \cdot \left(w \cdot h\right)\right)\right), \left(D \cdot D\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(c0, c0\right)\right)\right), \mathsf{*.f64}\left(w, \left(w \cdot h\right)\right)\right), \left(D \cdot D\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(c0, c0\right)\right)\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, h\right)\right)\right), \left(D \cdot D\right)\right) \]

      13. *-lowering-*.f64 31.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(c0, c0\right)\right)\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, h\right)\right)\right), \mathsf{*.f64}\left(D, \color{blue}{D}\right)\right) \]

   7. Applied egg-rr 31.8%

      \[\leadsto \color{blue}{\frac{\frac{d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)}{w \cdot \left(w \cdot h\right)}}{D \cdot D}} \]
   8. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \frac{\frac{d}{w} \cdot \frac{d \cdot \left(c0 \cdot c0\right)}{w \cdot h}}{\color{blue}{D} \cdot D} \]

      2. times-frac N/A

         \[\leadsto \frac{\frac{d}{w}}{D} \cdot \color{blue}{\frac{\frac{d \cdot \left(c0 \cdot c0\right)}{w \cdot h}}{D}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{d}{w}}{D}\right), \color{blue}{\left(\frac{\frac{d \cdot \left(c0 \cdot c0\right)}{w \cdot h}}{D}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{w}\right), D\right), \left(\frac{\color{blue}{\frac{d \cdot \left(c0 \cdot c0\right)}{w \cdot h}}}{D}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, w\right), D\right), \left(\frac{\frac{\color{blue}{d \cdot \left(c0 \cdot c0\right)}}{w \cdot h}}{D}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, w\right), D\right), \mathsf{/.f64}\left(\left(\frac{d \cdot \left(c0 \cdot c0\right)}{w \cdot h}\right), \color{blue}{D}\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, w\right), D\right), \mathsf{/.f64}\left(\left(\frac{\frac{d \cdot \left(c0 \cdot c0\right)}{w}}{h}\right), D\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, w\right), D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{d \cdot \left(c0 \cdot c0\right)}{w}\right), h\right), D\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, w\right), D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(d \cdot \left(c0 \cdot c0\right)\right), w\right), h\right), D\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, w\right), D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(c0 \cdot c0\right) \cdot d\right), w\right), h\right), D\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, w\right), D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(c0 \cdot \left(c0 \cdot d\right)\right), w\right), h\right), D\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, w\right), D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot d\right)\right), w\right), h\right), D\right)\right) \]

      13. *-lowering-*.f64 47.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, w\right), D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, d\right)\right), w\right), h\right), D\right)\right) \]

   9. Applied egg-rr 47.0%

      \[\leadsto \color{blue}{\frac{\frac{d}{w}}{D} \cdot \frac{\frac{\frac{c0 \cdot \left(c0 \cdot d\right)}{w}}{h}}{D}} \]

   if 2.5000000000000001e-143 < D < 0.880000000000000004

   1. Initial program 33.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Simplified 31.3%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{d \cdot \frac{\frac{c0 \cdot d}{w}}{h}}{D \cdot D} + \sqrt{\frac{\frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot \left(d \cdot \frac{\frac{c0 \cdot d}{w}}{h}\right)\right)}{D \cdot D}}{w \cdot h}}{D \cdot D} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot c0\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. mul0-lft N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(c0 \cdot \left(-1 + 1\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(\mathsf{neg}\left(\left(-1 + 1\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(\mathsf{neg}\left(0\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      13. metadata-eval 40.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 40.5%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(c0 \cdot 0\right)}}{2 \cdot w} \]
   8. Step-by-step derivation

      1. mul0-rgt N/A

         \[\leadsto \frac{c0 \cdot 0}{2 \cdot w} \]

      2. mul0-rgt N/A

         \[\leadsto \frac{0}{\color{blue}{2} \cdot w} \]

      3. div0 40.5%

         \[\leadsto 0 \]

   9. Applied egg-rr 40.5%

      \[\leadsto \color{blue}{0} \]

   if 0.880000000000000004 < D

   1. Initial program 25.4%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Simplified 22.9%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{d \cdot \frac{\frac{c0 \cdot d}{w}}{h}}{D \cdot D} + \sqrt{\frac{\frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot \left(d \cdot \frac{\frac{c0 \cdot d}{w}}{h}\right)\right)}{D \cdot D}}{w \cdot h}}{D \cdot D} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\left(2 \cdot \left(c0 \cdot {d}^{2}\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot {d}^{2}\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left({d}^{2}\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(d \cdot d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(w \cdot h\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. *-lowering-*.f64 37.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(w, h\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 37.1%

      \[\leadsto \frac{c0 \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}}{2 \cdot w} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. count-2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{\left(c0 \cdot d\right) \cdot d}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{\left(c0 \cdot d\right) \cdot d}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{\left(c0 \cdot d\right) \cdot d}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{\left(c0 \cdot d\right) \cdot d}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. times-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      12. distribute-rgt-out N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)}\right), \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   9. Applied egg-rr 62.6%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)}}{2 \cdot w} \]
   10. Step-by-step derivation

       1. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       2. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right) \cdot c0 + \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right) \cdot c0\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

       3. distribute-lft-out N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right) \cdot \left(c0 + c0\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot d}{D}\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       6. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \frac{1}{\frac{D}{c0 \cdot d}}\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       7. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{d}{D \cdot \left(w \cdot h\right)}}{\frac{D}{c0 \cdot d}}\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)}\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       9. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{d}{D}}{w \cdot h}\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{D}\right), \left(w \cdot h\right)\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \left(w \cdot h\right)\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(D, \left(c0 \cdot d\right)\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(c0, d\right)\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       15. count-2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(c0, d\right)\right)\right), \left(2 \cdot c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       16. *-lowering-*.f64 62.6%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(c0, d\right)\right)\right), \mathsf{*.f64}\left(2, c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   11. Applied egg-rr 62.6%

       \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{d}{D}}{w \cdot h}}{\frac{D}{c0 \cdot d}} \cdot \left(2 \cdot c0\right)}}{2 \cdot w} \]
   12. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \frac{\frac{\frac{d}{D}}{w \cdot h}}{\frac{D}{c0 \cdot d}} \cdot \color{blue}{\frac{2 \cdot c0}{2 \cdot w}} \]

       2. times-frac N/A

          \[\leadsto \frac{\frac{\frac{d}{D}}{w \cdot h}}{\frac{D}{c0 \cdot d}} \cdot \left(\frac{2}{2} \cdot \color{blue}{\frac{c0}{w}}\right) \]

       3. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{d}{D}}{w \cdot h}}{\frac{D}{c0 \cdot d}} \cdot \left(1 \cdot \frac{\color{blue}{c0}}{w}\right) \]

       4. *-lft-identity N/A

          \[\leadsto \frac{\frac{\frac{d}{D}}{w \cdot h}}{\frac{D}{c0 \cdot d}} \cdot \frac{c0}{\color{blue}{w}} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{d}{D}}{w \cdot h}}{\frac{D}{c0 \cdot d}}\right), \color{blue}{\left(\frac{c0}{w}\right)}\right) \]

   13. Applied egg-rr 64.9%

       \[\leadsto \color{blue}{\frac{\frac{d}{\frac{D}{c0}}}{\frac{h}{\frac{d}{w \cdot D}}} \cdot \frac{c0}{w}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 2.5 \cdot 10^{-143}:\\ \;\;\;\;\frac{\frac{\frac{c0 \cdot \left(c0 \cdot d\right)}{w}}{h}}{D} \cdot \frac{\frac{d}{w}}{D}\\ \mathbf{elif}\;D \leq 0.88:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\frac{D}{c0}}}{\frac{h}{\frac{d}{w \cdot D}}} \cdot \frac{c0}{w}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 44.0% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 6.5 \cdot 10^{-144}:\\ \;\;\;\;\frac{\frac{\frac{c0 \cdot \left(c0 \cdot d\right)}{w}}{h}}{D} \cdot \frac{\frac{d}{w}}{D}\\ \mathbf{elif}\;D \leq 0.00042:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{D}}{\left(w \cdot h\right) \cdot D} \cdot \left(\left(c0 \cdot d\right) \cdot \frac{c0}{w}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= D 6.5e-144)
   (* (/ (/ (/ (* c0 (* c0 d)) w) h) D) (/ (/ d w) D))
   (if (<= D 0.00042)
     0.0
     (* (/ (/ d D) (* (* w h) D)) (* (* c0 d) (/ c0 w))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (D <= 6.5e-144) {
		tmp = ((((c0 * (c0 * d)) / w) / h) / D) * ((d / w) / D);
	} else if (D <= 0.00042) {
		tmp = 0.0;
	} else {
		tmp = ((d / D) / ((w * h) * D)) * ((c0 * d) * (c0 / w));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (d <= 6.5d-144) then
        tmp = ((((c0 * (c0 * d_1)) / w) / h) / d) * ((d_1 / w) / d)
    else if (d <= 0.00042d0) then
        tmp = 0.0d0
    else
        tmp = ((d_1 / d) / ((w * h) * d)) * ((c0 * d_1) * (c0 / w))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (D <= 6.5e-144) {
		tmp = ((((c0 * (c0 * d)) / w) / h) / D) * ((d / w) / D);
	} else if (D <= 0.00042) {
		tmp = 0.0;
	} else {
		tmp = ((d / D) / ((w * h) * D)) * ((c0 * d) * (c0 / w));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if D <= 6.5e-144:
		tmp = ((((c0 * (c0 * d)) / w) / h) / D) * ((d / w) / D)
	elif D <= 0.00042:
		tmp = 0.0
	else:
		tmp = ((d / D) / ((w * h) * D)) * ((c0 * d) * (c0 / w))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (D <= 6.5e-144)
		tmp = Float64(Float64(Float64(Float64(Float64(c0 * Float64(c0 * d)) / w) / h) / D) * Float64(Float64(d / w) / D));
	elseif (D <= 0.00042)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(d / D) / Float64(Float64(w * h) * D)) * Float64(Float64(c0 * d) * Float64(c0 / w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (D <= 6.5e-144)
		tmp = ((((c0 * (c0 * d)) / w) / h) / D) * ((d / w) / D);
	elseif (D <= 0.00042)
		tmp = 0.0;
	else
		tmp = ((d / D) / ((w * h) * D)) * ((c0 * d) * (c0 / w));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[D, 6.5e-144], N[(N[(N[(N[(N[(c0 * N[(c0 * d), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision] / h), $MachinePrecision] / D), $MachinePrecision] * N[(N[(d / w), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision], If[LessEqual[D, 0.00042], 0.0, N[(N[(N[(d / D), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * N[(N[(c0 * d), $MachinePrecision] * N[(c0 / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if D < 6.49999999999999968e-144

