Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.3% → 66.3%

Time: 23.9s

Alternatives: 10

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: 24.3% 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: 66.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{\frac{\frac{\left(c0 \cdot d\right) \cdot -2}{\frac{h}{d}}}{w \cdot D}}{\frac{D}{\frac{c0}{w \cdot -2}}}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d} \cdot \frac{D}{d}\right)\\ \end{array} \end{array} \]

FPCore

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

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) * -2.0) / (h / d)) / (w * D)) / (D / (c0 / (w * -2.0)));
	} else {
		tmp = 0.25 * (((h * (D * (M * M))) / d) * (D / d));
	}
	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) * -2.0) / (h / d)) / (w * D)) / (D / (c0 / (w * -2.0)));
	} else {
		tmp = 0.25 * (((h * (D * (M * M))) / d) * (D / d));
	}
	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) * -2.0) / (h / d)) / (w * D)) / (D / (c0 / (w * -2.0)))
	else:
		tmp = 0.25 * (((h * (D * (M * M))) / d) * (D / d))
	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(Float64(Float64(Float64(c0 * d) * -2.0) / Float64(h / d)) / Float64(w * D)) / Float64(D / Float64(c0 / Float64(w * -2.0))));
	else
		tmp = Float64(0.25 * Float64(Float64(Float64(h * Float64(D * Float64(M * M))) / d) * Float64(D / d)));
	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) * -2.0) / (h / d)) / (w * D)) / (D / (c0 / (w * -2.0)));
	else
		tmp = 0.25 * (((h * (D * (M * M))) / d) * (D / d));
	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[(N[(N[(N[(c0 * d), $MachinePrecision] * -2.0), $MachinePrecision] / N[(h / d), $MachinePrecision]), $MachinePrecision] / N[(w * D), $MachinePrecision]), $MachinePrecision] / N[(D / N[(c0 / N[(w * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(N[(h * N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

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 77.7%

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

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

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

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

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

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

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

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

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

   3. Simplified 77.5%

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

   5. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

      6. associate-/r* N/A

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

      7. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      12. unpow2 N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 79.6%

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

   7. Simplified 79.6%

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

      1. associate-*r/ N/A

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

      2. associate-/r* N/A

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

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

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

   9. Applied egg-rr 81.8%

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

       1. associate-*r* N/A

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

       2. associate-/l* N/A

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

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

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

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

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

       5. /-lowering-/.f64 83.9%

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

   11. Applied egg-rr 83.9%

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

       1. associate-/l/ N/A

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

       2. div-inv N/A

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

       3. clear-num N/A

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

       4. associate-*r/ N/A

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

       5. *-commutative N/A

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

       6. associate-/r* N/A

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

       7. clear-num N/A

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

       8. frac-times N/A

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

       9. frac-2neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\frac{\frac{2 \cdot \left(\left(c0 \cdot d\right) \cdot \frac{d}{h}\right)}{w}}{D} \cdot 1\right)}{\color{blue}{\mathsf{neg}\left(D \cdot \frac{2 \cdot w}{c0}\right)}} \]

       10. *-rgt-identity N/A

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

       11. clear-num N/A

           \[\leadsto \frac{\mathsf{neg}\left(\frac{\frac{2 \cdot \left(\left(c0 \cdot d\right) \cdot \frac{d}{h}\right)}{w}}{D}\right)}{\mathsf{neg}\left(D \cdot \frac{1}{\frac{c0}{2 \cdot w}}\right)} \]

       12. div-inv N/A

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

       13. distribute-frac-neg2 N/A

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

   13. Applied egg-rr 86.9%

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

   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%

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

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

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

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

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

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

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

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

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

   3. Simplified 2.0%

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

   5. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\left({c0}^{2}\right), \color{blue}{\left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(c0 \cdot c0\right), \left(\color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right), \color{blue}{\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)}\right)\right) \]

   7. Simplified 20.8%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \left(\frac{0}{w} + \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}\right)} \]

   8. Taylor expanded in c0 around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right), \color{blue}{\left({d}^{2}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({D}^{2}\right), \left({M}^{2} \cdot h\right)\right), \left({\color{blue}{d}}^{2}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(D \cdot D\right), \left({M}^{2} \cdot h\right)\right), \left({d}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left({M}^{2} \cdot h\right)\right), \left({d}^{2}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot {M}^{2}\right)\right), \left({d}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right)\right) \]

      11. *-lowering-*.f64 51.8%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right)\right) \]

   10. Simplified 51.8%

       \[\leadsto \color{blue}{0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \]
   11. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left(\frac{D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\color{blue}{d} \cdot d}\right)\right) \]

       2. times-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left(\frac{D}{d} \cdot \color{blue}{\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d}}\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left(\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d} \cdot \color{blue}{\frac{D}{d}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d}\right), \color{blue}{\left(\frac{D}{d}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right), d\right), \left(\frac{\color{blue}{D}}{d}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(D \cdot \left(\left(M \cdot M\right) \cdot h\right)\right), d\right), \left(\frac{D}{d}\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(D \cdot \left(M \cdot M\right)\right) \cdot h\right), d\right), \left(\frac{D}{d}\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(h \cdot \left(D \cdot \left(M \cdot M\right)\right)\right), d\right), \left(\frac{D}{d}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(D \cdot \left(M \cdot M\right)\right)\right), d\right), \left(\frac{D}{d}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, \left(M \cdot M\right)\right)\right), d\right), \left(\frac{D}{d}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, M\right)\right)\right), d\right), \left(\frac{D}{d}\right)\right)\right) \]

       12. /-lowering-/.f64 66.7%

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, M\right)\right)\right), d\right), \mathsf{/.f64}\left(D, \color{blue}{d}\right)\right)\right) \]

   12. Applied egg-rr 66.7%

       \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d} \cdot \frac{D}{d}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 45.4% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := D \cdot \left(D \cdot 0.25\right)\\ \mathbf{if}\;d \leq 1.3 \cdot 10^{-107}:\\ \;\;\;\;\frac{\frac{\frac{d}{h} \cdot \left(\left(2 \cdot \left(c0 \cdot d\right)\right) \cdot \frac{\frac{c0}{2 \cdot w}}{w}\right)}{D}}{D}\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{-11}:\\ \;\;\;\;\frac{t\_0}{\frac{d \cdot d}{M}} \cdot \left(h \cdot M\right)\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{+116}:\\ \;\;\;\;\frac{\frac{2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{h}}{w \cdot D} \cdot \frac{c0}{2}}{w \cdot D}\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{+180}:\\ \;\;\;\;0.25 \cdot \left(\frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d} \cdot \frac{D}{d}\right)\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{+279}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\frac{c0 \cdot \left(d \cdot \left(c0 \cdot d\right)\right)}{D}}{D}}{w}}{h}}{w}\\ \mathbf{else}:\\ \;\;\;\;\frac{M \cdot M}{\frac{d}{h}} \cdot \frac{t\_0}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* D (* D 0.25))))
   (if (<= d 1.3e-107)
     (/ (/ (* (/ d h) (* (* 2.0 (* c0 d)) (/ (/ c0 (* 2.0 w)) w))) D) D)
     (if (<= d 1.45e-11)
       (* (/ t_0 (/ (* d d) M)) (* h M))
       (if (<= d 2.8e+116)
         (/ (* (/ (* 2.0 (/ (* c0 (* d d)) h)) (* w D)) (/ c0 2.0)) (* w D))
         (if (<= d 1.05e+180)
           (* 0.25 (* (/ (* h (* D (* M M))) d) (/ D d)))
           (if (<= d 3.2e+279)
             (/ (/ (/ (/ (/ (* c0 (* d (* c0 d))) D) D) w) h) w)
             (* (/ (* M M) (/ d h)) (/ t_0 d)))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = D * (D * 0.25);
	double tmp;
	if (d <= 1.3e-107) {
		tmp = (((d / h) * ((2.0 * (c0 * d)) * ((c0 / (2.0 * w)) / w))) / D) / D;
	} else if (d <= 1.45e-11) {
		tmp = (t_0 / ((d * d) / M)) * (h * M);
	} else if (d <= 2.8e+116) {
		tmp = (((2.0 * ((c0 * (d * d)) / h)) / (w * D)) * (c0 / 2.0)) / (w * D);
	} else if (d <= 1.05e+180) {
		tmp = 0.25 * (((h * (D * (M * M))) / d) * (D / d));
	} else if (d <= 3.2e+279) {
		tmp = (((((c0 * (d * (c0 * d))) / D) / D) / w) / h) / w;
	} else {
		tmp = ((M * M) / (d / h)) * (t_0 / 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) :: t_0
    real(8) :: tmp
    t_0 = d * (d * 0.25d0)
    if (d_1 <= 1.3d-107) then
        tmp = (((d_1 / h) * ((2.0d0 * (c0 * d_1)) * ((c0 / (2.0d0 * w)) / w))) / d) / d
    else if (d_1 <= 1.45d-11) then
        tmp = (t_0 / ((d_1 * d_1) / m)) * (h * m)
    else if (d_1 <= 2.8d+116) then
        tmp = (((2.0d0 * ((c0 * (d_1 * d_1)) / h)) / (w * d)) * (c0 / 2.0d0)) / (w * d)
    else if (d_1 <= 1.05d+180) then
        tmp = 0.25d0 * (((h * (d * (m * m))) / d_1) * (d / d_1))
    else if (d_1 <= 3.2d+279) then
        tmp = (((((c0 * (d_1 * (c0 * d_1))) / d) / d) / w) / h) / w
    else
        tmp = ((m * m) / (d_1 / h)) * (t_0 / d_1)
    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 * 0.25);
	double tmp;
	if (d <= 1.3e-107) {
		tmp = (((d / h) * ((2.0 * (c0 * d)) * ((c0 / (2.0 * w)) / w))) / D) / D;
	} else if (d <= 1.45e-11) {
		tmp = (t_0 / ((d * d) / M)) * (h * M);
	} else if (d <= 2.8e+116) {
		tmp = (((2.0 * ((c0 * (d * d)) / h)) / (w * D)) * (c0 / 2.0)) / (w * D);
	} else if (d <= 1.05e+180) {
		tmp = 0.25 * (((h * (D * (M * M))) / d) * (D / d));
	} else if (d <= 3.2e+279) {
		tmp = (((((c0 * (d * (c0 * d))) / D) / D) / w) / h) / w;
	} else {
		tmp = ((M * M) / (d / h)) * (t_0 / d);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = D * (D * 0.25)
	tmp = 0
	if d <= 1.3e-107:
		tmp = (((d / h) * ((2.0 * (c0 * d)) * ((c0 / (2.0 * w)) / w))) / D) / D
	elif d <= 1.45e-11:
		tmp = (t_0 / ((d * d) / M)) * (h * M)
	elif d <= 2.8e+116:
		tmp = (((2.0 * ((c0 * (d * d)) / h)) / (w * D)) * (c0 / 2.0)) / (w * D)
	elif d <= 1.05e+180:
		tmp = 0.25 * (((h * (D * (M * M))) / d) * (D / d))
	elif d <= 3.2e+279:
		tmp = (((((c0 * (d * (c0 * d))) / D) / D) / w) / h) / w
	else:
		tmp = ((M * M) / (d / h)) * (t_0 / d)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(D * Float64(D * 0.25))
	tmp = 0.0
	if (d <= 1.3e-107)
		tmp = Float64(Float64(Float64(Float64(d / h) * Float64(Float64(2.0 * Float64(c0 * d)) * Float64(Float64(c0 / Float64(2.0 * w)) / w))) / D) / D);
	elseif (d <= 1.45e-11)
		tmp = Float64(Float64(t_0 / Float64(Float64(d * d) / M)) * Float64(h * M));
	elseif (d <= 2.8e+116)
		tmp = Float64(Float64(Float64(Float64(2.0 * Float64(Float64(c0 * Float64(d * d)) / h)) / Float64(w * D)) * Float64(c0 / 2.0)) / Float64(w * D));
	elseif (d <= 1.05e+180)
		tmp = Float64(0.25 * Float64(Float64(Float64(h * Float64(D * Float64(M * M))) / d) * Float64(D / d)));
	elseif (d <= 3.2e+279)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(c0 * Float64(d * Float64(c0 * d))) / D) / D) / w) / h) / w);
	else
		tmp = Float64(Float64(Float64(M * M) / Float64(d / h)) * Float64(t_0 / d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = D * (D * 0.25);
	tmp = 0.0;
	if (d <= 1.3e-107)
		tmp = (((d / h) * ((2.0 * (c0 * d)) * ((c0 / (2.0 * w)) / w))) / D) / D;
	elseif (d <= 1.45e-11)
		tmp = (t_0 / ((d * d) / M)) * (h * M);
	elseif (d <= 2.8e+116)
		tmp = (((2.0 * ((c0 * (d * d)) / h)) / (w * D)) * (c0 / 2.0)) / (w * D);
	elseif (d <= 1.05e+180)
		tmp = 0.25 * (((h * (D * (M * M))) / d) * (D / d));
	elseif (d <= 3.2e+279)
		tmp = (((((c0 * (d * (c0 * d))) / D) / D) / w) / h) / w;
	else
		tmp = ((M * M) / (d / h)) * (t_0 / d);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(D * N[(D * 0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, 1.3e-107], N[(N[(N[(N[(d / h), $MachinePrecision] * N[(N[(2.0 * N[(c0 * d), $MachinePrecision]), $MachinePrecision] * N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision] / D), $MachinePrecision], If[LessEqual[d, 1.45e-11], N[(N[(t$95$0 / N[(N[(d * d), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision] * N[(h * M), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.8e+116], N[(N[(N[(N[(2.0 * N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision] / N[(w * D), $MachinePrecision]), $MachinePrecision] * N[(c0 / 2.0), $MachinePrecision]), $MachinePrecision] / N[(w * D), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.05e+180], N[(0.25 * N[(N[(N[(h * N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.2e+279], N[(N[(N[(N[(N[(N[(c0 * N[(d * N[(c0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision] / D), $MachinePrecision] / w), $MachinePrecision] / h), $MachinePrecision] / w), $MachinePrecision], N[(N[(N[(M * M), $MachinePrecision] / N[(d / h), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / d), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if d < 1.3e-107

   1. Initial program 26.5%

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

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

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

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

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

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

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

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

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

   3. Simplified 26.5%

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

   5. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

      6. associate-/r* N/A

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

      7. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      12. unpow2 N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 32.9%

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

   7. Simplified 32.9%

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

      1. associate-*r/ N/A

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

      2. associate-/r* N/A

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

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

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

   9. Applied egg-rr 42.1%

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

       1. associate-*r* N/A

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

       2. associate-/l* N/A

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

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

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

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

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

       5. /-lowering-/.f64 47.9%

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

   11. Applied egg-rr 47.9%

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

       1. associate-/r/ N/A

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

       2. associate-*r* N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

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

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

       7. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

       13. *-lowering-*.f64 52.0%

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

   13. Applied egg-rr 52.0%

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

   if 1.3e-107 < d < 1.45e-11

   1. Initial program 9.5%

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

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

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

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

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

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

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

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

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

   3. Simplified 9.5%

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

   5. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\left({c0}^{2}\right), \color{blue}{\left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(c0 \cdot c0\right), \left(\color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right), \color{blue}{\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)}\right)\right) \]