   1. Initial program 20.8%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({c0}^{2} \cdot {d}^{2}\right), \color{blue}{\left({D}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left({d}^{2} \cdot {c0}^{2}\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({d}^{2}\right), \left({c0}^{2}\right)\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(d \cdot d\right), \left({c0}^{2}\right)\right), \left({\color{blue}{D}}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \left({c0}^{2}\right)\right), \left({\color{blue}{D}}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \left(c0 \cdot c0\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(h \cdot {w}^{2}\right)}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{h} \cdot {w}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{h} \cdot {w}^{2}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \color{blue}{\left({w}^{2}\right)}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(w \cdot \color{blue}{w}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 20.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w, \color{blue}{w}\right)\right)\right)\right) \]

   5. Simplified 20.7%

      \[\leadsto \color{blue}{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{\left(h \cdot \left(w \cdot w\right)\right) \cdot \color{blue}{\left(D \cdot D\right)}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{h \cdot \left(w \cdot w\right)}}{\color{blue}{D \cdot D}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{h \cdot \left(w \cdot w\right)}\right), \color{blue}{\left(D \cdot D\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)\right), \left(h \cdot \left(w \cdot w\right)\right)\right), \left(\color{blue}{D} \cdot D\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)\right), \left(h \cdot \left(w \cdot w\right)\right)\right), \left(D \cdot D\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \left(d \cdot \left(c0 \cdot c0\right)\right)\right), \left(h \cdot \left(w \cdot w\right)\right)\right), \left(D \cdot D\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \left(c0 \cdot c0\right)\right)\right), \left(h \cdot \left(w \cdot w\right)\right)\right), \left(D \cdot D\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(c0, c0\right)\right)\right), \left(h \cdot \left(w \cdot w\right)\right)\right), \left(D \cdot D\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(c0, c0\right)\right)\right), \left(\left(w \cdot w\right) \cdot h\right)\right), \left(D \cdot D\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(c0, c0\right)\right)\right), \left(w \cdot \left(w \cdot h\right)\right)\right), \left(D \cdot D\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(c0, c0\right)\right)\right), \mathsf{*.f64}\left(w, \left(w \cdot h\right)\right)\right), \left(D \cdot D\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(c0, c0\right)\right)\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, h\right)\right)\right), \left(D \cdot D\right)\right) \]

      13. *-lowering-*.f64 31.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(c0, c0\right)\right)\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, h\right)\right)\right), \mathsf{*.f64}\left(D, \color{blue}{D}\right)\right) \]

   7. Applied egg-rr 31.8%

      \[\leadsto \color{blue}{\frac{\frac{d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)}{w \cdot \left(w \cdot h\right)}}{D \cdot D}} \]
   8. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \frac{\frac{d}{w} \cdot \frac{d \cdot \left(c0 \cdot c0\right)}{w \cdot h}}{\color{blue}{D} \cdot D} \]

      2. times-frac N/A

         \[\leadsto \frac{\frac{d}{w}}{D} \cdot \color{blue}{\frac{\frac{d \cdot \left(c0 \cdot c0\right)}{w \cdot h}}{D}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{d}{w}}{D}\right), \color{blue}{\left(\frac{\frac{d \cdot \left(c0 \cdot c0\right)}{w \cdot h}}{D}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{w}\right), D\right), \left(\frac{\color{blue}{\frac{d \cdot \left(c0 \cdot c0\right)}{w \cdot h}}}{D}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, w\right), D\right), \left(\frac{\frac{\color{blue}{d \cdot \left(c0 \cdot c0\right)}}{w \cdot h}}{D}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, w\right), D\right), \mathsf{/.f64}\left(\left(\frac{d \cdot \left(c0 \cdot c0\right)}{w \cdot h}\right), \color{blue}{D}\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, w\right), D\right), \mathsf{/.f64}\left(\left(\frac{\frac{d \cdot \left(c0 \cdot c0\right)}{w}}{h}\right), D\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, w\right), D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{d \cdot \left(c0 \cdot c0\right)}{w}\right), h\right), D\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, w\right), D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(d \cdot \left(c0 \cdot c0\right)\right), w\right), h\right), D\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, w\right), D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(c0 \cdot c0\right) \cdot d\right), w\right), h\right), D\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, w\right), D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(c0 \cdot \left(c0 \cdot d\right)\right), w\right), h\right), D\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, w\right), D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot d\right)\right), w\right), h\right), D\right)\right) \]

      13. *-lowering-*.f64 47.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, w\right), D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, d\right)\right), w\right), h\right), D\right)\right) \]

   9. Applied egg-rr 47.0%

      \[\leadsto \color{blue}{\frac{\frac{d}{w}}{D} \cdot \frac{\frac{\frac{c0 \cdot \left(c0 \cdot d\right)}{w}}{h}}{D}} \]

   if 6.49999999999999968e-144 < D < 4.2000000000000002e-4

   1. Initial program 34.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Simplified 31.9%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{d \cdot \frac{\frac{c0 \cdot d}{w}}{h}}{D \cdot D} + \sqrt{\frac{\frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot \left(d \cdot \frac{\frac{c0 \cdot d}{w}}{h}\right)\right)}{D \cdot D}}{w \cdot h}}{D \cdot D} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot c0\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. mul0-lft N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(c0 \cdot \left(-1 + 1\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(\mathsf{neg}\left(\left(-1 + 1\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(\mathsf{neg}\left(0\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      13. metadata-eval 41.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 41.6%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(c0 \cdot 0\right)}}{2 \cdot w} \]
   8. Step-by-step derivation

      1. mul0-rgt N/A

         \[\leadsto \frac{c0 \cdot 0}{2 \cdot w} \]

      2. mul0-rgt N/A

         \[\leadsto \frac{0}{\color{blue}{2} \cdot w} \]

      3. div0 41.6%

         \[\leadsto 0 \]

   9. Applied egg-rr 41.6%

      \[\leadsto \color{blue}{0} \]

   if 4.2000000000000002e-4 < D

   1. Initial program 25.0%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Simplified 22.5%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{d \cdot \frac{\frac{c0 \cdot d}{w}}{h}}{D \cdot D} + \sqrt{\frac{\frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot \left(d \cdot \frac{\frac{c0 \cdot d}{w}}{h}\right)\right)}{D \cdot D}}{w \cdot h}}{D \cdot D} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\left(2 \cdot \left(c0 \cdot {d}^{2}\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot {d}^{2}\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left({d}^{2}\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(d \cdot d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(w \cdot h\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. *-lowering-*.f64 36.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(w, h\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 36.4%

      \[\leadsto \frac{c0 \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}}{2 \cdot w} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. count-2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{\left(c0 \cdot d\right) \cdot d}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{\left(c0 \cdot d\right) \cdot d}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{\left(c0 \cdot d\right) \cdot d}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{\left(c0 \cdot d\right) \cdot d}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. times-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      12. distribute-rgt-out N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)}\right), \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   9. Applied egg-rr 63.5%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)}}{2 \cdot w} \]
   10. Step-by-step derivation

       1. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       2. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right) \cdot c0 + \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right) \cdot c0\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

       3. distribute-lft-out N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right) \cdot \left(c0 + c0\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot d}{D}\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       6. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \frac{1}{\frac{D}{c0 \cdot d}}\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       7. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{d}{D \cdot \left(w \cdot h\right)}}{\frac{D}{c0 \cdot d}}\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)}\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       9. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{d}{D}}{w \cdot h}\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{D}\right), \left(w \cdot h\right)\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \left(w \cdot h\right)\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(D, \left(c0 \cdot d\right)\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(c0, d\right)\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       15. count-2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(c0, d\right)\right)\right), \left(2 \cdot c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       16. *-lowering-*.f64 63.6%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(c0, d\right)\right)\right), \mathsf{*.f64}\left(2, c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   11. Applied egg-rr 63.6%

       \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{d}{D}}{w \cdot h}}{\frac{D}{c0 \cdot d}} \cdot \left(2 \cdot c0\right)}}{2 \cdot w} \]
   12. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \frac{\frac{\frac{d}{D}}{w \cdot h}}{\frac{D}{c0 \cdot d}} \cdot \color{blue}{\frac{2 \cdot c0}{2 \cdot w}} \]

       2. associate-/r/ N/A

          \[\leadsto \left(\frac{\frac{\frac{d}{D}}{w \cdot h}}{D} \cdot \left(c0 \cdot d\right)\right) \cdot \frac{\color{blue}{2 \cdot c0}}{2 \cdot w} \]

       3. associate-*l* N/A

          \[\leadsto \frac{\frac{\frac{d}{D}}{w \cdot h}}{D} \cdot \color{blue}{\left(\left(c0 \cdot d\right) \cdot \frac{2 \cdot c0}{2 \cdot w}\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{d}{D}}{w \cdot h}}{D}\right), \color{blue}{\left(\left(c0 \cdot d\right) \cdot \frac{2 \cdot c0}{2 \cdot w}\right)}\right) \]

       5. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{d}{D}}{D \cdot \left(w \cdot h\right)}\right), \left(\color{blue}{\left(c0 \cdot d\right)} \cdot \frac{2 \cdot c0}{2 \cdot w}\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{d}{D}}{\left(w \cdot h\right) \cdot D}\right), \left(\left(c0 \cdot \color{blue}{d}\right) \cdot \frac{2 \cdot c0}{2 \cdot w}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{D}\right), \left(\left(w \cdot h\right) \cdot D\right)\right), \left(\color{blue}{\left(c0 \cdot d\right)} \cdot \frac{2 \cdot c0}{2 \cdot w}\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \left(\left(w \cdot h\right) \cdot D\right)\right), \left(\left(\color{blue}{c0} \cdot d\right) \cdot \frac{2 \cdot c0}{2 \cdot w}\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \left(D \cdot \left(w \cdot h\right)\right)\right), \left(\left(c0 \cdot \color{blue}{d}\right) \cdot \frac{2 \cdot c0}{2 \cdot w}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(D, \left(w \cdot h\right)\right)\right), \left(\left(c0 \cdot \color{blue}{d}\right) \cdot \frac{2 \cdot c0}{2 \cdot w}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, h\right)\right)\right), \left(\left(c0 \cdot d\right) \cdot \frac{2 \cdot c0}{2 \cdot w}\right)\right) \]

       12. times-frac N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, h\right)\right)\right), \left(\left(c0 \cdot d\right) \cdot \left(\frac{2}{2} \cdot \color{blue}{\frac{c0}{w}}\right)\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, h\right)\right)\right), \left(\left(c0 \cdot d\right) \cdot \left(1 \cdot \frac{\color{blue}{c0}}{w}\right)\right)\right) \]

       14. *-lft-identity N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, h\right)\right)\right), \left(\left(c0 \cdot d\right) \cdot \frac{c0}{\color{blue}{w}}\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, h\right)\right)\right), \mathsf{*.f64}\left(\left(c0 \cdot d\right), \color{blue}{\left(\frac{c0}{w}\right)}\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, h\right)\right)\right), \mathsf{*.f64}\left(\left(d \cdot c0\right), \left(\frac{\color{blue}{c0}}{w}\right)\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, h\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, c0\right), \left(\frac{\color{blue}{c0}}{w}\right)\right)\right) \]

       18. /-lowering-/.f64 63.3%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, h\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{/.f64}\left(c0, \color{blue}{w}\right)\right)\right) \]

   13. Applied egg-rr 63.3%

       \[\leadsto \color{blue}{\frac{\frac{d}{D}}{D \cdot \left(w \cdot h\right)} \cdot \left(\left(d \cdot c0\right) \cdot \frac{c0}{w}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 6.5 \cdot 10^{-144}:\\ \;\;\;\;\frac{\frac{\frac{c0 \cdot \left(c0 \cdot d\right)}{w}}{h}}{D} \cdot \frac{\frac{d}{w}}{D}\\ \mathbf{elif}\;D \leq 0.00042:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{D}}{\left(w \cdot h\right) \cdot D} \cdot \left(\left(c0 \cdot d\right) \cdot \frac{c0}{w}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 43.9% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 3.5 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{d}{D}}{w \cdot h} \cdot \left(\frac{d}{\frac{D}{c0}} \cdot \frac{c0}{w}\right)\\ \mathbf{elif}\;D \leq 0.00024:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{D}}{\left(w \cdot h\right) \cdot D} \cdot \left(\left(c0 \cdot d\right) \cdot \frac{c0}{w}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= D 3.5e-145)
   (* (/ (/ d D) (* w h)) (* (/ d (/ D c0)) (/ c0 w)))
   (if (<= D 0.00024)
     0.0
     (* (/ (/ d D) (* (* w h) D)) (* (* c0 d) (/ c0 w))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (D <= 3.5e-145) {
		tmp = ((d / D) / (w * h)) * ((d / (D / c0)) * (c0 / w));
	} else if (D <= 0.00024) {
		tmp = 0.0;
	} else {
		tmp = ((d / D) / ((w * h) * D)) * ((c0 * d) * (c0 / w));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (d <= 3.5d-145) then
        tmp = ((d_1 / d) / (w * h)) * ((d_1 / (d / c0)) * (c0 / w))
    else if (d <= 0.00024d0) then
        tmp = 0.0d0
    else
        tmp = ((d_1 / d) / ((w * h) * d)) * ((c0 * d_1) * (c0 / w))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (D <= 3.5e-145) {
		tmp = ((d / D) / (w * h)) * ((d / (D / c0)) * (c0 / w));
	} else if (D <= 0.00024) {
		tmp = 0.0;
	} else {
		tmp = ((d / D) / ((w * h) * D)) * ((c0 * d) * (c0 / w));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if D <= 3.5e-145:
		tmp = ((d / D) / (w * h)) * ((d / (D / c0)) * (c0 / w))
	elif D <= 0.00024:
		tmp = 0.0
	else:
		tmp = ((d / D) / ((w * h) * D)) * ((c0 * d) * (c0 / w))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (D <= 3.5e-145)
		tmp = Float64(Float64(Float64(d / D) / Float64(w * h)) * Float64(Float64(d / Float64(D / c0)) * Float64(c0 / w)));
	elseif (D <= 0.00024)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(d / D) / Float64(Float64(w * h) * D)) * Float64(Float64(c0 * d) * Float64(c0 / w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (D <= 3.5e-145)
		tmp = ((d / D) / (w * h)) * ((d / (D / c0)) * (c0 / w));
	elseif (D <= 0.00024)
		tmp = 0.0;
	else
		tmp = ((d / D) / ((w * h) * D)) * ((c0 * d) * (c0 / w));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[D, 3.5e-145], N[(N[(N[(d / D), $MachinePrecision] / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[(N[(d / N[(D / c0), $MachinePrecision]), $MachinePrecision] * N[(c0 / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[D, 0.00024], 0.0, N[(N[(N[(d / D), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * N[(N[(c0 * d), $MachinePrecision] * N[(c0 / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if D < 3.49999999999999997e-145

   1. Initial program 20.8%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Simplified 20.4%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{d \cdot \frac{\frac{c0 \cdot d}{w}}{h}}{D \cdot D} + \sqrt{\frac{\frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot \left(d \cdot \frac{\frac{c0 \cdot d}{w}}{h}\right)\right)}{D \cdot D}}{w \cdot h}}{D \cdot D} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\left(2 \cdot \left(c0 \cdot {d}^{2}\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot {d}^{2}\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left({d}^{2}\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(d \cdot d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(w \cdot h\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. *-lowering-*.f64 26.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(w, h\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 26.9%

      \[\leadsto \frac{c0 \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}}{2 \cdot w} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. count-2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{\left(c0 \cdot d\right) \cdot d}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{\left(c0 \cdot d\right) \cdot d}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{\left(c0 \cdot d\right) \cdot d}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{\left(c0 \cdot d\right) \cdot d}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. times-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      12. distribute-rgt-out N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)}\right), \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   9. Applied egg-rr 48.4%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)}}{2 \cdot w} \]
   10. Step-by-step derivation

       1. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       2. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right) \cdot c0 + \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right) \cdot c0\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

       3. distribute-lft-out N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right) \cdot \left(c0 + c0\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot d}{D}\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       6. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \frac{1}{\frac{D}{c0 \cdot d}}\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       7. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{d}{D \cdot \left(w \cdot h\right)}}{\frac{D}{c0 \cdot d}}\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)}\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       9. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{d}{D}}{w \cdot h}\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{D}\right), \left(w \cdot h\right)\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \left(w \cdot h\right)\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(D, \left(c0 \cdot d\right)\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(c0, d\right)\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       15. count-2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(c0, d\right)\right)\right), \left(2 \cdot c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       16. *-lowering-*.f64 47.8%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(c0, d\right)\right)\right), \mathsf{*.f64}\left(2, c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   11. Applied egg-rr 47.8%

       \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{d}{D}}{w \cdot h}}{\frac{D}{c0 \cdot d}} \cdot \left(2 \cdot c0\right)}}{2 \cdot w} \]
   12. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \frac{\frac{\frac{d}{D}}{w \cdot h}}{\frac{D}{c0 \cdot d}} \cdot \color{blue}{\frac{2 \cdot c0}{2 \cdot w}} \]

       2. div-inv N/A

          \[\leadsto \left(\frac{\frac{d}{D}}{w \cdot h} \cdot \frac{1}{\frac{D}{c0 \cdot d}}\right) \cdot \frac{\color{blue}{2 \cdot c0}}{2 \cdot w} \]

       3. clear-num N/A

          \[\leadsto \left(\frac{\frac{d}{D}}{w \cdot h} \cdot \frac{c0 \cdot d}{D}\right) \cdot \frac{2 \cdot \color{blue}{c0}}{2 \cdot w} \]

       4. associate-*l* N/A

          \[\leadsto \frac{\frac{d}{D}}{w \cdot h} \cdot \color{blue}{\left(\frac{c0 \cdot d}{D} \cdot \frac{2 \cdot c0}{2 \cdot w}\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{d}{D}}{w \cdot h}\right), \color{blue}{\left(\frac{c0 \cdot d}{D} \cdot \frac{2 \cdot c0}{2 \cdot w}\right)}\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{D}\right), \left(w \cdot h\right)\right), \left(\color{blue}{\frac{c0 \cdot d}{D}} \cdot \frac{2 \cdot c0}{2 \cdot w}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \left(w \cdot h\right)\right), \left(\frac{\color{blue}{c0 \cdot d}}{D} \cdot \frac{2 \cdot c0}{2 \cdot w}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \left(\frac{c0 \cdot d}{\color{blue}{D}} \cdot \frac{2 \cdot c0}{2 \cdot w}\right)\right) \]