   7. Simplified 23.2%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \left(\frac{0}{w} + \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}\right)} \]

   8. Taylor expanded in c0 around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right), \color{blue}{\left({d}^{2}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({D}^{2}\right), \left({M}^{2} \cdot h\right)\right), \left({\color{blue}{d}}^{2}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(D \cdot D\right), \left({M}^{2} \cdot h\right)\right), \left({d}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left({M}^{2} \cdot h\right)\right), \left({d}^{2}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot {M}^{2}\right)\right), \left({d}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right)\right) \]

      11. *-lowering-*.f64 64.3%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right)\right) \]

   10. Simplified 64.3%

       \[\leadsto \color{blue}{0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \]
   11. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\frac{h \cdot \left(M \cdot M\right)}{d \cdot d}}\right) \]

       2. associate-*r* N/A

          \[\leadsto \frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \frac{\left(h \cdot M\right) \cdot M}{\color{blue}{d} \cdot d}\right) \]

       3. associate-*r/ N/A

          \[\leadsto \frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(\left(h \cdot M\right) \cdot \color{blue}{\frac{M}{d \cdot d}}\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\left(\left(h \cdot M\right) \cdot \frac{M}{d \cdot d}\right)} \]

       5. *-commutative N/A

          \[\leadsto \left(\left(D \cdot D\right) \cdot \frac{1}{4}\right) \cdot \left(\color{blue}{\left(h \cdot M\right)} \cdot \frac{M}{d \cdot d}\right) \]

       6. *-commutative N/A

          \[\leadsto \left(\left(D \cdot D\right) \cdot \frac{1}{4}\right) \cdot \left(\frac{M}{d \cdot d} \cdot \color{blue}{\left(h \cdot M\right)}\right) \]

       7. associate-*r* N/A

          \[\leadsto \left(\left(\left(D \cdot D\right) \cdot \frac{1}{4}\right) \cdot \frac{M}{d \cdot d}\right) \cdot \color{blue}{\left(h \cdot M\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(D \cdot D\right) \cdot \frac{1}{4}\right) \cdot \frac{M}{d \cdot d}\right), \color{blue}{\left(h \cdot M\right)}\right) \]

       9. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(D \cdot D\right) \cdot \frac{1}{4}\right) \cdot \frac{1}{\frac{d \cdot d}{M}}\right), \left(h \cdot M\right)\right) \]

       10. un-div-inv N/A

           \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(D \cdot D\right) \cdot \frac{1}{4}}{\frac{d \cdot d}{M}}\right), \left(\color{blue}{h} \cdot M\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(D \cdot D\right) \cdot \frac{1}{4}\right), \left(\frac{d \cdot d}{M}\right)\right), \left(\color{blue}{h} \cdot M\right)\right) \]

       12. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(D \cdot \left(D \cdot \frac{1}{4}\right)\right), \left(\frac{d \cdot d}{M}\right)\right), \left(h \cdot M\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(D \cdot \frac{1}{4}\right)\right), \left(\frac{d \cdot d}{M}\right)\right), \left(h \cdot M\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \frac{1}{4}\right)\right), \left(\frac{d \cdot d}{M}\right)\right), \left(h \cdot M\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \frac{1}{4}\right)\right), \mathsf{/.f64}\left(\left(d \cdot d\right), M\right)\right), \left(h \cdot M\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \frac{1}{4}\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, d\right), M\right)\right), \left(h \cdot M\right)\right) \]

       17. *-lowering-*.f64 73.3%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \frac{1}{4}\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, d\right), M\right)\right), \mathsf{*.f64}\left(h, \color{blue}{M}\right)\right) \]

   12. Applied egg-rr 73.3%

       \[\leadsto \color{blue}{\frac{D \cdot \left(D \cdot 0.25\right)}{\frac{d \cdot d}{M}} \cdot \left(h \cdot M\right)} \]

   if 1.45e-11 < d < 2.80000000000000004e116

   1. Initial program 42.9%

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

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

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

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

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

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

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

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

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

   3. Simplified 45.9%

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

   5. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

      6. associate-/r* N/A

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

      7. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      12. unpow2 N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 63.4%

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

   7. Simplified 63.4%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. associate-/r* N/A

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

      4. frac-times N/A

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

      5. *-commutative N/A

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

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

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

   9. Applied egg-rr 71.8%

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

   if 2.80000000000000004e116 < d < 1.05e180

   1. Initial program 15.4%

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

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

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

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

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

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

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

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

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

   3. Simplified 15.4%

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

   5. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\left({c0}^{2}\right), \color{blue}{\left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(c0 \cdot c0\right), \left(\color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right), \color{blue}{\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)}\right)\right) \]

   7. Simplified 23.5%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \left(\frac{0}{w} + \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}\right)} \]

   8. Taylor expanded in c0 around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right), \color{blue}{\left({d}^{2}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({D}^{2}\right), \left({M}^{2} \cdot h\right)\right), \left({\color{blue}{d}}^{2}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(D \cdot D\right), \left({M}^{2} \cdot h\right)\right), \left({d}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left({M}^{2} \cdot h\right)\right), \left({d}^{2}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot {M}^{2}\right)\right), \left({d}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right)\right) \]

      11. *-lowering-*.f64 70.2%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right)\right) \]

   10. Simplified 70.2%

       \[\leadsto \color{blue}{0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \]
   11. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left(\frac{D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\color{blue}{d} \cdot d}\right)\right) \]

       2. times-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left(\frac{D}{d} \cdot \color{blue}{\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d}}\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left(\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d} \cdot \color{blue}{\frac{D}{d}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d}\right), \color{blue}{\left(\frac{D}{d}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right), d\right), \left(\frac{\color{blue}{D}}{d}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(D \cdot \left(\left(M \cdot M\right) \cdot h\right)\right), d\right), \left(\frac{D}{d}\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(D \cdot \left(M \cdot M\right)\right) \cdot h\right), d\right), \left(\frac{D}{d}\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(h \cdot \left(D \cdot \left(M \cdot M\right)\right)\right), d\right), \left(\frac{D}{d}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(D \cdot \left(M \cdot M\right)\right)\right), d\right), \left(\frac{D}{d}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, \left(M \cdot M\right)\right)\right), d\right), \left(\frac{D}{d}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, M\right)\right)\right), d\right), \left(\frac{D}{d}\right)\right)\right) \]

       12. /-lowering-/.f64 62.7%

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, M\right)\right)\right), d\right), \mathsf{/.f64}\left(D, \color{blue}{d}\right)\right)\right) \]

   12. Applied egg-rr 62.7%

       \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d} \cdot \frac{D}{d}\right)} \]

   if 1.05e180 < d < 3.19999999999999988e279

   1. Initial program 33.6%

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

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

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

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

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

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

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

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

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

   3. Simplified 40.5%

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

   5. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

      6. associate-/r* N/A

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

      7. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      12. unpow2 N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 47.6%

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

   7. Simplified 47.6%

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

      1. associate-*r/ N/A

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

      2. associate-/r* N/A

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

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

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

   9. Applied egg-rr 51.1%

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

   10. Taylor expanded in c0 around 0

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

       1. associate-*r* N/A

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

       2. associate-/r* N/A

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

       3. unpow2 N/A

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

       4. associate-/r* N/A

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

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

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

   12. Simplified 64.2%

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

   if 3.19999999999999988e279 < d

   1. Initial program 20.0%

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

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

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

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

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

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

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

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

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

   3. Simplified 20.0%

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

   5. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\left({c0}^{2}\right), \color{blue}{\left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(c0 \cdot c0\right), \left(\color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right), \color{blue}{\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)}\right)\right) \]

   7. Simplified 20.2%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \left(\frac{0}{w} + \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}\right)} \]

   8. Taylor expanded in c0 around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right), \color{blue}{\left({d}^{2}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({D}^{2}\right), \left({M}^{2} \cdot h\right)\right), \left({\color{blue}{d}}^{2}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(D \cdot D\right), \left({M}^{2} \cdot h\right)\right), \left({d}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left({M}^{2} \cdot h\right)\right), \left({d}^{2}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot {M}^{2}\right)\right), \left({d}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right)\right) \]

      11. *-lowering-*.f64 70.3%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right)\right) \]

   10. Simplified 70.3%

       \[\leadsto \color{blue}{0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\color{blue}{d \cdot d}} \]

       2. associate-*r* N/A

          \[\leadsto \frac{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\color{blue}{d} \cdot d} \]

       3. *-commutative N/A

          \[\leadsto \frac{\left(\left(D \cdot D\right) \cdot \frac{1}{4}\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d} \]

       4. times-frac N/A

          \[\leadsto \frac{\left(D \cdot D\right) \cdot \frac{1}{4}}{d} \cdot \color{blue}{\frac{h \cdot \left(M \cdot M\right)}{d}} \]

       5. *-commutative N/A

          \[\leadsto \frac{h \cdot \left(M \cdot M\right)}{d} \cdot \color{blue}{\frac{\left(D \cdot D\right) \cdot \frac{1}{4}}{d}} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{h \cdot \left(M \cdot M\right)}{d}\right), \color{blue}{\left(\frac{\left(D \cdot D\right) \cdot \frac{1}{4}}{d}\right)}\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(M \cdot M\right) \cdot h}{d}\right), \left(\frac{\color{blue}{\left(D \cdot D\right)} \cdot \frac{1}{4}}{d}\right)\right) \]

       8. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(M \cdot M\right) \cdot \frac{h}{d}\right), \left(\frac{\color{blue}{\left(D \cdot D\right) \cdot \frac{1}{4}}}{d}\right)\right) \]

       9. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(M \cdot M\right) \cdot \frac{1}{\frac{d}{h}}\right), \left(\frac{\left(D \cdot D\right) \cdot \color{blue}{\frac{1}{4}}}{d}\right)\right) \]

       10. un-div-inv N/A

           \[\leadsto \mathsf{*.f64}\left(\left(\frac{M \cdot M}{\frac{d}{h}}\right), \left(\frac{\color{blue}{\left(D \cdot D\right) \cdot \frac{1}{4}}}{d}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot M\right), \left(\frac{d}{h}\right)\right), \left(\frac{\color{blue}{\left(D \cdot D\right) \cdot \frac{1}{4}}}{d}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, M\right), \left(\frac{d}{h}\right)\right), \left(\frac{\color{blue}{\left(D \cdot D\right)} \cdot \frac{1}{4}}{d}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{/.f64}\left(d, h\right)\right), \left(\frac{\left(D \cdot D\right) \cdot \color{blue}{\frac{1}{4}}}{d}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{/.f64}\left(d, h\right)\right), \mathsf{/.f64}\left(\left(\left(D \cdot D\right) \cdot \frac{1}{4}\right), \color{blue}{d}\right)\right) \]

       15. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{/.f64}\left(d, h\right)\right), \mathsf{/.f64}\left(\left(D \cdot \left(D \cdot \frac{1}{4}\right)\right), d\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{/.f64}\left(d, h\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(D \cdot \frac{1}{4}\right)\right), d\right)\right) \]

       17. *-lowering-*.f64 71.1%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{/.f64}\left(d, h\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \frac{1}{4}\right)\right), d\right)\right) \]

   12. Applied egg-rr 71.1%

       \[\leadsto \color{blue}{\frac{M \cdot M}{\frac{d}{h}} \cdot \frac{D \cdot \left(D \cdot 0.25\right)}{d}} \]
3. Recombined 6 regimes into one program.
4. Add Preprocessing

Alternative 3: 46.5% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 3.7 \cdot 10^{-17}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \frac{\frac{h \cdot \left(M \cdot M\right)}{d}}{d}\right)\right)\\ \mathbf{elif}\;d \leq 8.5 \cdot 10^{+114}:\\ \;\;\;\;\frac{\frac{2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{h}}{w \cdot D} \cdot \frac{c0}{2}}{w \cdot D}\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{+178}:\\ \;\;\;\;0.25 \cdot \left(\frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d} \cdot \frac{D}{d}\right)\\ \mathbf{elif}\;d \leq 5.6 \cdot 10^{+278}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\frac{c0 \cdot \left(d \cdot \left(c0 \cdot d\right)\right)}{D}}{D}}{w}}{h}}{w}\\ \mathbf{else}:\\ \;\;\;\;\frac{M \cdot M}{\frac{d}{h}} \cdot \frac{D \cdot \left(D \cdot 0.25\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= d 3.7e-17)
   (* 0.25 (* D (* D (/ (/ (* h (* M M)) d) d))))
   (if (<= d 8.5e+114)
     (/ (* (/ (* 2.0 (/ (* c0 (* d d)) h)) (* w D)) (/ c0 2.0)) (* w D))
     (if (<= d 2.2e+178)
       (* 0.25 (* (/ (* h (* D (* M M))) d) (/ D d)))
       (if (<= d 5.6e+278)
         (/ (/ (/ (/ (/ (* c0 (* d (* c0 d))) D) D) w) h) w)
         (* (/ (* M M) (/ d h)) (/ (* D (* D 0.25)) d)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (d <= 3.7e-17) {
		tmp = 0.25 * (D * (D * (((h * (M * M)) / d) / d)));
	} else if (d <= 8.5e+114) {
		tmp = (((2.0 * ((c0 * (d * d)) / h)) / (w * D)) * (c0 / 2.0)) / (w * D);
	} else if (d <= 2.2e+178) {
		tmp = 0.25 * (((h * (D * (M * M))) / d) * (D / d));
	} else if (d <= 5.6e+278) {
		tmp = (((((c0 * (d * (c0 * d))) / D) / D) / w) / h) / w;
	} else {
		tmp = ((M * M) / (d / h)) * ((D * (D * 0.25)) / 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 (d_1 <= 3.7d-17) then
        tmp = 0.25d0 * (d * (d * (((h * (m * m)) / d_1) / d_1)))
    else if (d_1 <= 8.5d+114) then
        tmp = (((2.0d0 * ((c0 * (d_1 * d_1)) / h)) / (w * d)) * (c0 / 2.0d0)) / (w * d)
    else if (d_1 <= 2.2d+178) then
        tmp = 0.25d0 * (((h * (d * (m * m))) / d_1) * (d / d_1))
    else if (d_1 <= 5.6d+278) then
        tmp = (((((c0 * (d_1 * (c0 * d_1))) / d) / d) / w) / h) / w
    else
        tmp = ((m * m) / (d_1 / h)) * ((d * (d * 0.25d0)) / d_1)
    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 <= 3.7e-17) {
		tmp = 0.25 * (D * (D * (((h * (M * M)) / d) / d)));
	} else if (d <= 8.5e+114) {
		tmp = (((2.0 * ((c0 * (d * d)) / h)) / (w * D)) * (c0 / 2.0)) / (w * D);
	} else if (d <= 2.2e+178) {
		tmp = 0.25 * (((h * (D * (M * M))) / d) * (D / d));
	} else if (d <= 5.6e+278) {
		tmp = (((((c0 * (d * (c0 * d))) / D) / D) / w) / h) / w;
	} else {
		tmp = ((M * M) / (d / h)) * ((D * (D * 0.25)) / d);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if d <= 3.7e-17:
		tmp = 0.25 * (D * (D * (((h * (M * M)) / d) / d)))
	elif d <= 8.5e+114:
		tmp = (((2.0 * ((c0 * (d * d)) / h)) / (w * D)) * (c0 / 2.0)) / (w * D)
	elif d <= 2.2e+178:
		tmp = 0.25 * (((h * (D * (M * M))) / d) * (D / d))
	elif d <= 5.6e+278:
		tmp = (((((c0 * (d * (c0 * d))) / D) / D) / w) / h) / w
	else:
		tmp = ((M * M) / (d / h)) * ((D * (D * 0.25)) / d)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (d <= 3.7e-17)
		tmp = Float64(0.25 * Float64(D * Float64(D * Float64(Float64(Float64(h * Float64(M * M)) / d) / d))));
	elseif (d <= 8.5e+114)
		tmp = Float64(Float64(Float64(Float64(2.0 * Float64(Float64(c0 * Float64(d * d)) / h)) / Float64(w * D)) * Float64(c0 / 2.0)) / Float64(w * D));
	elseif (d <= 2.2e+178)
		tmp = Float64(0.25 * Float64(Float64(Float64(h * Float64(D * Float64(M * M))) / d) * Float64(D / d)));
	elseif (d <= 5.6e+278)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(c0 * Float64(d * Float64(c0 * d))) / D) / D) / w) / h) / w);
	else
		tmp = Float64(Float64(Float64(M * M) / Float64(d / h)) * Float64(Float64(D * Float64(D * 0.25)) / d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (d <= 3.7e-17)
		tmp = 0.25 * (D * (D * (((h * (M * M)) / d) / d)));
	elseif (d <= 8.5e+114)
		tmp = (((2.0 * ((c0 * (d * d)) / h)) / (w * D)) * (c0 / 2.0)) / (w * D);
	elseif (d <= 2.2e+178)
		tmp = 0.25 * (((h * (D * (M * M))) / d) * (D / d));
	elseif (d <= 5.6e+278)
		tmp = (((((c0 * (d * (c0 * d))) / D) / D) / w) / h) / w;
	else
		tmp = ((M * M) / (d / h)) * ((D * (D * 0.25)) / d);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[d, 3.7e-17], N[(0.25 * N[(D * N[(D * N[(N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 8.5e+114], N[(N[(N[(N[(2.0 * N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision] / N[(w * D), $MachinePrecision]), $MachinePrecision] * N[(c0 / 2.0), $MachinePrecision]), $MachinePrecision] / N[(w * D), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.2e+178], N[(0.25 * N[(N[(N[(h * N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5.6e+278], N[(N[(N[(N[(N[(N[(c0 * N[(d * N[(c0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision] / D), $MachinePrecision] / w), $MachinePrecision] / h), $MachinePrecision] / w), $MachinePrecision], N[(N[(N[(M * M), $MachinePrecision] / N[(d / h), $MachinePrecision]), $MachinePrecision] * N[(N[(D * N[(D * 0.25), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if d < 3.6999999999999997e-17