       9. times-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \left(\frac{c0 \cdot d}{D} \cdot \left(\frac{2}{2} \cdot \color{blue}{\frac{c0}{w}}\right)\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \left(\frac{c0 \cdot d}{D} \cdot \left(1 \cdot \frac{\color{blue}{c0}}{w}\right)\right)\right) \]

       11. *-lft-identity N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \left(\frac{c0 \cdot d}{D} \cdot \frac{c0}{\color{blue}{w}}\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{*.f64}\left(\left(\frac{c0 \cdot d}{D}\right), \color{blue}{\left(\frac{c0}{w}\right)}\right)\right) \]

       13. clear-num N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{*.f64}\left(\left(\frac{1}{\frac{D}{c0 \cdot d}}\right), \left(\frac{\color{blue}{c0}}{w}\right)\right)\right) \]

       14. associate-/r* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{*.f64}\left(\left(\frac{1}{\frac{\frac{D}{c0}}{d}}\right), \left(\frac{c0}{w}\right)\right)\right) \]

       15. clear-num N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{*.f64}\left(\left(\frac{1}{\frac{1}{\frac{d}{\frac{D}{c0}}}}\right), \left(\frac{c0}{w}\right)\right)\right) \]

       16. remove-double-div N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{*.f64}\left(\left(\frac{d}{\frac{D}{c0}}\right), \left(\frac{\color{blue}{c0}}{w}\right)\right)\right) \]

       17. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(\frac{D}{c0}\right)\right), \left(\frac{\color{blue}{c0}}{w}\right)\right)\right) \]

       18. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{/.f64}\left(D, c0\right)\right), \left(\frac{c0}{w}\right)\right)\right) \]

       19. /-lowering-/.f64 44.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{/.f64}\left(D, c0\right)\right), \mathsf{/.f64}\left(c0, \color{blue}{w}\right)\right)\right) \]

   13. Applied egg-rr 44.8%

       \[\leadsto \color{blue}{\frac{\frac{d}{D}}{w \cdot h} \cdot \left(\frac{d}{\frac{D}{c0}} \cdot \frac{c0}{w}\right)} \]

   if 3.49999999999999997e-145 < D < 2.40000000000000006e-4

   1. Initial program 34.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Simplified 31.9%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{d \cdot \frac{\frac{c0 \cdot d}{w}}{h}}{D \cdot D} + \sqrt{\frac{\frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot \left(d \cdot \frac{\frac{c0 \cdot d}{w}}{h}\right)\right)}{D \cdot D}}{w \cdot h}}{D \cdot D} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot c0\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. mul0-lft N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(c0 \cdot \left(-1 + 1\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(\mathsf{neg}\left(\left(-1 + 1\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(\mathsf{neg}\left(0\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      13. metadata-eval 41.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 41.6%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(c0 \cdot 0\right)}}{2 \cdot w} \]
   8. Step-by-step derivation

      1. mul0-rgt N/A

         \[\leadsto \frac{c0 \cdot 0}{2 \cdot w} \]

      2. mul0-rgt N/A

         \[\leadsto \frac{0}{\color{blue}{2} \cdot w} \]

      3. div0 41.6%

         \[\leadsto 0 \]

   9. Applied egg-rr 41.6%

      \[\leadsto \color{blue}{0} \]

   if 2.40000000000000006e-4 < D

   1. Initial program 25.0%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Simplified 22.5%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{d \cdot \frac{\frac{c0 \cdot d}{w}}{h}}{D \cdot D} + \sqrt{\frac{\frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot \left(d \cdot \frac{\frac{c0 \cdot d}{w}}{h}\right)\right)}{D \cdot D}}{w \cdot h}}{D \cdot D} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\left(2 \cdot \left(c0 \cdot {d}^{2}\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot {d}^{2}\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left({d}^{2}\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(d \cdot d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(w \cdot h\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. *-lowering-*.f64 36.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(w, h\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 36.4%

      \[\leadsto \frac{c0 \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}}{2 \cdot w} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. count-2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{\left(c0 \cdot d\right) \cdot d}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{\left(c0 \cdot d\right) \cdot d}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{\left(c0 \cdot d\right) \cdot d}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{\left(c0 \cdot d\right) \cdot d}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. times-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      12. distribute-rgt-out N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)}\right), \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   9. Applied egg-rr 63.5%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)}}{2 \cdot w} \]
   10. Step-by-step derivation

       1. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       2. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right) \cdot c0 + \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right) \cdot c0\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

       3. distribute-lft-out N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right) \cdot \left(c0 + c0\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot d}{D}\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       6. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \frac{1}{\frac{D}{c0 \cdot d}}\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       7. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{d}{D \cdot \left(w \cdot h\right)}}{\frac{D}{c0 \cdot d}}\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)}\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       9. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{d}{D}}{w \cdot h}\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{D}\right), \left(w \cdot h\right)\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \left(w \cdot h\right)\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(D, \left(c0 \cdot d\right)\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(c0, d\right)\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       15. count-2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(c0, d\right)\right)\right), \left(2 \cdot c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       16. *-lowering-*.f64 63.6%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(c0, d\right)\right)\right), \mathsf{*.f64}\left(2, c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   11. Applied egg-rr 63.6%

       \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{d}{D}}{w \cdot h}}{\frac{D}{c0 \cdot d}} \cdot \left(2 \cdot c0\right)}}{2 \cdot w} \]
   12. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \frac{\frac{\frac{d}{D}}{w \cdot h}}{\frac{D}{c0 \cdot d}} \cdot \color{blue}{\frac{2 \cdot c0}{2 \cdot w}} \]

       2. associate-/r/ N/A

          \[\leadsto \left(\frac{\frac{\frac{d}{D}}{w \cdot h}}{D} \cdot \left(c0 \cdot d\right)\right) \cdot \frac{\color{blue}{2 \cdot c0}}{2 \cdot w} \]

       3. associate-*l* N/A

          \[\leadsto \frac{\frac{\frac{d}{D}}{w \cdot h}}{D} \cdot \color{blue}{\left(\left(c0 \cdot d\right) \cdot \frac{2 \cdot c0}{2 \cdot w}\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{d}{D}}{w \cdot h}}{D}\right), \color{blue}{\left(\left(c0 \cdot d\right) \cdot \frac{2 \cdot c0}{2 \cdot w}\right)}\right) \]

       5. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{d}{D}}{D \cdot \left(w \cdot h\right)}\right), \left(\color{blue}{\left(c0 \cdot d\right)} \cdot \frac{2 \cdot c0}{2 \cdot w}\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{d}{D}}{\left(w \cdot h\right) \cdot D}\right), \left(\left(c0 \cdot \color{blue}{d}\right) \cdot \frac{2 \cdot c0}{2 \cdot w}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{D}\right), \left(\left(w \cdot h\right) \cdot D\right)\right), \left(\color{blue}{\left(c0 \cdot d\right)} \cdot \frac{2 \cdot c0}{2 \cdot w}\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \left(\left(w \cdot h\right) \cdot D\right)\right), \left(\left(\color{blue}{c0} \cdot d\right) \cdot \frac{2 \cdot c0}{2 \cdot w}\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \left(D \cdot \left(w \cdot h\right)\right)\right), \left(\left(c0 \cdot \color{blue}{d}\right) \cdot \frac{2 \cdot c0}{2 \cdot w}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(D, \left(w \cdot h\right)\right)\right), \left(\left(c0 \cdot \color{blue}{d}\right) \cdot \frac{2 \cdot c0}{2 \cdot w}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, h\right)\right)\right), \left(\left(c0 \cdot d\right) \cdot \frac{2 \cdot c0}{2 \cdot w}\right)\right) \]

       12. times-frac N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, h\right)\right)\right), \left(\left(c0 \cdot d\right) \cdot \left(\frac{2}{2} \cdot \color{blue}{\frac{c0}{w}}\right)\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, h\right)\right)\right), \left(\left(c0 \cdot d\right) \cdot \left(1 \cdot \frac{\color{blue}{c0}}{w}\right)\right)\right) \]

       14. *-lft-identity N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, h\right)\right)\right), \left(\left(c0 \cdot d\right) \cdot \frac{c0}{\color{blue}{w}}\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, h\right)\right)\right), \mathsf{*.f64}\left(\left(c0 \cdot d\right), \color{blue}{\left(\frac{c0}{w}\right)}\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, h\right)\right)\right), \mathsf{*.f64}\left(\left(d \cdot c0\right), \left(\frac{\color{blue}{c0}}{w}\right)\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, h\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, c0\right), \left(\frac{\color{blue}{c0}}{w}\right)\right)\right) \]

       18. /-lowering-/.f64 63.3%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, h\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{/.f64}\left(c0, \color{blue}{w}\right)\right)\right) \]

   13. Applied egg-rr 63.3%

       \[\leadsto \color{blue}{\frac{\frac{d}{D}}{D \cdot \left(w \cdot h\right)} \cdot \left(\left(d \cdot c0\right) \cdot \frac{c0}{w}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 3.5 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{d}{D}}{w \cdot h} \cdot \left(\frac{d}{\frac{D}{c0}} \cdot \frac{c0}{w}\right)\\ \mathbf{elif}\;D \leq 0.00024:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{D}}{\left(w \cdot h\right) \cdot D} \cdot \left(\left(c0 \cdot d\right) \cdot \frac{c0}{w}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 44.0% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 8 \cdot 10^{-146}:\\ \;\;\;\;\frac{d}{D} \cdot \frac{\frac{\frac{\frac{c0 \cdot \left(c0 \cdot d\right)}{w}}{h}}{w}}{D}\\ \mathbf{elif}\;D \leq 0.00023:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{D}}{\left(w \cdot h\right) \cdot D} \cdot \left(\left(c0 \cdot d\right) \cdot \frac{c0}{w}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= D 8e-146)
   (* (/ d D) (/ (/ (/ (/ (* c0 (* c0 d)) w) h) w) D))
   (if (<= D 0.00023)
     0.0
     (* (/ (/ d D) (* (* w h) D)) (* (* c0 d) (/ c0 w))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (D <= 8e-146) {
		tmp = (d / D) * (((((c0 * (c0 * d)) / w) / h) / w) / D);
	} else if (D <= 0.00023) {
		tmp = 0.0;
	} else {
		tmp = ((d / D) / ((w * h) * D)) * ((c0 * d) * (c0 / w));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (d <= 8d-146) then
        tmp = (d_1 / d) * (((((c0 * (c0 * d_1)) / w) / h) / w) / d)
    else if (d <= 0.00023d0) then
        tmp = 0.0d0
    else
        tmp = ((d_1 / d) / ((w * h) * d)) * ((c0 * d_1) * (c0 / w))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (D <= 8e-146) {
		tmp = (d / D) * (((((c0 * (c0 * d)) / w) / h) / w) / D);
	} else if (D <= 0.00023) {
		tmp = 0.0;
	} else {
		tmp = ((d / D) / ((w * h) * D)) * ((c0 * d) * (c0 / w));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if D <= 8e-146:
		tmp = (d / D) * (((((c0 * (c0 * d)) / w) / h) / w) / D)
	elif D <= 0.00023:
		tmp = 0.0
	else:
		tmp = ((d / D) / ((w * h) * D)) * ((c0 * d) * (c0 / w))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (D <= 8e-146)
		tmp = Float64(Float64(d / D) * Float64(Float64(Float64(Float64(Float64(c0 * Float64(c0 * d)) / w) / h) / w) / D));
	elseif (D <= 0.00023)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(d / D) / Float64(Float64(w * h) * D)) * Float64(Float64(c0 * d) * Float64(c0 / w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (D <= 8e-146)
		tmp = (d / D) * (((((c0 * (c0 * d)) / w) / h) / w) / D);
	elseif (D <= 0.00023)
		tmp = 0.0;
	else
		tmp = ((d / D) / ((w * h) * D)) * ((c0 * d) * (c0 / w));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[D, 8e-146], N[(N[(d / D), $MachinePrecision] * N[(N[(N[(N[(N[(c0 * N[(c0 * d), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision] / h), $MachinePrecision] / w), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision], If[LessEqual[D, 0.00023], 0.0, N[(N[(N[(d / D), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * N[(N[(c0 * d), $MachinePrecision] * N[(c0 / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if D < 8.00000000000000021e-146

   1. Initial program 20.8%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({c0}^{2} \cdot {d}^{2}\right), \color{blue}{\left({D}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left({d}^{2} \cdot {c0}^{2}\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({d}^{2}\right), \left({c0}^{2}\right)\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(d \cdot d\right), \left({c0}^{2}\right)\right), \left({\color{blue}{D}}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \left({c0}^{2}\right)\right), \left({\color{blue}{D}}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \left(c0 \cdot c0\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(h \cdot {w}^{2}\right)}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{h} \cdot {w}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{h} \cdot {w}^{2}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \color{blue}{\left({w}^{2}\right)}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(w \cdot \color{blue}{w}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 20.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w, \color{blue}{w}\right)\right)\right)\right) \]

   5. Simplified 20.7%

      \[\leadsto \color{blue}{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{\left(h \cdot \left(w \cdot w\right)\right) \cdot \color{blue}{\left(D \cdot D\right)}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{h \cdot \left(w \cdot w\right)}}{\color{blue}{D \cdot D}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{h \cdot \left(w \cdot w\right)}\right), \color{blue}{\left(D \cdot D\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)\right), \left(h \cdot \left(w \cdot w\right)\right)\right), \left(\color{blue}{D} \cdot D\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)\right), \left(h \cdot \left(w \cdot w\right)\right)\right), \left(D \cdot D\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \left(d \cdot \left(c0 \cdot c0\right)\right)\right), \left(h \cdot \left(w \cdot w\right)\right)\right), \left(D \cdot D\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \left(c0 \cdot c0\right)\right)\right), \left(h \cdot \left(w \cdot w\right)\right)\right), \left(D \cdot D\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(c0, c0\right)\right)\right), \left(h \cdot \left(w \cdot w\right)\right)\right), \left(D \cdot D\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(c0, c0\right)\right)\right), \left(\left(w \cdot w\right) \cdot h\right)\right), \left(D \cdot D\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(c0, c0\right)\right)\right), \left(w \cdot \left(w \cdot h\right)\right)\right), \left(D \cdot D\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(c0, c0\right)\right)\right), \mathsf{*.f64}\left(w, \left(w \cdot h\right)\right)\right), \left(D \cdot D\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(c0, c0\right)\right)\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, h\right)\right)\right), \left(D \cdot D\right)\right) \]

      13. *-lowering-*.f64 31.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(c0, c0\right)\right)\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, h\right)\right)\right), \mathsf{*.f64}\left(D, \color{blue}{D}\right)\right) \]

   7. Applied egg-rr 31.8%

      \[\leadsto \color{blue}{\frac{\frac{d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)}{w \cdot \left(w \cdot h\right)}}{D \cdot D}} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{d \cdot \frac{d \cdot \left(c0 \cdot c0\right)}{w \cdot \left(w \cdot h\right)}}{\color{blue}{D} \cdot D} \]

      2. times-frac N/A

         \[\leadsto \frac{d}{D} \cdot \color{blue}{\frac{\frac{d \cdot \left(c0 \cdot c0\right)}{w \cdot \left(w \cdot h\right)}}{D}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{d}{D}\right), \color{blue}{\left(\frac{\frac{d \cdot \left(c0 \cdot c0\right)}{w \cdot \left(w \cdot h\right)}}{D}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, D\right), \left(\frac{\color{blue}{\frac{d \cdot \left(c0 \cdot c0\right)}{w \cdot \left(w \cdot h\right)}}}{D}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{/.f64}\left(\left(\frac{d \cdot \left(c0 \cdot c0\right)}{w \cdot \left(w \cdot h\right)}\right), \color{blue}{D}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{/.f64}\left(\left(\frac{d \cdot \left(c0 \cdot c0\right)}{\left(w \cdot h\right) \cdot w}\right), D\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{/.f64}\left(\left(\frac{\frac{d \cdot \left(c0 \cdot c0\right)}{w \cdot h}}{w}\right), D\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{d \cdot \left(c0 \cdot c0\right)}{w \cdot h}\right), w\right), D\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{d \cdot \left(c0 \cdot c0\right)}{w}}{h}\right), w\right), D\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{d \cdot \left(c0 \cdot c0\right)}{w}\right), h\right), w\right), D\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(d \cdot \left(c0 \cdot c0\right)\right), w\right), h\right), w\right), D\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(c0 \cdot c0\right) \cdot d\right), w\right), h\right), w\right), D\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(c0 \cdot \left(c0 \cdot d\right)\right), w\right), h\right), w\right), D\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot d\right)\right), w\right), h\right), w\right), D\right)\right) \]

      15. *-lowering-*.f64 45.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, d\right)\right), w\right), h\right), w\right), D\right)\right) \]

   9. Applied egg-rr 45.0%

      \[\leadsto \color{blue}{\frac{d}{D} \cdot \frac{\frac{\frac{\frac{c0 \cdot \left(c0 \cdot d\right)}{w}}{h}}{w}}{D}} \]

   if 8.00000000000000021e-146 < D < 2.3000000000000001e-4

   1. Initial program 34.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Simplified 31.9%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{d \cdot \frac{\frac{c0 \cdot d}{w}}{h}}{D \cdot D} + \sqrt{\frac{\frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot \left(d \cdot \frac{\frac{c0 \cdot d}{w}}{h}\right)\right)}{D \cdot D}}{w \cdot h}}{D \cdot D} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot c0\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. mul0-lft N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(c0 \cdot \left(-1 + 1\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(\mathsf{neg}\left(\left(-1 + 1\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(\mathsf{neg}\left(0\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      13. metadata-eval 41.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 41.6%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(c0 \cdot 0\right)}}{2 \cdot w} \]
   8. Step-by-step derivation

      1. mul0-rgt N/A

         \[\leadsto \frac{c0 \cdot 0}{2 \cdot w} \]

      2. mul0-rgt N/A

         \[\leadsto \frac{0}{\color{blue}{2} \cdot w} \]

      3. div0 41.6%

         \[\leadsto 0 \]

   9. Applied egg-rr 41.6%

      \[\leadsto \color{blue}{0} \]

   if 2.3000000000000001e-4 < D

   1. Initial program 25.0%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Simplified 22.5%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{d \cdot \frac{\frac{c0 \cdot d}{w}}{h}}{D \cdot D} + \sqrt{\frac{\frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot \left(d \cdot \frac{\frac{c0 \cdot d}{w}}{h}\right)\right)}{D \cdot D}}{w \cdot h}}{D \cdot D} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\left(2 \cdot \left(c0 \cdot {d}^{2}\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot {d}^{2}\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left({d}^{2}\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(d \cdot d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot w\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(w \cdot h\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. *-lowering-*.f64 36.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(w, h\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 36.4%

      \[\leadsto \frac{c0 \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}}{2 \cdot w} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. count-2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{\left(c0 \cdot d\right) \cdot d}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{\left(c0 \cdot d\right) \cdot d}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{\left(c0 \cdot d\right) \cdot d}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{\left(c0 \cdot d\right) \cdot d}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. times-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      12. distribute-rgt-out N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)}\right), \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   9. Applied egg-rr 63.5%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \left(\frac{c0 \cdot d}{D} + \frac{c0 \cdot d}{D}\right)\right)}}{2 \cdot w} \]
   10. Step-by-step derivation