   1. Initial program 24.6%

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

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

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

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

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

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

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

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

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

   3. Simplified 24.6%

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

   5. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\left({c0}^{2}\right), \color{blue}{\left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(c0 \cdot c0\right), \left(\color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right), \color{blue}{\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)}\right)\right) \]

   7. Simplified 14.9%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \left(\frac{0}{w} + \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}\right)} \]

   8. Taylor expanded in c0 around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right), \color{blue}{\left({d}^{2}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({D}^{2}\right), \left({M}^{2} \cdot h\right)\right), \left({\color{blue}{d}}^{2}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(D \cdot D\right), \left({M}^{2} \cdot h\right)\right), \left({d}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left({M}^{2} \cdot h\right)\right), \left({d}^{2}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot {M}^{2}\right)\right), \left({d}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right)\right) \]

      11. *-lowering-*.f64 37.6%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right)\right) \]

   10. Simplified 37.6%

       \[\leadsto \color{blue}{0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \]

   11. Taylor expanded in D around 0

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right)}\right) \]
   12. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot \color{blue}{\frac{{M}^{2} \cdot h}{{d}^{2}}}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left(\left(D \cdot D\right) \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{{d}^{2}}\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \color{blue}{\left(D \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \color{blue}{\left(D \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \color{blue}{\left(\frac{{M}^{2} \cdot h}{{d}^{2}}\right)}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(\frac{{M}^{2} \cdot h}{d \cdot \color{blue}{d}}\right)\right)\right)\right) \]

       7. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(\frac{\frac{{M}^{2} \cdot h}{d}}{\color{blue}{d}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\left(\frac{{M}^{2} \cdot h}{d}\right), \color{blue}{d}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({M}^{2} \cdot h\right), d\right), d\right)\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(h \cdot {M}^{2}\right), d\right), d\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left({M}^{2}\right)\right), d\right), d\right)\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(M \cdot M\right)\right), d\right), d\right)\right)\right)\right) \]

       13. *-lowering-*.f64 49.2%

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), d\right), d\right)\right)\right)\right) \]

   13. Simplified 49.2%

       \[\leadsto 0.25 \cdot \color{blue}{\left(D \cdot \left(D \cdot \frac{\frac{h \cdot \left(M \cdot M\right)}{d}}{d}\right)\right)} \]

   if 3.6999999999999997e-17 < d < 8.5000000000000001e114

   1. Initial program 40.6%

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

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

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

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

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

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

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

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

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

   3. Simplified 43.4%

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

   5. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

      6. associate-/r* N/A

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

      7. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      12. unpow2 N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 60.0%

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

   7. Simplified 60.0%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. associate-/r* N/A

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

      4. frac-times N/A

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

      5. *-commutative N/A

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

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

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

   9. Applied egg-rr 70.7%

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

   if 8.5000000000000001e114 < d < 2.19999999999999997e178

   1. Initial program 15.4%

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

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

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

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

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

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

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

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

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

   3. Simplified 15.4%

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

   5. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\left({c0}^{2}\right), \color{blue}{\left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(c0 \cdot c0\right), \left(\color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right), \color{blue}{\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)}\right)\right) \]

   7. Simplified 23.5%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \left(\frac{0}{w} + \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}\right)} \]

   8. Taylor expanded in c0 around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right), \color{blue}{\left({d}^{2}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({D}^{2}\right), \left({M}^{2} \cdot h\right)\right), \left({\color{blue}{d}}^{2}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(D \cdot D\right), \left({M}^{2} \cdot h\right)\right), \left({d}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left({M}^{2} \cdot h\right)\right), \left({d}^{2}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot {M}^{2}\right)\right), \left({d}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right)\right) \]

      11. *-lowering-*.f64 70.2%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right)\right) \]

   10. Simplified 70.2%

       \[\leadsto \color{blue}{0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \]
   11. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left(\frac{D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\color{blue}{d} \cdot d}\right)\right) \]

       2. times-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left(\frac{D}{d} \cdot \color{blue}{\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d}}\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left(\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d} \cdot \color{blue}{\frac{D}{d}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d}\right), \color{blue}{\left(\frac{D}{d}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right), d\right), \left(\frac{\color{blue}{D}}{d}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(D \cdot \left(\left(M \cdot M\right) \cdot h\right)\right), d\right), \left(\frac{D}{d}\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(D \cdot \left(M \cdot M\right)\right) \cdot h\right), d\right), \left(\frac{D}{d}\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(h \cdot \left(D \cdot \left(M \cdot M\right)\right)\right), d\right), \left(\frac{D}{d}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(D \cdot \left(M \cdot M\right)\right)\right), d\right), \left(\frac{D}{d}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, \left(M \cdot M\right)\right)\right), d\right), \left(\frac{D}{d}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, M\right)\right)\right), d\right), \left(\frac{D}{d}\right)\right)\right) \]

       12. /-lowering-/.f64 62.7%

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, M\right)\right)\right), d\right), \mathsf{/.f64}\left(D, \color{blue}{d}\right)\right)\right) \]

   12. Applied egg-rr 62.7%

       \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d} \cdot \frac{D}{d}\right)} \]

   if 2.19999999999999997e178 < d < 5.6000000000000002e278

   1. Initial program 33.6%

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

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

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

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

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

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

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

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

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

   3. Simplified 40.5%

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

   5. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

      6. associate-/r* N/A

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

      7. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      12. unpow2 N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 47.6%

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

   7. Simplified 47.6%

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

      1. associate-*r/ N/A

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

      2. associate-/r* N/A

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

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

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

   9. Applied egg-rr 51.1%

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

   10. Taylor expanded in c0 around 0

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

       1. associate-*r* N/A

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

       2. associate-/r* N/A

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

       3. unpow2 N/A

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

       4. associate-/r* N/A

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

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

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

   12. Simplified 64.2%

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

   if 5.6000000000000002e278 < d

   1. Initial program 20.0%

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

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

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

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

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

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

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

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

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

   3. Simplified 20.0%

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

   5. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\left({c0}^{2}\right), \color{blue}{\left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(c0 \cdot c0\right), \left(\color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right), \color{blue}{\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)}\right)\right) \]

   7. Simplified 20.2%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \left(\frac{0}{w} + \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}\right)} \]

   8. Taylor expanded in c0 around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right), \color{blue}{\left({d}^{2}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({D}^{2}\right), \left({M}^{2} \cdot h\right)\right), \left({\color{blue}{d}}^{2}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(D \cdot D\right), \left({M}^{2} \cdot h\right)\right), \left({d}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left({M}^{2} \cdot h\right)\right), \left({d}^{2}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot {M}^{2}\right)\right), \left({d}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right)\right) \]

      11. *-lowering-*.f64 70.3%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right)\right) \]

   10. Simplified 70.3%

       \[\leadsto \color{blue}{0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\color{blue}{d \cdot d}} \]

       2. associate-*r* N/A

          \[\leadsto \frac{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\color{blue}{d} \cdot d} \]

       3. *-commutative N/A

          \[\leadsto \frac{\left(\left(D \cdot D\right) \cdot \frac{1}{4}\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d} \]

       4. times-frac N/A

          \[\leadsto \frac{\left(D \cdot D\right) \cdot \frac{1}{4}}{d} \cdot \color{blue}{\frac{h \cdot \left(M \cdot M\right)}{d}} \]

       5. *-commutative N/A

          \[\leadsto \frac{h \cdot \left(M \cdot M\right)}{d} \cdot \color{blue}{\frac{\left(D \cdot D\right) \cdot \frac{1}{4}}{d}} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{h \cdot \left(M \cdot M\right)}{d}\right), \color{blue}{\left(\frac{\left(D \cdot D\right) \cdot \frac{1}{4}}{d}\right)}\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(M \cdot M\right) \cdot h}{d}\right), \left(\frac{\color{blue}{\left(D \cdot D\right)} \cdot \frac{1}{4}}{d}\right)\right) \]

       8. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(M \cdot M\right) \cdot \frac{h}{d}\right), \left(\frac{\color{blue}{\left(D \cdot D\right) \cdot \frac{1}{4}}}{d}\right)\right) \]

       9. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(M \cdot M\right) \cdot \frac{1}{\frac{d}{h}}\right), \left(\frac{\left(D \cdot D\right) \cdot \color{blue}{\frac{1}{4}}}{d}\right)\right) \]

       10. un-div-inv N/A

           \[\leadsto \mathsf{*.f64}\left(\left(\frac{M \cdot M}{\frac{d}{h}}\right), \left(\frac{\color{blue}{\left(D \cdot D\right) \cdot \frac{1}{4}}}{d}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot M\right), \left(\frac{d}{h}\right)\right), \left(\frac{\color{blue}{\left(D \cdot D\right) \cdot \frac{1}{4}}}{d}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, M\right), \left(\frac{d}{h}\right)\right), \left(\frac{\color{blue}{\left(D \cdot D\right)} \cdot \frac{1}{4}}{d}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{/.f64}\left(d, h\right)\right), \left(\frac{\left(D \cdot D\right) \cdot \color{blue}{\frac{1}{4}}}{d}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{/.f64}\left(d, h\right)\right), \mathsf{/.f64}\left(\left(\left(D \cdot D\right) \cdot \frac{1}{4}\right), \color{blue}{d}\right)\right) \]

       15. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{/.f64}\left(d, h\right)\right), \mathsf{/.f64}\left(\left(D \cdot \left(D \cdot \frac{1}{4}\right)\right), d\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{/.f64}\left(d, h\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(D \cdot \frac{1}{4}\right)\right), d\right)\right) \]

       17. *-lowering-*.f64 71.1%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{/.f64}\left(d, h\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \frac{1}{4}\right)\right), d\right)\right) \]

   12. Applied egg-rr 71.1%

       \[\leadsto \color{blue}{\frac{M \cdot M}{\frac{d}{h}} \cdot \frac{D \cdot \left(D \cdot 0.25\right)}{d}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 4: 46.1% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 1.04 \cdot 10^{-16}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \frac{\frac{h \cdot \left(M \cdot M\right)}{d}}{d}\right)\right)\\ \mathbf{elif}\;d \leq 6.6 \cdot 10^{+115}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \frac{\frac{\frac{2 \cdot \left(c0 \cdot d\right)}{\frac{h}{d}}}{w \cdot D}}{D}\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{+178}:\\ \;\;\;\;0.25 \cdot \left(\frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d} \cdot \frac{D}{d}\right)\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{+279}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\frac{c0 \cdot \left(d \cdot \left(c0 \cdot d\right)\right)}{D}}{D}}{w}}{h}}{w}\\ \mathbf{else}:\\ \;\;\;\;\frac{M \cdot M}{\frac{d}{h}} \cdot \frac{D \cdot \left(D \cdot 0.25\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= d 1.04e-16)
   (* 0.25 (* D (* D (/ (/ (* h (* M M)) d) d))))
   (if (<= d 6.6e+115)
     (* (/ c0 (* 2.0 w)) (/ (/ (/ (* 2.0 (* c0 d)) (/ h d)) (* w D)) D))
     (if (<= d 2.4e+178)
       (* 0.25 (* (/ (* h (* D (* M M))) d) (/ D d)))
       (if (<= d 3.5e+279)
         (/ (/ (/ (/ (/ (* c0 (* d (* c0 d))) D) D) w) h) w)
         (* (/ (* M M) (/ d h)) (/ (* D (* D 0.25)) d)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (d <= 1.04e-16) {
		tmp = 0.25 * (D * (D * (((h * (M * M)) / d) / d)));
	} else if (d <= 6.6e+115) {
		tmp = (c0 / (2.0 * w)) * ((((2.0 * (c0 * d)) / (h / d)) / (w * D)) / D);
	} else if (d <= 2.4e+178) {
		tmp = 0.25 * (((h * (D * (M * M))) / d) * (D / d));
	} else if (d <= 3.5e+279) {
		tmp = (((((c0 * (d * (c0 * d))) / D) / D) / w) / h) / w;
	} else {
		tmp = ((M * M) / (d / h)) * ((D * (D * 0.25)) / 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 (d_1 <= 1.04d-16) then
        tmp = 0.25d0 * (d * (d * (((h * (m * m)) / d_1) / d_1)))
    else if (d_1 <= 6.6d+115) then
        tmp = (c0 / (2.0d0 * w)) * ((((2.0d0 * (c0 * d_1)) / (h / d_1)) / (w * d)) / d)
    else if (d_1 <= 2.4d+178) then
        tmp = 0.25d0 * (((h * (d * (m * m))) / d_1) * (d / d_1))
    else if (d_1 <= 3.5d+279) then
        tmp = (((((c0 * (d_1 * (c0 * d_1))) / d) / d) / w) / h) / w
    else
        tmp = ((m * m) / (d_1 / h)) * ((d * (d * 0.25d0)) / d_1)
    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 <= 1.04e-16) {
		tmp = 0.25 * (D * (D * (((h * (M * M)) / d) / d)));
	} else if (d <= 6.6e+115) {
		tmp = (c0 / (2.0 * w)) * ((((2.0 * (c0 * d)) / (h / d)) / (w * D)) / D);
	} else if (d <= 2.4e+178) {
		tmp = 0.25 * (((h * (D * (M * M))) / d) * (D / d));
	} else if (d <= 3.5e+279) {
		tmp = (((((c0 * (d * (c0 * d))) / D) / D) / w) / h) / w;
	} else {
		tmp = ((M * M) / (d / h)) * ((D * (D * 0.25)) / d);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if d <= 1.04e-16:
		tmp = 0.25 * (D * (D * (((h * (M * M)) / d) / d)))
	elif d <= 6.6e+115:
		tmp = (c0 / (2.0 * w)) * ((((2.0 * (c0 * d)) / (h / d)) / (w * D)) / D)
	elif d <= 2.4e+178:
		tmp = 0.25 * (((h * (D * (M * M))) / d) * (D / d))
	elif d <= 3.5e+279:
		tmp = (((((c0 * (d * (c0 * d))) / D) / D) / w) / h) / w
	else:
		tmp = ((M * M) / (d / h)) * ((D * (D * 0.25)) / d)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (d <= 1.04e-16)
		tmp = Float64(0.25 * Float64(D * Float64(D * Float64(Float64(Float64(h * Float64(M * M)) / d) / d))));
	elseif (d <= 6.6e+115)
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(Float64(Float64(Float64(2.0 * Float64(c0 * d)) / Float64(h / d)) / Float64(w * D)) / D));
	elseif (d <= 2.4e+178)
		tmp = Float64(0.25 * Float64(Float64(Float64(h * Float64(D * Float64(M * M))) / d) * Float64(D / d)));
	elseif (d <= 3.5e+279)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(c0 * Float64(d * Float64(c0 * d))) / D) / D) / w) / h) / w);
	else
		tmp = Float64(Float64(Float64(M * M) / Float64(d / h)) * Float64(Float64(D * Float64(D * 0.25)) / d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (d <= 1.04e-16)
		tmp = 0.25 * (D * (D * (((h * (M * M)) / d) / d)));
	elseif (d <= 6.6e+115)
		tmp = (c0 / (2.0 * w)) * ((((2.0 * (c0 * d)) / (h / d)) / (w * D)) / D);
	elseif (d <= 2.4e+178)
		tmp = 0.25 * (((h * (D * (M * M))) / d) * (D / d));
	elseif (d <= 3.5e+279)
		tmp = (((((c0 * (d * (c0 * d))) / D) / D) / w) / h) / w;
	else
		tmp = ((M * M) / (d / h)) * ((D * (D * 0.25)) / d);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[d, 1.04e-16], N[(0.25 * N[(D * N[(D * N[(N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 6.6e+115], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * N[(c0 * d), $MachinePrecision]), $MachinePrecision] / N[(h / d), $MachinePrecision]), $MachinePrecision] / N[(w * D), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.4e+178], N[(0.25 * N[(N[(N[(h * N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.5e+279], N[(N[(N[(N[(N[(N[(c0 * N[(d * N[(c0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision] / D), $MachinePrecision] / w), $MachinePrecision] / h), $MachinePrecision] / w), $MachinePrecision], N[(N[(N[(M * M), $MachinePrecision] / N[(d / h), $MachinePrecision]), $MachinePrecision] * N[(N[(D * N[(D * 0.25), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if d < 1.04000000000000001e-16

   1. Initial program 24.6%

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

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

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

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

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

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

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

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

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

   3. Simplified 24.6%

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

   5. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\left({c0}^{2}\right), \color{blue}{\left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(c0 \cdot c0\right), \left(\color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right), \color{blue}{\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)}\right)\right) \]

   7. Simplified 14.9%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \left(\frac{0}{w} + \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}\right)} \]

   8. Taylor expanded in c0 around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right), \color{blue}{\left({d}^{2}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({D}^{2}\right), \left({M}^{2} \cdot h\right)\right), \left({\color{blue}{d}}^{2}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(D \cdot D\right), \left({M}^{2} \cdot h\right)\right), \left({d}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left({M}^{2} \cdot h\right)\right), \left({d}^{2}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot {M}^{2}\right)\right), \left({d}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right)\right) \]

      11. *-lowering-*.f64 37.6%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right)\right) \]

   10. Simplified 37.6%

       \[\leadsto \color{blue}{0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \]

   11. Taylor expanded in D around 0

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right)}\right) \]
   12. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot \color{blue}{\frac{{M}^{2} \cdot h}{{d}^{2}}}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left(\left(D \cdot D\right) \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{{d}^{2}}\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \color{blue}{\left(D \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \color{blue}{\left(D \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \color{blue}{\left(\frac{{M}^{2} \cdot h}{{d}^{2}}\right)}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(\frac{{M}^{2} \cdot h}{d \cdot \color{blue}{d}}\right)\right)\right)\right) \]

       7. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(\frac{\frac{{M}^{2} \cdot h}{d}}{\color{blue}{d}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\left(\frac{{M}^{2} \cdot h}{d}\right), \color{blue}{d}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({M}^{2} \cdot h\right), d\right), d\right)\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(h \cdot {M}^{2}\right), d\right), d\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left({M}^{2}\right)\right), d\right), d\right)\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(M \cdot M\right)\right), d\right), d\right)\right)\right)\right) \]

       13. *-lowering-*.f64 49.2%

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), d\right), d\right)\right)\right)\right) \]

   13. Simplified 49.2%

       \[\leadsto 0.25 \cdot \color{blue}{\left(D \cdot \left(D \cdot \frac{\frac{h \cdot \left(M \cdot M\right)}{d}}{d}\right)\right)} \]

   if 1.04000000000000001e-16 < d < 6.6000000000000001e115

   1. Initial program 40.6%

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

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

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

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

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

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

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

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

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

   3. Simplified 43.4%

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

   5. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

      6. associate-/r* N/A

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

      7. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      12. unpow2 N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 60.0%

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

   7. Simplified 60.0%

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. /-lowering-/.f64 60.1%

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

   9. Applied egg-rr 60.1%

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

       1. associate-/r* N/A

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

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

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

       3. associate-/l/ N/A

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

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

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

       5. associate-*r* N/A

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

       6. clear-num N/A

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

       7. un-div-inv N/A

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

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

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

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

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

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

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

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

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

       12. *-commutative N/A

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

       13. *-lowering-*.f64 70.7%

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

   11. Applied egg-rr 70.7%

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

   if 6.6000000000000001e115 < d < 2.4e178

   1. Initial program 15.4%

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

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

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

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

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

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

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

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

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

   3. Simplified 15.4%

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

   5. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\left({c0}^{2}\right), \color{blue}{\left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(c0 \cdot c0\right), \left(\color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right), \color{blue}{\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)}\right)\right) \]

   7. Simplified 23.5%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \left(\frac{0}{w} + \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}\right)} \]

   8. Taylor expanded in c0 around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right), \color{blue}{\left({d}^{2}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({D}^{2}\right), \left({M}^{2} \cdot h\right)\right), \left({\color{blue}{d}}^{2}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(D \cdot D\right), \left({M}^{2} \cdot h\right)\right), \left({d}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left({M}^{2} \cdot h\right)\right), \left({d}^{2}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot {M}^{2}\right)\right), \left({d}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right)\right) \]

      11. *-lowering-*.f64 70.2%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right)\right) \]

   10. Simplified 70.2%

       \[\leadsto \color{blue}{0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \]
   11. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left(\frac{D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\color{blue}{d} \cdot d}\right)\right) \]

       2. times-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left(\frac{D}{d} \cdot \color{blue}{\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d}}\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left(\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d} \cdot \color{blue}{\frac{D}{d}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d}\right), \color{blue}{\left(\frac{D}{d}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right), d\right), \left(\frac{\color{blue}{D}}{d}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(D \cdot \left(\left(M \cdot M\right) \cdot h\right)\right), d\right), \left(\frac{D}{d}\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(D \cdot \left(M \cdot M\right)\right) \cdot h\right), d\right), \left(\frac{D}{d}\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(h \cdot \left(D \cdot \left(M \cdot M\right)\right)\right), d\right), \left(\frac{D}{d}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(D \cdot \left(M \cdot M\right)\right)\right), d\right), \left(\frac{D}{d}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, \left(M \cdot M\right)\right)\right), d\right), \left(\frac{D}{d}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, M\right)\right)\right), d\right), \left(\frac{D}{d}\right)\right)\right) \]

       12. /-lowering-/.f64 62.7%

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, M\right)\right)\right), d\right), \mathsf{/.f64}\left(D, \color{blue}{d}\right)\right)\right) \]

   12. Applied egg-rr 62.7%

       \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d} \cdot \frac{D}{d}\right)} \]

   if 2.4e178 < d < 3.50000000000000001e279

   1. Initial program 33.6%

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

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

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

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

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

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

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

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

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

   3. Simplified 40.5%

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

   5. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

      6. associate-/r* N/A

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

      7. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      12. unpow2 N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 47.6%

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

   7. Simplified 47.6%

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

      1. associate-*r/ N/A

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

      2. associate-/r* N/A

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

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

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

   9. Applied egg-rr 51.1%

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

   10. Taylor expanded in c0 around 0

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

       1. associate-*r* N/A

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

       2. associate-/r* N/A

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

       3. unpow2 N/A

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

       4. associate-/r* N/A

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

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

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

   12. Simplified 64.2%

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

   if 3.50000000000000001e279 < d

   1. Initial program 20.0%

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

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

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

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

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

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

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

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

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

   3. Simplified 20.0%

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

   5. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\left({c0}^{2}\right), \color{blue}{\left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(c0 \cdot c0\right), \left(\color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right), \color{blue}{\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)}\right)\right) \]

   7. Simplified 20.2%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \left(\frac{0}{w} + \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}\right)} \]

   8. Taylor expanded in c0 around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right), \color{blue}{\left({d}^{2}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({D}^{2}\right), \left({M}^{2} \cdot h\right)\right), \left({\color{blue}{d}}^{2}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(D \cdot D\right), \left({M}^{2} \cdot h\right)\right), \left({d}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left({M}^{2} \cdot h\right)\right), \left({d}^{2}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot {M}^{2}\right)\right), \left({d}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right)\right) \]

      11. *-lowering-*.f64 70.3%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right)\right) \]

   10. Simplified 70.3%

       \[\leadsto \color{blue}{0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\color{blue}{d \cdot d}} \]

       2. associate-*r* N/A

          \[\leadsto \frac{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\color{blue}{d} \cdot d} \]

       3. *-commutative N/A

          \[\leadsto \frac{\left(\left(D \cdot D\right) \cdot \frac{1}{4}\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d} \]

       4. times-frac N/A

          \[\leadsto \frac{\left(D \cdot D\right) \cdot \frac{1}{4}}{d} \cdot \color{blue}{\frac{h \cdot \left(M \cdot M\right)}{d}} \]

       5. *-commutative N/A

          \[\leadsto \frac{h \cdot \left(M \cdot M\right)}{d} \cdot \color{blue}{\frac{\left(D \cdot D\right) \cdot \frac{1}{4}}{d}} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{h \cdot \left(M \cdot M\right)}{d}\right), \color{blue}{\left(\frac{\left(D \cdot D\right) \cdot \frac{1}{4}}{d}\right)}\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(M \cdot M\right) \cdot h}{d}\right), \left(\frac{\color{blue}{\left(D \cdot D\right)} \cdot \frac{1}{4}}{d}\right)\right) \]

       8. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(M \cdot M\right) \cdot \frac{h}{d}\right), \left(\frac{\color{blue}{\left(D \cdot D\right) \cdot \frac{1}{4}}}{d}\right)\right) \]

       9. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(M \cdot M\right) \cdot \frac{1}{\frac{d}{h}}\right), \left(\frac{\left(D \cdot D\right) \cdot \color{blue}{\frac{1}{4}}}{d}\right)\right) \]

       10. un-div-inv N/A

           \[\leadsto \mathsf{*.f64}\left(\left(\frac{M \cdot M}{\frac{d}{h}}\right), \left(\frac{\color{blue}{\left(D \cdot D\right) \cdot \frac{1}{4}}}{d}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot M\right), \left(\frac{d}{h}\right)\right), \left(\frac{\color{blue}{\left(D \cdot D\right) \cdot \frac{1}{4}}}{d}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, M\right), \left(\frac{d}{h}\right)\right), \left(\frac{\color{blue}{\left(D \cdot D\right)} \cdot \frac{1}{4}}{d}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{/.f64}\left(d, h\right)\right), \left(\frac{\left(D \cdot D\right) \cdot \color{blue}{\frac{1}{4}}}{d}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{/.f64}\left(d, h\right)\right), \mathsf{/.f64}\left(\left(\left(D \cdot D\right) \cdot \frac{1}{4}\right), \color{blue}{d}\right)\right) \]

       15. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{/.f64}\left(d, h\right)\right), \mathsf{/.f64}\left(\left(D \cdot \left(D \cdot \frac{1}{4}\right)\right), d\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{/.f64}\left(d, h\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(D \cdot \frac{1}{4}\right)\right), d\right)\right) \]

       17. *-lowering-*.f64 71.1%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{/.f64}\left(d, h\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \frac{1}{4}\right)\right), d\right)\right) \]