       1. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)} + \frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       2. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right) \cdot c0 + \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right) \cdot c0\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

       3. distribute-lft-out N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right) \cdot \left(c0 + c0\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{c0 \cdot d}{D} \cdot \frac{d}{D \cdot \left(w \cdot h\right)}\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, w\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot d}{D}\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       6. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)} \cdot \frac{1}{\frac{D}{c0 \cdot d}}\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       7. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{d}{D \cdot \left(w \cdot h\right)}}{\frac{D}{c0 \cdot d}}\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{D \cdot \left(w \cdot h\right)}\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       9. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{d}{D}}{w \cdot h}\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{D}\right), \left(w \cdot h\right)\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \left(w \cdot h\right)\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \left(\frac{D}{c0 \cdot d}\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(D, \left(c0 \cdot d\right)\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(c0, d\right)\right)\right), \left(c0 + c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       15. count-2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(c0, d\right)\right)\right), \left(2 \cdot c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

       16. *-lowering-*.f64 63.6%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(c0, d\right)\right)\right), \mathsf{*.f64}\left(2, c0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   11. Applied egg-rr 63.6%

       \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{d}{D}}{w \cdot h}}{\frac{D}{c0 \cdot d}} \cdot \left(2 \cdot c0\right)}}{2 \cdot w} \]
   12. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \frac{\frac{\frac{d}{D}}{w \cdot h}}{\frac{D}{c0 \cdot d}} \cdot \color{blue}{\frac{2 \cdot c0}{2 \cdot w}} \]

       2. associate-/r/ N/A

          \[\leadsto \left(\frac{\frac{\frac{d}{D}}{w \cdot h}}{D} \cdot \left(c0 \cdot d\right)\right) \cdot \frac{\color{blue}{2 \cdot c0}}{2 \cdot w} \]

       3. associate-*l* N/A

          \[\leadsto \frac{\frac{\frac{d}{D}}{w \cdot h}}{D} \cdot \color{blue}{\left(\left(c0 \cdot d\right) \cdot \frac{2 \cdot c0}{2 \cdot w}\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{d}{D}}{w \cdot h}}{D}\right), \color{blue}{\left(\left(c0 \cdot d\right) \cdot \frac{2 \cdot c0}{2 \cdot w}\right)}\right) \]

       5. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{d}{D}}{D \cdot \left(w \cdot h\right)}\right), \left(\color{blue}{\left(c0 \cdot d\right)} \cdot \frac{2 \cdot c0}{2 \cdot w}\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{d}{D}}{\left(w \cdot h\right) \cdot D}\right), \left(\left(c0 \cdot \color{blue}{d}\right) \cdot \frac{2 \cdot c0}{2 \cdot w}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{D}\right), \left(\left(w \cdot h\right) \cdot D\right)\right), \left(\color{blue}{\left(c0 \cdot d\right)} \cdot \frac{2 \cdot c0}{2 \cdot w}\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \left(\left(w \cdot h\right) \cdot D\right)\right), \left(\left(\color{blue}{c0} \cdot d\right) \cdot \frac{2 \cdot c0}{2 \cdot w}\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \left(D \cdot \left(w \cdot h\right)\right)\right), \left(\left(c0 \cdot \color{blue}{d}\right) \cdot \frac{2 \cdot c0}{2 \cdot w}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(D, \left(w \cdot h\right)\right)\right), \left(\left(c0 \cdot \color{blue}{d}\right) \cdot \frac{2 \cdot c0}{2 \cdot w}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, h\right)\right)\right), \left(\left(c0 \cdot d\right) \cdot \frac{2 \cdot c0}{2 \cdot w}\right)\right) \]

       12. times-frac N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, h\right)\right)\right), \left(\left(c0 \cdot d\right) \cdot \left(\frac{2}{2} \cdot \color{blue}{\frac{c0}{w}}\right)\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, h\right)\right)\right), \left(\left(c0 \cdot d\right) \cdot \left(1 \cdot \frac{\color{blue}{c0}}{w}\right)\right)\right) \]

       14. *-lft-identity N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, h\right)\right)\right), \left(\left(c0 \cdot d\right) \cdot \frac{c0}{\color{blue}{w}}\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, h\right)\right)\right), \mathsf{*.f64}\left(\left(c0 \cdot d\right), \color{blue}{\left(\frac{c0}{w}\right)}\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, h\right)\right)\right), \mathsf{*.f64}\left(\left(d \cdot c0\right), \left(\frac{\color{blue}{c0}}{w}\right)\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, h\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, c0\right), \left(\frac{\color{blue}{c0}}{w}\right)\right)\right) \]

       18. /-lowering-/.f64 63.3%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, h\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{/.f64}\left(c0, \color{blue}{w}\right)\right)\right) \]

   13. Applied egg-rr 63.3%

       \[\leadsto \color{blue}{\frac{\frac{d}{D}}{D \cdot \left(w \cdot h\right)} \cdot \left(\left(d \cdot c0\right) \cdot \frac{c0}{w}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 8 \cdot 10^{-146}:\\ \;\;\;\;\frac{d}{D} \cdot \frac{\frac{\frac{\frac{c0 \cdot \left(c0 \cdot d\right)}{w}}{h}}{w}}{D}\\ \mathbf{elif}\;D \leq 0.00023:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{D}}{\left(w \cdot h\right) \cdot D} \cdot \left(\left(c0 \cdot d\right) \cdot \frac{c0}{w}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 40.9% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 1.5 \cdot 10^{-224}:\\ \;\;\;\;d \cdot \frac{d}{\frac{D \cdot \left(D \cdot \left(w \cdot \left(w \cdot h\right)\right)\right)}{c0 \cdot c0}}\\ \mathbf{elif}\;M \leq 8 \cdot 10^{-105}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(c0 \cdot \left(c0 \cdot d\right)\right) \cdot \frac{d}{w \cdot \left(D \cdot \left(h \cdot \left(w \cdot D\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= M 1.5e-224)
   (* d (/ d (/ (* D (* D (* w (* w h)))) (* c0 c0))))
   (if (<= M 8e-105) 0.0 (* (* c0 (* c0 d)) (/ d (* w (* D (* h (* w D)))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 1.5e-224) {
		tmp = d * (d / ((D * (D * (w * (w * h)))) / (c0 * c0)));
	} else if (M <= 8e-105) {
		tmp = 0.0;
	} else {
		tmp = (c0 * (c0 * d)) * (d / (w * (D * (h * (w * D)))));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 1.5d-224) then
        tmp = d_1 * (d_1 / ((d * (d * (w * (w * h)))) / (c0 * c0)))
    else if (m <= 8d-105) then
        tmp = 0.0d0
    else
        tmp = (c0 * (c0 * d_1)) * (d_1 / (w * (d * (h * (w * d)))))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 1.5e-224) {
		tmp = d * (d / ((D * (D * (w * (w * h)))) / (c0 * c0)));
	} else if (M <= 8e-105) {
		tmp = 0.0;
	} else {
		tmp = (c0 * (c0 * d)) * (d / (w * (D * (h * (w * D)))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if M <= 1.5e-224:
		tmp = d * (d / ((D * (D * (w * (w * h)))) / (c0 * c0)))
	elif M <= 8e-105:
		tmp = 0.0
	else:
		tmp = (c0 * (c0 * d)) * (d / (w * (D * (h * (w * D)))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (M <= 1.5e-224)
		tmp = Float64(d * Float64(d / Float64(Float64(D * Float64(D * Float64(w * Float64(w * h)))) / Float64(c0 * c0))));
	elseif (M <= 8e-105)
		tmp = 0.0;
	else
		tmp = Float64(Float64(c0 * Float64(c0 * d)) * Float64(d / Float64(w * Float64(D * Float64(h * Float64(w * D))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (M <= 1.5e-224)
		tmp = d * (d / ((D * (D * (w * (w * h)))) / (c0 * c0)));
	elseif (M <= 8e-105)
		tmp = 0.0;
	else
		tmp = (c0 * (c0 * d)) * (d / (w * (D * (h * (w * D)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[M, 1.5e-224], N[(d * N[(d / N[(N[(D * N[(D * N[(w * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c0 * c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 8e-105], 0.0, N[(N[(c0 * N[(c0 * d), $MachinePrecision]), $MachinePrecision] * N[(d / N[(w * N[(D * N[(h * N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if M < 1.49999999999999991e-224

   1. Initial program 23.8%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({c0}^{2} \cdot {d}^{2}\right), \color{blue}{\left({D}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left({d}^{2} \cdot {c0}^{2}\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({d}^{2}\right), \left({c0}^{2}\right)\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(d \cdot d\right), \left({c0}^{2}\right)\right), \left({\color{blue}{D}}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \left({c0}^{2}\right)\right), \left({\color{blue}{D}}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \left(c0 \cdot c0\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(h \cdot {w}^{2}\right)}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{h} \cdot {w}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{h} \cdot {w}^{2}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \color{blue}{\left({w}^{2}\right)}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(w \cdot \color{blue}{w}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 24.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w, \color{blue}{w}\right)\right)\right)\right) \]

   5. Simplified 24.1%

      \[\leadsto \color{blue}{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \left(d \cdot d\right) \cdot \color{blue}{\frac{c0 \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}} \]