   12. Applied egg-rr 71.1%

       \[\leadsto \color{blue}{\frac{M \cdot M}{\frac{d}{h}} \cdot \frac{D \cdot \left(D \cdot 0.25\right)}{d}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 5: 40.3% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := h \cdot \left(M \cdot M\right)\\ t_1 := \frac{d \cdot \left(d \cdot \frac{\frac{\frac{\frac{c0 \cdot c0}{w}}{w}}{h}}{D}\right)}{D}\\ \mathbf{if}\;M \leq 2.95 \cdot 10^{-227}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 10^{-185}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;M \leq 2.3 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{0.25}{d}}{\frac{d}{\left(D \cdot D\right) \cdot t\_0}}\\ \mathbf{elif}\;M \leq 4.2 \cdot 10^{+127}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \frac{\frac{t\_0}{d}}{d}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* h (* M M)))
        (t_1 (/ (* d (* d (/ (/ (/ (/ (* c0 c0) w) w) h) D))) D)))
   (if (<= M 2.95e-227)
     0.0
     (if (<= M 1e-185)
       t_1
       (if (<= M 2.3e-9)
         (/ (/ 0.25 d) (/ d (* (* D D) t_0)))
         (if (<= M 4.2e+127) t_1 (* 0.25 (* D (* D (/ (/ t_0 d) d))))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = h * (M * M);
	double t_1 = (d * (d * (((((c0 * c0) / w) / w) / h) / D))) / D;
	double tmp;
	if (M <= 2.95e-227) {
		tmp = 0.0;
	} else if (M <= 1e-185) {
		tmp = t_1;
	} else if (M <= 2.3e-9) {
		tmp = (0.25 / d) / (d / ((D * D) * t_0));
	} else if (M <= 4.2e+127) {
		tmp = t_1;
	} else {
		tmp = 0.25 * (D * (D * ((t_0 / d) / 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = h * (m * m)
    t_1 = (d_1 * (d_1 * (((((c0 * c0) / w) / w) / h) / d))) / d
    if (m <= 2.95d-227) then
        tmp = 0.0d0
    else if (m <= 1d-185) then
        tmp = t_1
    else if (m <= 2.3d-9) then
        tmp = (0.25d0 / d_1) / (d_1 / ((d * d) * t_0))
    else if (m <= 4.2d+127) then
        tmp = t_1
    else
        tmp = 0.25d0 * (d * (d * ((t_0 / d_1) / d_1)))
    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 = h * (M * M);
	double t_1 = (d * (d * (((((c0 * c0) / w) / w) / h) / D))) / D;
	double tmp;
	if (M <= 2.95e-227) {
		tmp = 0.0;
	} else if (M <= 1e-185) {
		tmp = t_1;
	} else if (M <= 2.3e-9) {
		tmp = (0.25 / d) / (d / ((D * D) * t_0));
	} else if (M <= 4.2e+127) {
		tmp = t_1;
	} else {
		tmp = 0.25 * (D * (D * ((t_0 / d) / d)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = h * (M * M)
	t_1 = (d * (d * (((((c0 * c0) / w) / w) / h) / D))) / D
	tmp = 0
	if M <= 2.95e-227:
		tmp = 0.0
	elif M <= 1e-185:
		tmp = t_1
	elif M <= 2.3e-9:
		tmp = (0.25 / d) / (d / ((D * D) * t_0))
	elif M <= 4.2e+127:
		tmp = t_1
	else:
		tmp = 0.25 * (D * (D * ((t_0 / d) / d)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(h * Float64(M * M))
	t_1 = Float64(Float64(d * Float64(d * Float64(Float64(Float64(Float64(Float64(c0 * c0) / w) / w) / h) / D))) / D)
	tmp = 0.0
	if (M <= 2.95e-227)
		tmp = 0.0;
	elseif (M <= 1e-185)
		tmp = t_1;
	elseif (M <= 2.3e-9)
		tmp = Float64(Float64(0.25 / d) / Float64(d / Float64(Float64(D * D) * t_0)));
	elseif (M <= 4.2e+127)
		tmp = t_1;
	else
		tmp = Float64(0.25 * Float64(D * Float64(D * Float64(Float64(t_0 / d) / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = h * (M * M);
	t_1 = (d * (d * (((((c0 * c0) / w) / w) / h) / D))) / D;
	tmp = 0.0;
	if (M <= 2.95e-227)
		tmp = 0.0;
	elseif (M <= 1e-185)
		tmp = t_1;
	elseif (M <= 2.3e-9)
		tmp = (0.25 / d) / (d / ((D * D) * t_0));
	elseif (M <= 4.2e+127)
		tmp = t_1;
	else
		tmp = 0.25 * (D * (D * ((t_0 / d) / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(d * N[(d * N[(N[(N[(N[(N[(c0 * c0), $MachinePrecision] / w), $MachinePrecision] / w), $MachinePrecision] / h), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]}, If[LessEqual[M, 2.95e-227], 0.0, If[LessEqual[M, 1e-185], t$95$1, If[LessEqual[M, 2.3e-9], N[(N[(0.25 / d), $MachinePrecision] / N[(d / N[(N[(D * D), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 4.2e+127], t$95$1, N[(0.25 * N[(D * N[(D * N[(N[(t$95$0 / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if M < 2.9499999999999998e-227

   1. Initial program 28.6%

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

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

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

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

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

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

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

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

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

   3. Simplified 29.9%

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

   5. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right) \cdot {c0}^{2}}{w} \]

      2. distribute-lft1-in N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right) \cdot {c0}^{2}}{w} \]

      3. metadata-eval N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\left(0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right) \cdot {c0}^{2}}{w} \]

      4. mul0-lft N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{0 \cdot {c0}^{2}}{w} \]

      5. metadata-eval N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\left(\frac{-1}{2} \cdot 0\right) \cdot {c0}^{2}}{w} \]

      6. mul0-lft N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\left(\frac{-1}{2} \cdot \left(0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot {c0}^{2}}{w} \]

      7. metadata-eval N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\left(\frac{-1}{2} \cdot \left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot {c0}^{2}}{w} \]

      8. distribute-lft1-in N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\left(\frac{-1}{2} \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot {c0}^{2}}{w} \]

      9. associate-*l/ N/A

         \[\leadsto \frac{-1}{2} \cdot \left(\frac{\frac{-1}{2} \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w} \cdot \color{blue}{{c0}^{2}}\right) \]

   7. Simplified 30.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{\left(c0 \cdot c0\right) \cdot 0}{w}} \]
   8. Step-by-step derivation

      1. mul0-rgt N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{0}{w} \]

      2. div0 N/A

         \[\leadsto \frac{-1}{2} \cdot 0 \]

      3. metadata-eval 38.8%

         \[\leadsto 0 \]

   9. Applied egg-rr 38.8%

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

   if 2.9499999999999998e-227 < M < 9.9999999999999999e-186 or 2.2999999999999999e-9 < M < 4.19999999999999983e127

   1. Initial program 40.3%

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

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

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

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

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

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

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

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

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

   3. Simplified 43.2%

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

   5. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

      6. associate-/r* N/A

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

      7. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      12. unpow2 N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 43.7%

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

   7. Simplified 43.7%

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

      1. associate-*r/ N/A

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

      2. associate-/r* N/A

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

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

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

   9. Applied egg-rr 47.0%

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

       1. associate-*r* N/A

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

       2. associate-/l* N/A

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

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

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

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

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

       5. /-lowering-/.f64 52.6%

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

   11. Applied egg-rr 52.6%

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

   12. Taylor expanded in c0 around 0

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. times-frac N/A

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

       7. *-commutative N/A

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

       8. associate-*r/ N/A

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

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

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

   14. Simplified 53.3%

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

   if 9.9999999999999999e-186 < M < 2.2999999999999999e-9

   1. Initial program 23.9%

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

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

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

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

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

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

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

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

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

   3. Simplified 23.8%

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

   5. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\left({c0}^{2}\right), \color{blue}{\left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(c0 \cdot c0\right), \left(\color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right), \color{blue}{\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)}\right)\right) \]

   7. Simplified 26.1%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \left(\frac{0}{w} + \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}\right)} \]

   8. Taylor expanded in c0 around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right), \color{blue}{\left({d}^{2}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({D}^{2}\right), \left({M}^{2} \cdot h\right)\right), \left({\color{blue}{d}}^{2}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(D \cdot D\right), \left({M}^{2} \cdot h\right)\right), \left({d}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left({M}^{2} \cdot h\right)\right), \left({d}^{2}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot {M}^{2}\right)\right), \left({d}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right)\right) \]

      11. *-lowering-*.f64 56.0%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right)\right) \]

   10. Simplified 56.0%

       \[\leadsto \color{blue}{0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\color{blue}{d \cdot d}} \]

       2. times-frac N/A

          \[\leadsto \frac{\frac{1}{4}}{d} \cdot \color{blue}{\frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d}} \]

       3. clear-num N/A

          \[\leadsto \frac{\frac{1}{4}}{d} \cdot \frac{1}{\color{blue}{\frac{d}{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}}} \]

       4. un-div-inv N/A

          \[\leadsto \frac{\frac{\frac{1}{4}}{d}}{\color{blue}{\frac{d}{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}}} \]

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

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{4}}{d}\right), \color{blue}{\left(\frac{d}{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}\right)}\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, d\right), \left(\frac{\color{blue}{d}}{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, d\right), \mathsf{/.f64}\left(d, \color{blue}{\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, d\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(\left(D \cdot D\right), \color{blue}{\left(h \cdot \left(M \cdot M\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, d\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{h} \cdot \left(M \cdot M\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, d\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \color{blue}{\left(M \cdot M\right)}\right)\right)\right)\right) \]

       11. *-lowering-*.f64 57.1%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, d\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \color{blue}{M}\right)\right)\right)\right)\right) \]

   12. Applied egg-rr 57.1%

       \[\leadsto \color{blue}{\frac{\frac{0.25}{d}}{\frac{d}{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}}} \]

   if 4.19999999999999983e127 < M

   1. Initial program 0.0%

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

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

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

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

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

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

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

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

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

   3. Simplified 0.0%

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

   5. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\left({c0}^{2}\right), \color{blue}{\left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(c0 \cdot c0\right), \left(\color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right), \color{blue}{\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)}\right)\right) \]

   7. Simplified 0.9%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \left(\frac{0}{w} + \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}\right)} \]

   8. Taylor expanded in c0 around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right), \color{blue}{\left({d}^{2}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({D}^{2}\right), \left({M}^{2} \cdot h\right)\right), \left({\color{blue}{d}}^{2}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(D \cdot D\right), \left({M}^{2} \cdot h\right)\right), \left({d}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left({M}^{2} \cdot h\right)\right), \left({d}^{2}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot {M}^{2}\right)\right), \left({d}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right)\right) \]

      11. *-lowering-*.f64 34.3%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right)\right) \]

   10. Simplified 34.3%

       \[\leadsto \color{blue}{0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \]

   11. Taylor expanded in D around 0

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right)}\right) \]
   12. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot \color{blue}{\frac{{M}^{2} \cdot h}{{d}^{2}}}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left(\left(D \cdot D\right) \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{{d}^{2}}\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \color{blue}{\left(D \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \color{blue}{\left(D \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \color{blue}{\left(\frac{{M}^{2} \cdot h}{{d}^{2}}\right)}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(\frac{{M}^{2} \cdot h}{d \cdot \color{blue}{d}}\right)\right)\right)\right) \]

       7. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(\frac{\frac{{M}^{2} \cdot h}{d}}{\color{blue}{d}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\left(\frac{{M}^{2} \cdot h}{d}\right), \color{blue}{d}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({M}^{2} \cdot h\right), d\right), d\right)\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(h \cdot {M}^{2}\right), d\right), d\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left({M}^{2}\right)\right), d\right), d\right)\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(M \cdot M\right)\right), d\right), d\right)\right)\right)\right) \]

       13. *-lowering-*.f64 62.8%

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), d\right), d\right)\right)\right)\right) \]

   13. Simplified 62.8%

       \[\leadsto 0.25 \cdot \color{blue}{\left(D \cdot \left(D \cdot \frac{\frac{h \cdot \left(M \cdot M\right)}{d}}{d}\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.95 \cdot 10^{-227}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 10^{-185}:\\ \;\;\;\;\frac{d \cdot \left(d \cdot \frac{\frac{\frac{\frac{c0 \cdot c0}{w}}{w}}{h}}{D}\right)}{D}\\ \mathbf{elif}\;M \leq 2.3 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{0.25}{d}}{\frac{d}{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}}\\ \mathbf{elif}\;M \leq 4.2 \cdot 10^{+127}:\\ \;\;\;\;\frac{d \cdot \left(d \cdot \frac{\frac{\frac{\frac{c0 \cdot c0}{w}}{w}}{h}}{D}\right)}{D}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \frac{\frac{h \cdot \left(M \cdot M\right)}{d}}{d}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 40.4% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 1.18 \cdot 10^{-228}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 8 \cdot 10^{-185}:\\ \;\;\;\;\frac{d \cdot \left(d \cdot \frac{\frac{\frac{\frac{c0 \cdot c0}{w}}{w}}{h}}{D}\right)}{D}\\ \mathbf{elif}\;M \leq 2.3 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{0.25}{d}}{\frac{d}{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.5}{w}}{\frac{\frac{w}{2 \cdot \left(c0 \cdot d\right)}}{\frac{d}{h}}} \cdot \frac{c0}{D}}{D}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= M 1.18e-228)
   0.0
   (if (<= M 8e-185)
     (/ (* d (* d (/ (/ (/ (/ (* c0 c0) w) w) h) D))) D)
     (if (<= M 2.3e-10)
       (/ (/ 0.25 d) (/ d (* (* D D) (* h (* M M)))))
       (/ (* (/ (/ 0.5 w) (/ (/ w (* 2.0 (* c0 d))) (/ d h))) (/ c0 D)) D)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 1.18e-228) {
		tmp = 0.0;
	} else if (M <= 8e-185) {
		tmp = (d * (d * (((((c0 * c0) / w) / w) / h) / D))) / D;
	} else if (M <= 2.3e-10) {
		tmp = (0.25 / d) / (d / ((D * D) * (h * (M * M))));
	} else {
		tmp = (((0.5 / w) / ((w / (2.0 * (c0 * d))) / (d / h))) * (c0 / D)) / 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 <= 1.18d-228) then
        tmp = 0.0d0
    else if (m <= 8d-185) then
        tmp = (d_1 * (d_1 * (((((c0 * c0) / w) / w) / h) / d))) / d
    else if (m <= 2.3d-10) then
        tmp = (0.25d0 / d_1) / (d_1 / ((d * d) * (h * (m * m))))
    else
        tmp = (((0.5d0 / w) / ((w / (2.0d0 * (c0 * d_1))) / (d_1 / h))) * (c0 / d)) / 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 <= 1.18e-228) {
		tmp = 0.0;
	} else if (M <= 8e-185) {
		tmp = (d * (d * (((((c0 * c0) / w) / w) / h) / D))) / D;
	} else if (M <= 2.3e-10) {
		tmp = (0.25 / d) / (d / ((D * D) * (h * (M * M))));
	} else {
		tmp = (((0.5 / w) / ((w / (2.0 * (c0 * d))) / (d / h))) * (c0 / D)) / D;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if M <= 1.18e-228:
		tmp = 0.0
	elif M <= 8e-185:
		tmp = (d * (d * (((((c0 * c0) / w) / w) / h) / D))) / D
	elif M <= 2.3e-10:
		tmp = (0.25 / d) / (d / ((D * D) * (h * (M * M))))
	else:
		tmp = (((0.5 / w) / ((w / (2.0 * (c0 * d))) / (d / h))) * (c0 / D)) / D
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (M <= 1.18e-228)
		tmp = 0.0;
	elseif (M <= 8e-185)
		tmp = Float64(Float64(d * Float64(d * Float64(Float64(Float64(Float64(Float64(c0 * c0) / w) / w) / h) / D))) / D);
	elseif (M <= 2.3e-10)
		tmp = Float64(Float64(0.25 / d) / Float64(d / Float64(Float64(D * D) * Float64(h * Float64(M * M)))));
	else
		tmp = Float64(Float64(Float64(Float64(0.5 / w) / Float64(Float64(w / Float64(2.0 * Float64(c0 * d))) / Float64(d / h))) * Float64(c0 / D)) / D);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (M <= 1.18e-228)
		tmp = 0.0;
	elseif (M <= 8e-185)
		tmp = (d * (d * (((((c0 * c0) / w) / w) / h) / D))) / D;
	elseif (M <= 2.3e-10)
		tmp = (0.25 / d) / (d / ((D * D) * (h * (M * M))));
	else
		tmp = (((0.5 / w) / ((w / (2.0 * (c0 * d))) / (d / h))) * (c0 / D)) / D;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[M, 1.18e-228], 0.0, If[LessEqual[M, 8e-185], N[(N[(d * N[(d * N[(N[(N[(N[(N[(c0 * c0), $MachinePrecision] / w), $MachinePrecision] / w), $MachinePrecision] / h), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision], If[LessEqual[M, 2.3e-10], N[(N[(0.25 / d), $MachinePrecision] / N[(d / N[(N[(D * D), $MachinePrecision] * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.5 / w), $MachinePrecision] / N[(N[(w / N[(2.0 * N[(c0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c0 / D), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if M < 1.1800000000000001e-228

   1. Initial program 28.6%

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

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

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

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

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

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

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

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

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

   3. Simplified 29.9%

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

   5. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right) \cdot {c0}^{2}}{w} \]

      2. distribute-lft1-in N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right) \cdot {c0}^{2}}{w} \]

      3. metadata-eval N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\left(0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right) \cdot {c0}^{2}}{w} \]

      4. mul0-lft N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{0 \cdot {c0}^{2}}{w} \]

      5. metadata-eval N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\left(\frac{-1}{2} \cdot 0\right) \cdot {c0}^{2}}{w} \]

      6. mul0-lft N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\left(\frac{-1}{2} \cdot \left(0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot {c0}^{2}}{w} \]

      7. metadata-eval N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\left(\frac{-1}{2} \cdot \left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot {c0}^{2}}{w} \]

      8. distribute-lft1-in N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\left(\frac{-1}{2} \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot {c0}^{2}}{w} \]

      9. associate-*l/ N/A

         \[\leadsto \frac{-1}{2} \cdot \left(\frac{\frac{-1}{2} \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w} \cdot \color{blue}{{c0}^{2}}\right) \]

   7. Simplified 30.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{\left(c0 \cdot c0\right) \cdot 0}{w}} \]
   8. Step-by-step derivation

      1. mul0-rgt N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{0}{w} \]

      2. div0 N/A

         \[\leadsto \frac{-1}{2} \cdot 0 \]

      3. metadata-eval 38.8%

         \[\leadsto 0 \]

   9. Applied egg-rr 38.8%

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

   if 1.1800000000000001e-228 < M < 7.9999999999999999e-185

   1. Initial program 41.4%

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

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

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \left(2 \cdot w\right)\right), \left(\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{+.f64}\left(\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right), \color{blue}{\left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}\right)\right) \]

   3. Simplified 47.5%

      \[\leadsto \color{blue}{\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)}}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)} - M \cdot M}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\color{blue}{{D}^{2} \cdot \left(h \cdot w\right)}}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\left(h \cdot w\right) \cdot \color{blue}{{D}^{2}}}\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{h \cdot w}}{\color{blue}{{D}^{2}}}\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{2 \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}}{{\color{blue}{D}}^{2}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(2 \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right), \color{blue}{\left({D}^{2}\right)}\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(2 \cdot \frac{\frac{c0 \cdot {d}^{2}}{h}}{w}\right), \left({D}^{2}\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{c0 \cdot {d}^{2}}{h}}{w}\right), \left({\color{blue}{D}}^{2}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \frac{c0 \cdot {d}^{2}}{h}\right), w\right), \left({\color{blue}{D}}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{c0 \cdot {d}^{2}}{h}\right)\right), w\right), \left({D}^{2}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(c0 \cdot {d}^{2}\right), h\right)\right), w\right), \left({D}^{2}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left({d}^{2}\right)\right), h\right)\right), w\right), \left({D}^{2}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(d \cdot d\right)\right), h\right)\right), w\right), \left({D}^{2}\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right), h\right)\right), w\right), \left({D}^{2}\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right), h\right)\right), w\right), \left(D \cdot \color{blue}{D}\right)\right)\right) \]

      15. *-lowering-*.f64 48.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right), h\right)\right), w\right), \mathsf{*.f64}\left(D, \color{blue}{D}\right)\right)\right) \]

   7. Simplified 48.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{\frac{2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{h}}{w}}{D \cdot D}} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{h}}{w}}{\color{blue}{D \cdot D}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{h}}{w}}{D}}{\color{blue}{D}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{h}}{w}}{D}\right), \color{blue}{D}\right) \]

   9. Applied egg-rr 53.8%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{c0}{2 \cdot w}}{\frac{w}{2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{h}}}}{D}}{D}} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(w, \mathsf{*.f64}\left(2, \left(\frac{\left(c0 \cdot d\right) \cdot d}{h}\right)\right)\right)\right), D\right), D\right) \]

       2. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(w, \mathsf{*.f64}\left(2, \left(\left(c0 \cdot d\right) \cdot \frac{d}{h}\right)\right)\right)\right), D\right), D\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(w, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(c0 \cdot d\right), \left(\frac{d}{h}\right)\right)\right)\right)\right), D\right), D\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(w, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, d\right), \left(\frac{d}{h}\right)\right)\right)\right)\right), D\right), D\right) \]

       5. /-lowering-/.f64 59.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(w, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, d\right), \mathsf{/.f64}\left(d, h\right)\right)\right)\right)\right), D\right), D\right) \]

   11. Applied egg-rr 59.3%

       \[\leadsto \frac{\frac{\frac{\frac{c0}{2 \cdot w}}{\frac{w}{2 \cdot \color{blue}{\left(\left(c0 \cdot d\right) \cdot \frac{d}{h}\right)}}}}{D}}{D} \]

   12. Taylor expanded in c0 around 0

       \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]
   13. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{{c0}^{2} \cdot \left(d \cdot d\right)}{{D}^{\color{blue}{2}} \cdot \left(h \cdot {w}^{2}\right)} \]

       2. associate-*r* N/A

          \[\leadsto \frac{\left({c0}^{2} \cdot d\right) \cdot d}{\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)} \]

       3. unpow2 N/A

          \[\leadsto \frac{\left({c0}^{2} \cdot d\right) \cdot d}{\left(D \cdot D\right) \cdot \left(\color{blue}{h} \cdot {w}^{2}\right)} \]

       4. associate-*l* N/A

          \[\leadsto \frac{\left({c0}^{2} \cdot d\right) \cdot d}{D \cdot \color{blue}{\left(D \cdot \left(h \cdot {w}^{2}\right)\right)}} \]

       5. *-commutative N/A

          \[\leadsto \frac{\left({c0}^{2} \cdot d\right) \cdot d}{\left(D \cdot \left(h \cdot {w}^{2}\right)\right) \cdot \color{blue}{D}} \]

       6. times-frac N/A

          \[\leadsto \frac{{c0}^{2} \cdot d}{D \cdot \left(h \cdot {w}^{2}\right)} \cdot \color{blue}{\frac{d}{D}} \]

       7. *-commutative N/A

          \[\leadsto \frac{d \cdot {c0}^{2}}{D \cdot \left(h \cdot {w}^{2}\right)} \cdot \frac{d}{D} \]

       8. associate-*r/ N/A

          \[\leadsto \frac{\frac{d \cdot {c0}^{2}}{D \cdot \left(h \cdot {w}^{2}\right)} \cdot d}{\color{blue}{D}} \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{d \cdot {c0}^{2}}{D \cdot \left(h \cdot {w}^{2}\right)} \cdot d\right), \color{blue}{D}\right) \]

   14. Simplified 71.1%

       \[\leadsto \color{blue}{\frac{\left(d \cdot \frac{\frac{\frac{\frac{c0 \cdot c0}{w}}{w}}{h}}{D}\right) \cdot d}{D}} \]

   if 7.9999999999999999e-185 < M < 2.30000000000000007e-10

   1. Initial program 23.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{c0}{2 \cdot w}\right), \color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \left(2 \cdot w\right)\right), \left(\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{+.f64}\left(\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right), \color{blue}{\left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}\right)\right) \]

   3. Simplified 23.8%

      \[\leadsto \color{blue}{\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)}}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)} - M \cdot M}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({c0}^{2}\right), \color{blue}{\left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(c0 \cdot c0\right), \left(\color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right), \color{blue}{\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)}\right)\right) \]

   7. Simplified 26.1%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \left(\frac{0}{w} + \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}\right)} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right), \color{blue}{\left({d}^{2}\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({D}^{2}\right), \left({M}^{2} \cdot h\right)\right), \left({\color{blue}{d}}^{2}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(D \cdot D\right), \left({M}^{2} \cdot h\right)\right), \left({d}^{2}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left({M}^{2} \cdot h\right)\right), \left({d}^{2}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot {M}^{2}\right)\right), \left({d}^{2}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right)\right) \]

      11. *-lowering-*.f64 56.0%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right)\right) \]

   10. Simplified 56.0%

       \[\leadsto \color{blue}{0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\color{blue}{d \cdot d}} \]

       2. times-frac N/A

          \[\leadsto \frac{\frac{1}{4}}{d} \cdot \color{blue}{\frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d}} \]

       3. clear-num N/A

          \[\leadsto \frac{\frac{1}{4}}{d} \cdot \frac{1}{\color{blue}{\frac{d}{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}}} \]

       4. un-div-inv N/A

          \[\leadsto \frac{\frac{\frac{1}{4}}{d}}{\color{blue}{\frac{d}{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}}} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{4}}{d}\right), \color{blue}{\left(\frac{d}{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}\right)}\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, d\right), \left(\frac{\color{blue}{d}}{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, d\right), \mathsf{/.f64}\left(d, \color{blue}{\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, d\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(\left(D \cdot D\right), \color{blue}{\left(h \cdot \left(M \cdot M\right)\right)}\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, d\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{h} \cdot \left(M \cdot M\right)\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, d\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \color{blue}{\left(M \cdot M\right)}\right)\right)\right)\right) \]

       11. *-lowering-*.f64 57.1%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, d\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \color{blue}{M}\right)\right)\right)\right)\right) \]

   12. Applied egg-rr 57.1%

       \[\leadsto \color{blue}{\frac{\frac{0.25}{d}}{\frac{d}{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}}} \]

   if 2.30000000000000007e-10 < M

   1. Initial program 19.7%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{c0}{2 \cdot w}\right), \color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \left(2 \cdot w\right)\right), \left(\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{+.f64}\left(\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right), \color{blue}{\left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}\right)\right) \]

   3. Simplified 19.6%

      \[\leadsto \color{blue}{\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)}}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)} - M \cdot M}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\color{blue}{{D}^{2} \cdot \left(h \cdot w\right)}}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\left(h \cdot w\right) \cdot \color{blue}{{D}^{2}}}\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{h \cdot w}}{\color{blue}{{D}^{2}}}\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{2 \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}}{{\color{blue}{D}}^{2}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(2 \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right), \color{blue}{\left({D}^{2}\right)}\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(2 \cdot \frac{\frac{c0 \cdot {d}^{2}}{h}}{w}\right), \left({D}^{2}\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{c0 \cdot {d}^{2}}{h}}{w}\right), \left({\color{blue}{D}}^{2}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \frac{c0 \cdot {d}^{2}}{h}\right), w\right), \left({\color{blue}{D}}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{c0 \cdot {d}^{2}}{h}\right)\right), w\right), \left({D}^{2}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(c0 \cdot {d}^{2}\right), h\right)\right), w\right), \left({D}^{2}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left({d}^{2}\right)\right), h\right)\right), w\right), \left({D}^{2}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(d \cdot d\right)\right), h\right)\right), w\right), \left({D}^{2}\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right), h\right)\right), w\right), \left({D}^{2}\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right), h\right)\right), w\right), \left(D \cdot \color{blue}{D}\right)\right)\right) \]

      15. *-lowering-*.f64 40.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right), h\right)\right), w\right), \mathsf{*.f64}\left(D, \color{blue}{D}\right)\right)\right) \]

   7. Simplified 40.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{\frac{2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{h}}{w}}{D \cdot D}} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{h}}{w}}{\color{blue}{D \cdot D}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{h}}{w}}{D}}{\color{blue}{D}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{h}}{w}}{D}\right), \color{blue}{D}\right) \]

   9. Applied egg-rr 46.1%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{c0}{2 \cdot w}}{\frac{w}{2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{h}}}}{D}}{D}} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(w, \mathsf{*.f64}\left(2, \left(\frac{\left(c0 \cdot d\right) \cdot d}{h}\right)\right)\right)\right), D\right), D\right) \]

       2. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(w, \mathsf{*.f64}\left(2, \left(\left(c0 \cdot d\right) \cdot \frac{d}{h}\right)\right)\right)\right), D\right), D\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(w, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(c0 \cdot d\right), \left(\frac{d}{h}\right)\right)\right)\right)\right), D\right), D\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(w, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, d\right), \left(\frac{d}{h}\right)\right)\right)\right)\right), D\right), D\right) \]

       5. /-lowering-/.f64 51.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(w, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, d\right), \mathsf{/.f64}\left(d, h\right)\right)\right)\right)\right), D\right), D\right) \]

   11. Applied egg-rr 51.4%

       \[\leadsto \frac{\frac{\frac{\frac{c0}{2 \cdot w}}{\frac{w}{2 \cdot \color{blue}{\left(\left(c0 \cdot d\right) \cdot \frac{d}{h}\right)}}}}{D}}{D} \]
   12. Step-by-step derivation

       1. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{c0}{2 \cdot w}}{\frac{w}{2 \cdot \left(\left(c0 \cdot d\right) \cdot \frac{d}{h}\right)}} \cdot \frac{1}{D}\right), D\right) \]

       2. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{c0}{\frac{w}{2 \cdot \left(\left(c0 \cdot d\right) \cdot \frac{d}{h}\right)} \cdot \left(2 \cdot w\right)} \cdot \frac{1}{D}\right), D\right) \]

       3. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{c0 \cdot \frac{1}{D}}{\frac{w}{2 \cdot \left(\left(c0 \cdot d\right) \cdot \frac{d}{h}\right)} \cdot \left(2 \cdot w\right)}\right), D\right) \]

       4. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(c0 \cdot \frac{1}{D}\right) \cdot \frac{1}{\frac{w}{2 \cdot \left(\left(c0 \cdot d\right) \cdot \frac{d}{h}\right)} \cdot \left(2 \cdot w\right)}\right), D\right) \]

       5. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{c0}{D} \cdot \frac{1}{\frac{w}{2 \cdot \left(\left(c0 \cdot d\right) \cdot \frac{d}{h}\right)} \cdot \left(2 \cdot w\right)}\right), D\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\frac{w}{2 \cdot \left(\left(c0 \cdot d\right) \cdot \frac{d}{h}\right)} \cdot \left(2 \cdot w\right)} \cdot \frac{c0}{D}\right), D\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{\frac{w}{2 \cdot \left(\left(c0 \cdot d\right) \cdot \frac{d}{h}\right)} \cdot \left(2 \cdot w\right)}\right), \left(\frac{c0}{D}\right)\right), D\right) \]

   13. Applied egg-rr 51.4%

       \[\leadsto \frac{\color{blue}{\frac{\frac{0.5}{w}}{\frac{\frac{w}{2 \cdot \left(c0 \cdot d\right)}}{\frac{d}{h}}} \cdot \frac{c0}{D}}}{D} \]
3. Recombined 4 regimes into one program.