      2. associate-*l* N/A

         \[\leadsto d \cdot \color{blue}{\left(d \cdot \frac{c0 \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \color{blue}{\left(d \cdot \frac{c0 \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)}\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(d, \left(d \cdot \frac{1}{\color{blue}{\frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}{c0 \cdot c0}}}\right)\right) \]

      5. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(d, \left(\frac{d}{\color{blue}{\frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}{c0 \cdot c0}}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{/.f64}\left(d, \color{blue}{\left(\frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}{c0 \cdot c0}\right)}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{/.f64}\left(d, \mathsf{/.f64}\left(\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)\right), \color{blue}{\left(c0 \cdot c0\right)}\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{/.f64}\left(d, \mathsf{/.f64}\left(\left(D \cdot \left(D \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)\right), \left(\color{blue}{c0} \cdot c0\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{/.f64}\left(d, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(D \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)\right), \left(\color{blue}{c0} \cdot c0\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(d, \mathsf{/.f64}\left(d, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(h \cdot \left(w \cdot w\right)\right)\right)\right), \left(c0 \cdot c0\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(d, \mathsf{/.f64}\left(d, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(\left(w \cdot w\right) \cdot h\right)\right)\right), \left(c0 \cdot c0\right)\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(d, \mathsf{/.f64}\left(d, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(w \cdot \left(w \cdot h\right)\right)\right)\right), \left(c0 \cdot c0\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(d, \mathsf{/.f64}\left(d, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, \left(w \cdot h\right)\right)\right)\right), \left(c0 \cdot c0\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(d, \mathsf{/.f64}\left(d, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, h\right)\right)\right)\right), \left(c0 \cdot c0\right)\right)\right)\right) \]

      15. *-lowering-*.f64 44.7%

          \[\leadsto \mathsf{*.f64}\left(d, \mathsf{/.f64}\left(d, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, h\right)\right)\right)\right), \mathsf{*.f64}\left(c0, \color{blue}{c0}\right)\right)\right)\right) \]

   7. Applied egg-rr 44.7%

      \[\leadsto \color{blue}{d \cdot \frac{d}{\frac{D \cdot \left(D \cdot \left(w \cdot \left(w \cdot h\right)\right)\right)}{c0 \cdot c0}}} \]

   if 1.49999999999999991e-224 < M < 7.99999999999999972e-105

   1. Initial program 27.2%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Simplified 24.8%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{d \cdot \frac{\frac{c0 \cdot d}{w}}{h}}{D \cdot D} + \sqrt{\frac{\frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot \left(d \cdot \frac{\frac{c0 \cdot d}{w}}{h}\right)\right)}{D \cdot D}}{w \cdot h}}{D \cdot D} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot c0\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. mul0-lft N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(c0 \cdot \left(-1 + 1\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(\mathsf{neg}\left(\left(-1 + 1\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(\mathsf{neg}\left(0\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      13. metadata-eval 47.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 47.4%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(c0 \cdot 0\right)}}{2 \cdot w} \]
   8. Step-by-step derivation

      1. mul0-rgt N/A

         \[\leadsto \frac{c0 \cdot 0}{2 \cdot w} \]

      2. mul0-rgt N/A

         \[\leadsto \frac{0}{\color{blue}{2} \cdot w} \]

      3. div0 47.4%

         \[\leadsto 0 \]

   9. Applied egg-rr 47.4%

      \[\leadsto \color{blue}{0} \]

   if 7.99999999999999972e-105 < M

   1. Initial program 20.1%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({c0}^{2} \cdot {d}^{2}\right), \color{blue}{\left({D}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left({d}^{2} \cdot {c0}^{2}\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({d}^{2}\right), \left({c0}^{2}\right)\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(d \cdot d\right), \left({c0}^{2}\right)\right), \left({\color{blue}{D}}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \left({c0}^{2}\right)\right), \left({\color{blue}{D}}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \left(c0 \cdot c0\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(h \cdot {w}^{2}\right)}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{h} \cdot {w}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{h} \cdot {w}^{2}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \color{blue}{\left({w}^{2}\right)}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(w \cdot \color{blue}{w}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 23.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w, \color{blue}{w}\right)\right)\right)\right) \]

   5. Simplified 23.8%

      \[\leadsto \color{blue}{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{\left(h \cdot \left(w \cdot w\right)\right) \cdot \color{blue}{\left(D \cdot D\right)}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{h \cdot \left(w \cdot w\right)}}{\color{blue}{D \cdot D}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{h \cdot \left(w \cdot w\right)}\right), \color{blue}{\left(D \cdot D\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)\right), \left(h \cdot \left(w \cdot w\right)\right)\right), \left(\color{blue}{D} \cdot D\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)\right), \left(h \cdot \left(w \cdot w\right)\right)\right), \left(D \cdot D\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \left(d \cdot \left(c0 \cdot c0\right)\right)\right), \left(h \cdot \left(w \cdot w\right)\right)\right), \left(D \cdot D\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \left(c0 \cdot c0\right)\right)\right), \left(h \cdot \left(w \cdot w\right)\right)\right), \left(D \cdot D\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(c0, c0\right)\right)\right), \left(h \cdot \left(w \cdot w\right)\right)\right), \left(D \cdot D\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(c0, c0\right)\right)\right), \left(\left(w \cdot w\right) \cdot h\right)\right), \left(D \cdot D\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(c0, c0\right)\right)\right), \left(w \cdot \left(w \cdot h\right)\right)\right), \left(D \cdot D\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(c0, c0\right)\right)\right), \mathsf{*.f64}\left(w, \left(w \cdot h\right)\right)\right), \left(D \cdot D\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(c0, c0\right)\right)\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, h\right)\right)\right), \left(D \cdot D\right)\right) \]

      13. *-lowering-*.f64 33.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(c0, c0\right)\right)\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, h\right)\right)\right), \mathsf{*.f64}\left(D, \color{blue}{D}\right)\right) \]

   7. Applied egg-rr 33.3%

      \[\leadsto \color{blue}{\frac{\frac{d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)}{w \cdot \left(w \cdot h\right)}}{D \cdot D}} \]
   8. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)}{\color{blue}{\left(D \cdot D\right) \cdot \left(w \cdot \left(w \cdot h\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(d \cdot \left(c0 \cdot c0\right)\right) \cdot d}{\color{blue}{\left(D \cdot D\right)} \cdot \left(w \cdot \left(w \cdot h\right)\right)} \]

      3. associate-/l* N/A

         \[\leadsto \left(d \cdot \left(c0 \cdot c0\right)\right) \cdot \color{blue}{\frac{d}{\left(D \cdot D\right) \cdot \left(w \cdot \left(w \cdot h\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(d \cdot \left(c0 \cdot c0\right)\right), \color{blue}{\left(\frac{d}{\left(D \cdot D\right) \cdot \left(w \cdot \left(w \cdot h\right)\right)}\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(c0 \cdot c0\right) \cdot d\right), \left(\frac{\color{blue}{d}}{\left(D \cdot D\right) \cdot \left(w \cdot \left(w \cdot h\right)\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(c0 \cdot \left(c0 \cdot d\right)\right), \left(\frac{\color{blue}{d}}{\left(D \cdot D\right) \cdot \left(w \cdot \left(w \cdot h\right)\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot d\right)\right), \left(\frac{\color{blue}{d}}{\left(D \cdot D\right) \cdot \left(w \cdot \left(w \cdot h\right)\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, d\right)\right), \left(\frac{d}{\left(D \cdot D\right) \cdot \left(w \cdot \left(w \cdot h\right)\right)}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, d\right)\right), \mathsf{/.f64}\left(d, \color{blue}{\left(\left(D \cdot D\right) \cdot \left(w \cdot \left(w \cdot h\right)\right)\right)}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, d\right)\right), \mathsf{/.f64}\left(d, \left(\left(w \cdot \left(w \cdot h\right)\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, d\right)\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(\left(w \cdot \left(w \cdot h\right)\right), \color{blue}{\left(D \cdot D\right)}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, d\right)\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, \left(w \cdot h\right)\right), \left(\color{blue}{D} \cdot D\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, d\right)\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, h\right)\right), \left(D \cdot D\right)\right)\right)\right) \]

      14. *-lowering-*.f64 36.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, d\right)\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, h\right)\right), \mathsf{*.f64}\left(D, \color{blue}{D}\right)\right)\right)\right) \]

   9. Applied egg-rr 36.8%

      \[\leadsto \color{blue}{\left(c0 \cdot \left(c0 \cdot d\right)\right) \cdot \frac{d}{\left(w \cdot \left(w \cdot h\right)\right) \cdot \left(D \cdot D\right)}} \]

   10. Taylor expanded in d around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, d\right)\right), \color{blue}{\left(\frac{d}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}\right)}\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, d\right)\right), \mathsf{/.f64}\left(d, \color{blue}{\left({D}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, d\right)\right), \mathsf{/.f64}\left(d, \left({D}^{2} \cdot \left(h \cdot \left(w \cdot \color{blue}{w}\right)\right)\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, d\right)\right), \mathsf{/.f64}\left(d, \left({D}^{2} \cdot \left(\left(h \cdot w\right) \cdot \color{blue}{w}\right)\right)\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, d\right)\right), \mathsf{/.f64}\left(d, \left(\left({D}^{2} \cdot \left(h \cdot w\right)\right) \cdot \color{blue}{w}\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, d\right)\right), \mathsf{/.f64}\left(d, \left(w \cdot \color{blue}{\left({D}^{2} \cdot \left(h \cdot w\right)\right)}\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, d\right)\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(w, \color{blue}{\left({D}^{2} \cdot \left(h \cdot w\right)\right)}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, d\right)\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(w, \left(\left(D \cdot D\right) \cdot \left(\color{blue}{h} \cdot w\right)\right)\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, d\right)\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(w, \left(D \cdot \color{blue}{\left(D \cdot \left(h \cdot w\right)\right)}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, d\right)\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(D, \color{blue}{\left(D \cdot \left(h \cdot w\right)\right)}\right)\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, d\right)\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(D, \left(\left(h \cdot w\right) \cdot \color{blue}{D}\right)\right)\right)\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, d\right)\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(D, \left(h \cdot \color{blue}{\left(w \cdot D\right)}\right)\right)\right)\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, d\right)\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(D, \left(h \cdot \left(D \cdot \color{blue}{w}\right)\right)\right)\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, d\right)\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(h, \color{blue}{\left(D \cdot w\right)}\right)\right)\right)\right)\right) \]