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.18 \cdot 10^{-228}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 8 \cdot 10^{-185}:\\ \;\;\;\;\frac{d \cdot \left(d \cdot \frac{\frac{\frac{\frac{c0 \cdot c0}{w}}{w}}{h}}{D}\right)}{D}\\ \mathbf{elif}\;M \leq 2.3 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{0.25}{d}}{\frac{d}{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.5}{w}}{\frac{\frac{w}{2 \cdot \left(c0 \cdot d\right)}}{\frac{d}{h}}} \cdot \frac{c0}{D}}{D}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 40.2% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 2.6 \cdot 10^{-227}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 5.5 \cdot 10^{-186}:\\ \;\;\;\;\frac{d \cdot \left(d \cdot \frac{\frac{\frac{\frac{c0 \cdot c0}{w}}{w}}{h}}{D}\right)}{D}\\ \mathbf{elif}\;M \leq 1.9 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{0.25}{d}}{\frac{d}{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{c0 \cdot \frac{\frac{d \cdot \left(c0 \cdot d\right)}{w}}{w}}{h}}{D}}{D}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= M 2.6e-227)
   0.0
   (if (<= M 5.5e-186)
     (/ (* d (* d (/ (/ (/ (/ (* c0 c0) w) w) h) D))) D)
     (if (<= M 1.9e-29)
       (/ (/ 0.25 d) (/ d (* (* D D) (* h (* M M)))))
       (/ (/ (/ (* c0 (/ (/ (* d (* c0 d)) w) w)) h) D) D)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 2.6e-227) {
		tmp = 0.0;
	} else if (M <= 5.5e-186) {
		tmp = (d * (d * (((((c0 * c0) / w) / w) / h) / D))) / D;
	} else if (M <= 1.9e-29) {
		tmp = (0.25 / d) / (d / ((D * D) * (h * (M * M))));
	} else {
		tmp = (((c0 * (((d * (c0 * d)) / w) / w)) / h) / D) / 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 <= 2.6d-227) then
        tmp = 0.0d0
    else if (m <= 5.5d-186) then
        tmp = (d_1 * (d_1 * (((((c0 * c0) / w) / w) / h) / d))) / d
    else if (m <= 1.9d-29) then
        tmp = (0.25d0 / d_1) / (d_1 / ((d * d) * (h * (m * m))))
    else
        tmp = (((c0 * (((d_1 * (c0 * d_1)) / w) / w)) / h) / d) / 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 <= 2.6e-227) {
		tmp = 0.0;
	} else if (M <= 5.5e-186) {
		tmp = (d * (d * (((((c0 * c0) / w) / w) / h) / D))) / D;
	} else if (M <= 1.9e-29) {
		tmp = (0.25 / d) / (d / ((D * D) * (h * (M * M))));
	} else {
		tmp = (((c0 * (((d * (c0 * d)) / w) / w)) / h) / D) / D;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if M <= 2.6e-227:
		tmp = 0.0
	elif M <= 5.5e-186:
		tmp = (d * (d * (((((c0 * c0) / w) / w) / h) / D))) / D
	elif M <= 1.9e-29:
		tmp = (0.25 / d) / (d / ((D * D) * (h * (M * M))))
	else:
		tmp = (((c0 * (((d * (c0 * d)) / w) / w)) / h) / D) / D
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (M <= 2.6e-227)
		tmp = 0.0;
	elseif (M <= 5.5e-186)
		tmp = Float64(Float64(d * Float64(d * Float64(Float64(Float64(Float64(Float64(c0 * c0) / w) / w) / h) / D))) / D);
	elseif (M <= 1.9e-29)
		tmp = Float64(Float64(0.25 / d) / Float64(d / Float64(Float64(D * D) * Float64(h * Float64(M * M)))));
	else
		tmp = Float64(Float64(Float64(Float64(c0 * Float64(Float64(Float64(d * Float64(c0 * d)) / w) / w)) / h) / D) / D);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (M <= 2.6e-227)
		tmp = 0.0;
	elseif (M <= 5.5e-186)
		tmp = (d * (d * (((((c0 * c0) / w) / w) / h) / D))) / D;
	elseif (M <= 1.9e-29)
		tmp = (0.25 / d) / (d / ((D * D) * (h * (M * M))));
	else
		tmp = (((c0 * (((d * (c0 * d)) / w) / w)) / h) / D) / D;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[M, 2.6e-227], 0.0, If[LessEqual[M, 5.5e-186], N[(N[(d * N[(d * N[(N[(N[(N[(N[(c0 * c0), $MachinePrecision] / w), $MachinePrecision] / w), $MachinePrecision] / h), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision], If[LessEqual[M, 1.9e-29], N[(N[(0.25 / d), $MachinePrecision] / N[(d / N[(N[(D * D), $MachinePrecision] * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(c0 * N[(N[(N[(d * N[(c0 * d), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision] / D), $MachinePrecision] / D), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if M < 2.60000000000000011e-227

   1. Initial program 28.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{c0}{2 \cdot w}\right), \color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \left(2 \cdot w\right)\right), \left(\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{+.f64}\left(\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right), \color{blue}{\left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}\right)\right) \]

   3. Simplified 29.9%

      \[\leadsto \color{blue}{\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)}}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)} - M \cdot M}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{{c0}^{2} \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right) \cdot {c0}^{2}}{w} \]

      2. distribute-lft1-in N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right) \cdot {c0}^{2}}{w} \]

      3. metadata-eval N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\left(0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right) \cdot {c0}^{2}}{w} \]

      4. mul0-lft N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{0 \cdot {c0}^{2}}{w} \]

      5. metadata-eval N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\left(\frac{-1}{2} \cdot 0\right) \cdot {c0}^{2}}{w} \]

      6. mul0-lft N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\left(\frac{-1}{2} \cdot \left(0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot {c0}^{2}}{w} \]

      7. metadata-eval N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\left(\frac{-1}{2} \cdot \left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot {c0}^{2}}{w} \]

      8. distribute-lft1-in N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\left(\frac{-1}{2} \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot {c0}^{2}}{w} \]

      9. associate-*l/ N/A

         \[\leadsto \frac{-1}{2} \cdot \left(\frac{\frac{-1}{2} \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w} \cdot \color{blue}{{c0}^{2}}\right) \]

   7. Simplified 30.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{\left(c0 \cdot c0\right) \cdot 0}{w}} \]
   8. Step-by-step derivation

      1. mul0-rgt N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{0}{w} \]

      2. div0 N/A

         \[\leadsto \frac{-1}{2} \cdot 0 \]

      3. metadata-eval 38.8%

         \[\leadsto 0 \]

   9. Applied egg-rr 38.8%

      \[\leadsto \color{blue}{0} \]

   if 2.60000000000000011e-227 < M < 5.5000000000000001e-186

   1. Initial program 41.4%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{c0}{2 \cdot w}\right), \color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \left(2 \cdot w\right)\right), \left(\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{+.f64}\left(\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right), \color{blue}{\left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}\right)\right) \]

   3. Simplified 47.5%

      \[\leadsto \color{blue}{\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)}}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)} - M \cdot M}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\color{blue}{{D}^{2} \cdot \left(h \cdot w\right)}}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\left(h \cdot w\right) \cdot \color{blue}{{D}^{2}}}\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{h \cdot w}}{\color{blue}{{D}^{2}}}\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{2 \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}}{{\color{blue}{D}}^{2}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(2 \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right), \color{blue}{\left({D}^{2}\right)}\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(2 \cdot \frac{\frac{c0 \cdot {d}^{2}}{h}}{w}\right), \left({D}^{2}\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{c0 \cdot {d}^{2}}{h}}{w}\right), \left({\color{blue}{D}}^{2}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \frac{c0 \cdot {d}^{2}}{h}\right), w\right), \left({\color{blue}{D}}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{c0 \cdot {d}^{2}}{h}\right)\right), w\right), \left({D}^{2}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(c0 \cdot {d}^{2}\right), h\right)\right), w\right), \left({D}^{2}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left({d}^{2}\right)\right), h\right)\right), w\right), \left({D}^{2}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(d \cdot d\right)\right), h\right)\right), w\right), \left({D}^{2}\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right), h\right)\right), w\right), \left({D}^{2}\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right), h\right)\right), w\right), \left(D \cdot \color{blue}{D}\right)\right)\right) \]

      15. *-lowering-*.f64 48.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right), h\right)\right), w\right), \mathsf{*.f64}\left(D, \color{blue}{D}\right)\right)\right) \]

   7. Simplified 48.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{\frac{2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{h}}{w}}{D \cdot D}} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{h}}{w}}{\color{blue}{D \cdot D}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{h}}{w}}{D}}{\color{blue}{D}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{h}}{w}}{D}\right), \color{blue}{D}\right) \]

   9. Applied egg-rr 53.8%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{c0}{2 \cdot w}}{\frac{w}{2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{h}}}}{D}}{D}} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(w, \mathsf{*.f64}\left(2, \left(\frac{\left(c0 \cdot d\right) \cdot d}{h}\right)\right)\right)\right), D\right), D\right) \]

       2. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(w, \mathsf{*.f64}\left(2, \left(\left(c0 \cdot d\right) \cdot \frac{d}{h}\right)\right)\right)\right), D\right), D\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(w, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(c0 \cdot d\right), \left(\frac{d}{h}\right)\right)\right)\right)\right), D\right), D\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(w, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, d\right), \left(\frac{d}{h}\right)\right)\right)\right)\right), D\right), D\right) \]

       5. /-lowering-/.f64 59.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(w, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, d\right), \mathsf{/.f64}\left(d, h\right)\right)\right)\right)\right), D\right), D\right) \]

   11. Applied egg-rr 59.3%

       \[\leadsto \frac{\frac{\frac{\frac{c0}{2 \cdot w}}{\frac{w}{2 \cdot \color{blue}{\left(\left(c0 \cdot d\right) \cdot \frac{d}{h}\right)}}}}{D}}{D} \]

   12. Taylor expanded in c0 around 0

       \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]
   13. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{{c0}^{2} \cdot \left(d \cdot d\right)}{{D}^{\color{blue}{2}} \cdot \left(h \cdot {w}^{2}\right)} \]

       2. associate-*r* N/A

          \[\leadsto \frac{\left({c0}^{2} \cdot d\right) \cdot d}{\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)} \]

       3. unpow2 N/A

          \[\leadsto \frac{\left({c0}^{2} \cdot d\right) \cdot d}{\left(D \cdot D\right) \cdot \left(\color{blue}{h} \cdot {w}^{2}\right)} \]

       4. associate-*l* N/A

          \[\leadsto \frac{\left({c0}^{2} \cdot d\right) \cdot d}{D \cdot \color{blue}{\left(D \cdot \left(h \cdot {w}^{2}\right)\right)}} \]

       5. *-commutative N/A

          \[\leadsto \frac{\left({c0}^{2} \cdot d\right) \cdot d}{\left(D \cdot \left(h \cdot {w}^{2}\right)\right) \cdot \color{blue}{D}} \]

       6. times-frac N/A

          \[\leadsto \frac{{c0}^{2} \cdot d}{D \cdot \left(h \cdot {w}^{2}\right)} \cdot \color{blue}{\frac{d}{D}} \]

       7. *-commutative N/A

          \[\leadsto \frac{d \cdot {c0}^{2}}{D \cdot \left(h \cdot {w}^{2}\right)} \cdot \frac{d}{D} \]

       8. associate-*r/ N/A

          \[\leadsto \frac{\frac{d \cdot {c0}^{2}}{D \cdot \left(h \cdot {w}^{2}\right)} \cdot d}{\color{blue}{D}} \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{d \cdot {c0}^{2}}{D \cdot \left(h \cdot {w}^{2}\right)} \cdot d\right), \color{blue}{D}\right) \]

   14. Simplified 71.1%

       \[\leadsto \color{blue}{\frac{\left(d \cdot \frac{\frac{\frac{\frac{c0 \cdot c0}{w}}{w}}{h}}{D}\right) \cdot d}{D}} \]

   if 5.5000000000000001e-186 < M < 1.89999999999999988e-29

   1. Initial program 23.3%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{c0}{2 \cdot w}\right), \color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \left(2 \cdot w\right)\right), \left(\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{+.f64}\left(\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right), \color{blue}{\left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}\right)\right) \]

   3. Simplified 23.1%

      \[\leadsto \color{blue}{\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)}}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)} - M \cdot M}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({c0}^{2}\right), \color{blue}{\left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(c0 \cdot c0\right), \left(\color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right), \color{blue}{\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)}\right)\right) \]

   7. Simplified 27.6%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \left(\frac{0}{w} + \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}\right)} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right), \color{blue}{\left({d}^{2}\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({D}^{2}\right), \left({M}^{2} \cdot h\right)\right), \left({\color{blue}{d}}^{2}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(D \cdot D\right), \left({M}^{2} \cdot h\right)\right), \left({d}^{2}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left({M}^{2} \cdot h\right)\right), \left({d}^{2}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot {M}^{2}\right)\right), \left({d}^{2}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right)\right) \]

      11. *-lowering-*.f64 55.2%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right)\right) \]

   10. Simplified 55.2%

       \[\leadsto \color{blue}{0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\color{blue}{d \cdot d}} \]

       2. times-frac N/A

          \[\leadsto \frac{\frac{1}{4}}{d} \cdot \color{blue}{\frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d}} \]

       3. clear-num N/A

          \[\leadsto \frac{\frac{1}{4}}{d} \cdot \frac{1}{\color{blue}{\frac{d}{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}}} \]

       4. un-div-inv N/A

          \[\leadsto \frac{\frac{\frac{1}{4}}{d}}{\color{blue}{\frac{d}{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}}} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{4}}{d}\right), \color{blue}{\left(\frac{d}{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}\right)}\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, d\right), \left(\frac{\color{blue}{d}}{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, d\right), \mathsf{/.f64}\left(d, \color{blue}{\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, d\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(\left(D \cdot D\right), \color{blue}{\left(h \cdot \left(M \cdot M\right)\right)}\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, d\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{h} \cdot \left(M \cdot M\right)\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, d\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \color{blue}{\left(M \cdot M\right)}\right)\right)\right)\right) \]

       11. *-lowering-*.f64 56.5%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, d\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \color{blue}{M}\right)\right)\right)\right)\right) \]

   12. Applied egg-rr 56.5%

       \[\leadsto \color{blue}{\frac{\frac{0.25}{d}}{\frac{d}{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}}} \]

   if 1.89999999999999988e-29 < M

   1. Initial program 20.7%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{c0}{2 \cdot w}\right), \color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \left(2 \cdot w\right)\right), \left(\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{+.f64}\left(\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right), \color{blue}{\left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}\right)\right) \]

   3. Simplified 20.6%

      \[\leadsto \color{blue}{\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)}}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)} - M \cdot M}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\color{blue}{{D}^{2} \cdot \left(h \cdot w\right)}}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\left(h \cdot w\right) \cdot \color{blue}{{D}^{2}}}\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{h \cdot w}}{\color{blue}{{D}^{2}}}\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{2 \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}}{{\color{blue}{D}}^{2}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(2 \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right), \color{blue}{\left({D}^{2}\right)}\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(2 \cdot \frac{\frac{c0 \cdot {d}^{2}}{h}}{w}\right), \left({D}^{2}\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{c0 \cdot {d}^{2}}{h}}{w}\right), \left({\color{blue}{D}}^{2}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \frac{c0 \cdot {d}^{2}}{h}\right), w\right), \left({\color{blue}{D}}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{c0 \cdot {d}^{2}}{h}\right)\right), w\right), \left({D}^{2}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(c0 \cdot {d}^{2}\right), h\right)\right), w\right), \left({D}^{2}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left({d}^{2}\right)\right), h\right)\right), w\right), \left({D}^{2}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(d \cdot d\right)\right), h\right)\right), w\right), \left({D}^{2}\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right), h\right)\right), w\right), \left({D}^{2}\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right), h\right)\right), w\right), \left(D \cdot \color{blue}{D}\right)\right)\right) \]

      15. *-lowering-*.f64 39.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right), h\right)\right), w\right), \mathsf{*.f64}\left(D, \color{blue}{D}\right)\right)\right) \]

   7. Simplified 39.7%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{\frac{2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{h}}{w}}{D \cdot D}} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{h}}{w}}{\color{blue}{D \cdot D}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{h}}{w}}{D}}{\color{blue}{D}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{h}}{w}}{D}\right), \color{blue}{D}\right) \]

   9. Applied egg-rr 45.2%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{c0}{2 \cdot w}}{\frac{w}{2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{h}}}}{D}}{D}} \]

   10. Taylor expanded in c0 around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{{c0}^{2} \cdot {d}^{2}}{D \cdot \left(h \cdot {w}^{2}\right)}\right)}, D\right) \]
   11. Step-by-step derivation

       1. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{{c0}^{2} \cdot {d}^{2}}{h \cdot {w}^{2}}}{D}\right), D\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{{c0}^{2} \cdot {d}^{2}}{h \cdot {w}^{2}}\right), D\right), D\right) \]

   12. Simplified 46.3%

       \[\leadsto \frac{\color{blue}{\frac{\frac{c0 \cdot \frac{\frac{d \cdot \left(c0 \cdot d\right)}{w}}{w}}{h}}{D}}}{D} \]
3. Recombined 4 regimes into one program.