       14. *-lowering-*.f64 43.1%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, d\right)\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, \color{blue}{w}\right)\right)\right)\right)\right)\right) \]

   12. Simplified 43.1%

       \[\leadsto \left(c0 \cdot \left(c0 \cdot d\right)\right) \cdot \color{blue}{\frac{d}{w \cdot \left(D \cdot \left(h \cdot \left(D \cdot w\right)\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.5 \cdot 10^{-224}:\\ \;\;\;\;d \cdot \frac{d}{\frac{D \cdot \left(D \cdot \left(w \cdot \left(w \cdot h\right)\right)\right)}{c0 \cdot c0}}\\ \mathbf{elif}\;M \leq 8 \cdot 10^{-105}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(c0 \cdot \left(c0 \cdot d\right)\right) \cdot \frac{d}{w \cdot \left(D \cdot \left(h \cdot \left(w \cdot D\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 38.8% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := d \cdot \frac{d}{\frac{D \cdot \left(D \cdot \left(w \cdot \left(w \cdot h\right)\right)\right)}{c0 \cdot c0}}\\ \mathbf{if}\;D \leq 3 \cdot 10^{-145}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;D \leq 0.00023:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* d (/ d (/ (* D (* D (* w (* w h)))) (* c0 c0))))))
   (if (<= D 3e-145) t_0 (if (<= D 0.00023) 0.0 t_0))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = d * (d / ((D * (D * (w * (w * h)))) / (c0 * c0)));
	double tmp;
	if (D <= 3e-145) {
		tmp = t_0;
	} else if (D <= 0.00023) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d_1 * (d_1 / ((d * (d * (w * (w * h)))) / (c0 * c0)))
    if (d <= 3d-145) then
        tmp = t_0
    else if (d <= 0.00023d0) then
        tmp = 0.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = d * (d / ((D * (D * (w * (w * h)))) / (c0 * c0)));
	double tmp;
	if (D <= 3e-145) {
		tmp = t_0;
	} else if (D <= 0.00023) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = d * (d / ((D * (D * (w * (w * h)))) / (c0 * c0)))
	tmp = 0
	if D <= 3e-145:
		tmp = t_0
	elif D <= 0.00023:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(d * Float64(d / Float64(Float64(D * Float64(D * Float64(w * Float64(w * h)))) / Float64(c0 * c0))))
	tmp = 0.0
	if (D <= 3e-145)
		tmp = t_0;
	elseif (D <= 0.00023)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = d * (d / ((D * (D * (w * (w * h)))) / (c0 * c0)));
	tmp = 0.0;
	if (D <= 3e-145)
		tmp = t_0;
	elseif (D <= 0.00023)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(d * N[(d / N[(N[(D * N[(D * N[(w * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c0 * c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[D, 3e-145], t$95$0, If[LessEqual[D, 0.00023], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if D < 2.99999999999999992e-145 or 2.3000000000000001e-4 < D

   1. Initial program 21.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({c0}^{2} \cdot {d}^{2}\right), \color{blue}{\left({D}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left({d}^{2} \cdot {c0}^{2}\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({d}^{2}\right), \left({c0}^{2}\right)\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(d \cdot d\right), \left({c0}^{2}\right)\right), \left({\color{blue}{D}}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \left({c0}^{2}\right)\right), \left({\color{blue}{D}}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \left(c0 \cdot c0\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(h \cdot {w}^{2}\right)}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{h} \cdot {w}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{h} \cdot {w}^{2}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \color{blue}{\left({w}^{2}\right)}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(w \cdot \color{blue}{w}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 22.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w, \color{blue}{w}\right)\right)\right)\right) \]

   5. Simplified 22.8%

      \[\leadsto \color{blue}{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \left(d \cdot d\right) \cdot \color{blue}{\frac{c0 \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}} \]

      2. associate-*l* N/A

         \[\leadsto d \cdot \color{blue}{\left(d \cdot \frac{c0 \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \color{blue}{\left(d \cdot \frac{c0 \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)}\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(d, \left(d \cdot \frac{1}{\color{blue}{\frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}{c0 \cdot c0}}}\right)\right) \]

      5. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(d, \left(\frac{d}{\color{blue}{\frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}{c0 \cdot c0}}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{/.f64}\left(d, \color{blue}{\left(\frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}{c0 \cdot c0}\right)}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{/.f64}\left(d, \mathsf{/.f64}\left(\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)\right), \color{blue}{\left(c0 \cdot c0\right)}\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{/.f64}\left(d, \mathsf{/.f64}\left(\left(D \cdot \left(D \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)\right), \left(\color{blue}{c0} \cdot c0\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{/.f64}\left(d, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(D \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)\right), \left(\color{blue}{c0} \cdot c0\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(d, \mathsf{/.f64}\left(d, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(h \cdot \left(w \cdot w\right)\right)\right)\right), \left(c0 \cdot c0\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(d, \mathsf{/.f64}\left(d, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(\left(w \cdot w\right) \cdot h\right)\right)\right), \left(c0 \cdot c0\right)\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(d, \mathsf{/.f64}\left(d, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(w \cdot \left(w \cdot h\right)\right)\right)\right), \left(c0 \cdot c0\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(d, \mathsf{/.f64}\left(d, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, \left(w \cdot h\right)\right)\right)\right), \left(c0 \cdot c0\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(d, \mathsf{/.f64}\left(d, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, h\right)\right)\right)\right), \left(c0 \cdot c0\right)\right)\right)\right) \]

      15. *-lowering-*.f64 42.6%

          \[\leadsto \mathsf{*.f64}\left(d, \mathsf{/.f64}\left(d, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, h\right)\right)\right)\right), \mathsf{*.f64}\left(c0, \color{blue}{c0}\right)\right)\right)\right) \]

   7. Applied egg-rr 42.6%

      \[\leadsto \color{blue}{d \cdot \frac{d}{\frac{D \cdot \left(D \cdot \left(w \cdot \left(w \cdot h\right)\right)\right)}{c0 \cdot c0}}} \]

   if 2.99999999999999992e-145 < D < 2.3000000000000001e-4

   1. Initial program 34.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Simplified 31.9%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{d \cdot \frac{\frac{c0 \cdot d}{w}}{h}}{D \cdot D} + \sqrt{\frac{\frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot \left(d \cdot \frac{\frac{c0 \cdot d}{w}}{h}\right)\right)}{D \cdot D}}{w \cdot h}}{D \cdot D} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot c0\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. mul0-lft N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(c0 \cdot \left(-1 + 1\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(\mathsf{neg}\left(\left(-1 + 1\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(\mathsf{neg}\left(0\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      13. metadata-eval 41.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 41.6%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(c0 \cdot 0\right)}}{2 \cdot w} \]
   8. Step-by-step derivation

      1. mul0-rgt N/A

         \[\leadsto \frac{c0 \cdot 0}{2 \cdot w} \]

      2. mul0-rgt N/A

         \[\leadsto \frac{0}{\color{blue}{2} \cdot w} \]

      3. div0 41.6%

         \[\leadsto 0 \]

   9. Applied egg-rr 41.6%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 33.7% accurate, 151.0× speedup

Math

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

FPCore

(FPCore (c0 w h D d M) :precision binary64 0.0)

C

double code(double c0, double w, double h, double D, double d, double M) {
	return 0.0;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = 0.0d0
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return 0.0;
}

Python

def code(c0, w, h, D, d, M):
	return 0.0

Julia

function code(c0, w, h, D, d, M)
	return 0.0
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = 0.0;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := 0.0

Derivation

1. Initial program 23.3%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

3. Simplified 22.3%

   \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{d \cdot \frac{\frac{c0 \cdot d}{w}}{h}}{D \cdot D} + \sqrt{\frac{\frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot \left(d \cdot \frac{\frac{c0 \cdot d}{w}}{h}\right)\right)}{D \cdot D}}{w \cdot h}}{D \cdot D} - M \cdot M}\right)}{2 \cdot w}} \]
4. Add Preprocessing

5. Taylor expanded in c0 around -inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot c0\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   3. distribute-lft1-in N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   5. mul0-lft N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. distribute-lft-neg-in N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(c0 \cdot \left(-1 + 1\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   8. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(\mathsf{neg}\left(\left(-1 + 1\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(\mathsf{neg}\left(0\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   13. metadata-eval 34.0%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

7. Simplified 34.0%

   \[\leadsto \frac{c0 \cdot \color{blue}{\left(c0 \cdot 0\right)}}{2 \cdot w} \]
8. Step-by-step derivation

   1. mul0-rgt N/A

      \[\leadsto \frac{c0 \cdot 0}{2 \cdot w} \]

   2. mul0-rgt N/A

      \[\leadsto \frac{0}{\color{blue}{2} \cdot w} \]

   3. div0 34.0%

      \[\leadsto 0 \]

9. Applied egg-rr 34.0%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024164 
(FPCore (c0 w h D d M)
  :name "Henrywood and Agarwal, Equation (13)"
  :precision binary64
  (* (/ c0 (* 2.0 w)) (+ (/ (* c0 (* d d)) (* (* w h) (* D D))) (sqrt (- (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M))))))