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.6 \cdot 10^{-227}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 5.5 \cdot 10^{-186}:\\ \;\;\;\;\frac{d \cdot \left(d \cdot \frac{\frac{\frac{\frac{c0 \cdot c0}{w}}{w}}{h}}{D}\right)}{D}\\ \mathbf{elif}\;M \leq 1.9 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{0.25}{d}}{\frac{d}{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{c0 \cdot \frac{\frac{d \cdot \left(c0 \cdot d\right)}{w}}{w}}{h}}{D}}{D}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 47.7% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 3.5 \cdot 10^{+123}:\\ \;\;\;\;0.25 \cdot \left(\frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d} \cdot \frac{D}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \frac{\frac{h \cdot \left(M \cdot M\right)}{d}}{d}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= M 3.5e+123)
   (* 0.25 (* (/ (* h (* D (* M M))) d) (/ D d)))
   (* 0.25 (* D (* D (/ (/ (* h (* M M)) d) d))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 3.5e+123) {
		tmp = 0.25 * (((h * (D * (M * M))) / d) * (D / d));
	} else {
		tmp = 0.25 * (D * (D * (((h * (M * M)) / d) / 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 <= 3.5d+123) then
        tmp = 0.25d0 * (((h * (d * (m * m))) / d_1) * (d / d_1))
    else
        tmp = 0.25d0 * (d * (d * (((h * (m * m)) / d_1) / d_1)))
    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 <= 3.5e+123) {
		tmp = 0.25 * (((h * (D * (M * M))) / d) * (D / d));
	} else {
		tmp = 0.25 * (D * (D * (((h * (M * M)) / d) / d)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if M <= 3.5e+123:
		tmp = 0.25 * (((h * (D * (M * M))) / d) * (D / d))
	else:
		tmp = 0.25 * (D * (D * (((h * (M * M)) / d) / d)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (M <= 3.5e+123)
		tmp = Float64(0.25 * Float64(Float64(Float64(h * Float64(D * Float64(M * M))) / d) * Float64(D / d)));
	else
		tmp = Float64(0.25 * Float64(D * Float64(D * Float64(Float64(Float64(h * Float64(M * M)) / d) / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (M <= 3.5e+123)
		tmp = 0.25 * (((h * (D * (M * M))) / d) * (D / d));
	else
		tmp = 0.25 * (D * (D * (((h * (M * M)) / d) / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[M, 3.5e+123], N[(0.25 * N[(N[(N[(h * N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(D * N[(D * N[(N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3.5e123

   1. Initial program 29.1%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{c0}{2 \cdot w}\right), \color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \left(2 \cdot w\right)\right), \left(\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{+.f64}\left(\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right), \color{blue}{\left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}\right)\right) \]

   3. Simplified 30.4%

      \[\leadsto \color{blue}{\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)}}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)} - M \cdot M}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({c0}^{2}\right), \color{blue}{\left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(c0 \cdot c0\right), \left(\color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right), \color{blue}{\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)}\right)\right) \]

   7. Simplified 17.8%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \left(\frac{0}{w} + \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}\right)} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right), \color{blue}{\left({d}^{2}\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({D}^{2}\right), \left({M}^{2} \cdot h\right)\right), \left({\color{blue}{d}}^{2}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(D \cdot D\right), \left({M}^{2} \cdot h\right)\right), \left({d}^{2}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left({M}^{2} \cdot h\right)\right), \left({d}^{2}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot {M}^{2}\right)\right), \left({d}^{2}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right)\right) \]

      11. *-lowering-*.f64 40.6%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right)\right) \]

   10. Simplified 40.6%

       \[\leadsto \color{blue}{0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \]
   11. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left(\frac{D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\color{blue}{d} \cdot d}\right)\right) \]

       2. times-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left(\frac{D}{d} \cdot \color{blue}{\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d}}\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left(\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d} \cdot \color{blue}{\frac{D}{d}}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d}\right), \color{blue}{\left(\frac{D}{d}\right)}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right), d\right), \left(\frac{\color{blue}{D}}{d}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(D \cdot \left(\left(M \cdot M\right) \cdot h\right)\right), d\right), \left(\frac{D}{d}\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(D \cdot \left(M \cdot M\right)\right) \cdot h\right), d\right), \left(\frac{D}{d}\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(h \cdot \left(D \cdot \left(M \cdot M\right)\right)\right), d\right), \left(\frac{D}{d}\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(D \cdot \left(M \cdot M\right)\right)\right), d\right), \left(\frac{D}{d}\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, \left(M \cdot M\right)\right)\right), d\right), \left(\frac{D}{d}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, M\right)\right)\right), d\right), \left(\frac{D}{d}\right)\right)\right) \]

       12. /-lowering-/.f64 49.4%

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, M\right)\right)\right), d\right), \mathsf{/.f64}\left(D, \color{blue}{d}\right)\right)\right) \]

   12. Applied egg-rr 49.4%

       \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{h \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d} \cdot \frac{D}{d}\right)} \]

   if 3.5e123 < M

   1. Initial program 5.3%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{c0}{2 \cdot w}\right), \color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \left(2 \cdot w\right)\right), \left(\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{+.f64}\left(\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right), \color{blue}{\left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}\right)\right) \]

   3. Simplified 5.3%

      \[\leadsto \color{blue}{\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)}}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)} - M \cdot M}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({c0}^{2}\right), \color{blue}{\left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(c0 \cdot c0\right), \left(\color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right), \color{blue}{\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)}\right)\right) \]

   7. Simplified 0.9%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \left(\frac{0}{w} + \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}\right)} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right), \color{blue}{\left({d}^{2}\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({D}^{2}\right), \left({M}^{2} \cdot h\right)\right), \left({\color{blue}{d}}^{2}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(D \cdot D\right), \left({M}^{2} \cdot h\right)\right), \left({d}^{2}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left({M}^{2} \cdot h\right)\right), \left({d}^{2}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot {M}^{2}\right)\right), \left({d}^{2}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right)\right) \]

      11. *-lowering-*.f64 32.5%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right)\right) \]

   10. Simplified 32.5%

       \[\leadsto \color{blue}{0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \]

   11. Taylor expanded in D around 0

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right)}\right) \]
   12. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot \color{blue}{\frac{{M}^{2} \cdot h}{{d}^{2}}}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left(\left(D \cdot D\right) \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{{d}^{2}}\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \color{blue}{\left(D \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \color{blue}{\left(D \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \color{blue}{\left(\frac{{M}^{2} \cdot h}{{d}^{2}}\right)}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(\frac{{M}^{2} \cdot h}{d \cdot \color{blue}{d}}\right)\right)\right)\right) \]

       7. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(\frac{\frac{{M}^{2} \cdot h}{d}}{\color{blue}{d}}\right)\right)\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\left(\frac{{M}^{2} \cdot h}{d}\right), \color{blue}{d}\right)\right)\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({M}^{2} \cdot h\right), d\right), d\right)\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(h \cdot {M}^{2}\right), d\right), d\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left({M}^{2}\right)\right), d\right), d\right)\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(M \cdot M\right)\right), d\right), d\right)\right)\right)\right) \]

       13. *-lowering-*.f64 64.8%

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), d\right), d\right)\right)\right)\right) \]

   13. Simplified 64.8%

       \[\leadsto 0.25 \cdot \color{blue}{\left(D \cdot \left(D \cdot \frac{\frac{h \cdot \left(M \cdot M\right)}{d}}{d}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 47.6% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ 0.25 \cdot \left(D \cdot \left(D \cdot \frac{\frac{h \cdot \left(M \cdot M\right)}{d}}{d}\right)\right) \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (* 0.25 (* D (* D (/ (/ (* h (* M M)) d) d)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return 0.25 * (D * (D * (((h * (M * M)) / d) / d)));
}

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 = 0.25d0 * (d * (d * (((h * (m * m)) / d_1) / d_1)))
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return 0.25 * (D * (D * (((h * (M * M)) / d) / d)));
}

Python

def code(c0, w, h, D, d, M):
	return 0.25 * (D * (D * (((h * (M * M)) / d) / d)))

Julia

function code(c0, w, h, D, d, M)
	return Float64(0.25 * Float64(D * Float64(D * Float64(Float64(Float64(h * Float64(M * M)) / d) / d))))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = 0.25 * (D * (D * (((h * (M * M)) / d) / d)));
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(0.25 * N[(D * N[(D * N[(N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 27.3%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
2. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{c0}{2 \cdot w}\right), \color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \left(2 \cdot w\right)\right), \left(\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{+.f64}\left(\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right), \color{blue}{\left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}\right)\right) \]

3. Simplified 28.5%

   \[\leadsto \color{blue}{\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)}}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)} - M \cdot M}\right)} \]
4. Add Preprocessing

5. Taylor expanded in c0 around -inf

   \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({c0}^{2}\right), \color{blue}{\left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(c0 \cdot c0\right), \left(\color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right), \color{blue}{\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)}\right)\right) \]

7. Simplified 16.6%

   \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \left(\frac{0}{w} + \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}\right)} \]

8. Taylor expanded in c0 around 0

   \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
9. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right)}\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right), \color{blue}{\left({d}^{2}\right)}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({D}^{2}\right), \left({M}^{2} \cdot h\right)\right), \left({\color{blue}{d}}^{2}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(D \cdot D\right), \left({M}^{2} \cdot h\right)\right), \left({d}^{2}\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left({M}^{2} \cdot h\right)\right), \left({d}^{2}\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot {M}^{2}\right)\right), \left({d}^{2}\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \left({d}^{2}\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right)\right) \]

   11. *-lowering-*.f64 40.0%

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right)\right) \]

10. Simplified 40.0%

    \[\leadsto \color{blue}{0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \]

11. Taylor expanded in D around 0

    \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right)}\right) \]
12. Step-by-step derivation

    1. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot \color{blue}{\frac{{M}^{2} \cdot h}{{d}^{2}}}\right)\right) \]

    2. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left(\left(D \cdot D\right) \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{{d}^{2}}\right)\right) \]

    3. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \color{blue}{\left(D \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)}\right)\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \color{blue}{\left(D \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)}\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \color{blue}{\left(\frac{{M}^{2} \cdot h}{{d}^{2}}\right)}\right)\right)\right) \]

    6. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(\frac{{M}^{2} \cdot h}{d \cdot \color{blue}{d}}\right)\right)\right)\right) \]

    7. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(\frac{\frac{{M}^{2} \cdot h}{d}}{\color{blue}{d}}\right)\right)\right)\right) \]

    8. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\left(\frac{{M}^{2} \cdot h}{d}\right), \color{blue}{d}\right)\right)\right)\right) \]

    9. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({M}^{2} \cdot h\right), d\right), d\right)\right)\right)\right) \]

    10. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(h \cdot {M}^{2}\right), d\right), d\right)\right)\right)\right) \]

    11. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left({M}^{2}\right)\right), d\right), d\right)\right)\right)\right) \]

    12. unpow2 N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(M \cdot M\right)\right), d\right), d\right)\right)\right)\right) \]

    13. *-lowering-*.f64 50.3%

        \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), d\right), d\right)\right)\right)\right) \]

13. Simplified 50.3%

    \[\leadsto 0.25 \cdot \color{blue}{\left(D \cdot \left(D \cdot \frac{\frac{h \cdot \left(M \cdot M\right)}{d}}{d}\right)\right)} \]
14. Add Preprocessing

Alternative 10: 34.2% 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 27.3%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
2. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{c0}{2 \cdot w}\right), \color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \left(2 \cdot w\right)\right), \left(\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{+.f64}\left(\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right), \color{blue}{\left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}\right)\right) \]

3. Simplified 28.5%

   \[\leadsto \color{blue}{\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)}}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)} - M \cdot M}\right)} \]
4. Add Preprocessing

5. Taylor expanded in c0 around -inf

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{{c0}^{2} \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{-1}{2} \cdot \frac{\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right) \cdot {c0}^{2}}{w} \]

   2. distribute-lft1-in N/A

      \[\leadsto \frac{-1}{2} \cdot \frac{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right) \cdot {c0}^{2}}{w} \]

   3. metadata-eval N/A

      \[\leadsto \frac{-1}{2} \cdot \frac{\left(0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right) \cdot {c0}^{2}}{w} \]

   4. mul0-lft N/A

      \[\leadsto \frac{-1}{2} \cdot \frac{0 \cdot {c0}^{2}}{w} \]

   5. metadata-eval N/A

      \[\leadsto \frac{-1}{2} \cdot \frac{\left(\frac{-1}{2} \cdot 0\right) \cdot {c0}^{2}}{w} \]

   6. mul0-lft N/A

      \[\leadsto \frac{-1}{2} \cdot \frac{\left(\frac{-1}{2} \cdot \left(0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot {c0}^{2}}{w} \]

   7. metadata-eval N/A

      \[\leadsto \frac{-1}{2} \cdot \frac{\left(\frac{-1}{2} \cdot \left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot {c0}^{2}}{w} \]

   8. distribute-lft1-in N/A

      \[\leadsto \frac{-1}{2} \cdot \frac{\left(\frac{-1}{2} \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot {c0}^{2}}{w} \]

   9. associate-*l/ N/A

      \[\leadsto \frac{-1}{2} \cdot \left(\frac{\frac{-1}{2} \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w} \cdot \color{blue}{{c0}^{2}}\right) \]

7. Simplified 27.1%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{\left(c0 \cdot c0\right) \cdot 0}{w}} \]
8. Step-by-step derivation

   1. mul0-rgt N/A

      \[\leadsto \frac{-1}{2} \cdot \frac{0}{w} \]

   2. div0 N/A

      \[\leadsto \frac{-1}{2} \cdot 0 \]

   3. metadata-eval 36.6%

      \[\leadsto 0 \]

9. Applied egg-rr 36.6%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024138 
(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))))))