Henrywood and Agarwal, Equation (13)

Percentage Accurate: 25.0% → 68.6%

Time: 21.3s

Alternatives: 12

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: 25.0% 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: 68.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{\frac{d \cdot \frac{c0}{w \cdot h}}{D}}{w \cdot \frac{D}{c0 \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{h}{d} \cdot \left(-0.25 \cdot \left(\left(M \cdot \left(D \cdot M\right)\right) \cdot \left(0 - D\right)\right)\right)}{d}\\ \end{array} \end{array} \]

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)
     (/ (/ (* d (/ c0 (* w h))) D) (* w (/ D (* c0 d))))
     (/ (* (/ h d) (* -0.25 (* (* M (* D M)) (- 0.0 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 = ((d * (c0 / (w * h))) / D) / (w * (D / (c0 * d)));
	} else {
		tmp = ((h / d) * (-0.25 * ((M * (D * M)) * (0.0 - 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 = ((d * (c0 / (w * h))) / D) / (w * (D / (c0 * d)));
	} else {
		tmp = ((h / d) * (-0.25 * ((M * (D * M)) * (0.0 - 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 = ((d * (c0 / (w * h))) / D) / (w * (D / (c0 * d)))
	else:
		tmp = ((h / d) * (-0.25 * ((M * (D * M)) * (0.0 - 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(d * Float64(c0 / Float64(w * h))) / D) / Float64(w * Float64(D / Float64(c0 * d))));
	else
		tmp = Float64(Float64(Float64(h / d) * Float64(-0.25 * Float64(Float64(M * Float64(D * M)) * Float64(0.0 - 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 = ((d * (c0 / (w * h))) / D) / (w * (D / (c0 * d)));
	else
		tmp = ((h / d) * (-0.25 * ((M * (D * M)) * (0.0 - 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[(d * N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision] / N[(w * N[(D / N[(c0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(h / d), $MachinePrecision] * N[(-0.25 * N[(N[(M * N[(D * M), $MachinePrecision]), $MachinePrecision] * N[(0.0 - D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $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 69.8%

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

   3. Taylor expanded in c0 around inf

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. unpow2 N/A

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 53.5%

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

   5. Simplified 53.5%

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

      1. unswap-sqr N/A

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

      2. associate-*l* N/A

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

      3. times-frac N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

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

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

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

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 72.9%

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

   7. Applied egg-rr 72.9%

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

      1. associate-/r* N/A

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

      2. div-inv N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

      5. associate-*r/ N/A

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

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

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

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

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

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

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

      11. *-commutative N/A

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

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

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

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

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

      14. /-lowering-/.f64 79.0%

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

   9. Applied egg-rr 79.0%

      \[\leadsto \frac{d \cdot c0}{D} \cdot \color{blue}{\left(\frac{d}{\frac{w \cdot h}{c0}} \cdot \frac{\frac{1}{D}}{w}\right)} \]
   10. Step-by-step derivation

       1. clear-num N/A

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

       2. associate-*r/ N/A

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

       3. frac-times N/A

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

       4. *-lft-identity N/A

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

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

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

       6. un-div-inv N/A

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

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

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

       8. div-inv N/A

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

       9. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

       15. *-lowering-*.f64 80.3%

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

   11. Applied egg-rr 80.3%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   4. 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. +-commutative N/A

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

      5. distribute-lft1-in N/A

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

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0}{w}\right)\right) \]

      8. div0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot 0\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + 0\right)\right) \]

      10. metadata-eval N/A

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

   5. Simplified 16.4%

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

   6. Taylor expanded in c0 around 0

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

      1. associate-*r/ N/A

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

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

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

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

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

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

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

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

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

      17. *-commutative N/A

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

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

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

      19. unpow2 N/A

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

      20. *-lowering-*.f64 53.2%

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

   8. Simplified 53.2%

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

      1. associate-/r* N/A

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

      2. frac-2neg N/A

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

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

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

   10. Applied egg-rr 62.1%

       \[\leadsto \color{blue}{\frac{\left(\left(D \cdot \left(\left(D \cdot M\right) \cdot M\right)\right) \cdot -0.25\right) \cdot \frac{h}{d}}{0 - d}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{\frac{d \cdot \frac{c0}{w \cdot h}}{D}}{w \cdot \frac{D}{c0 \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{h}{d} \cdot \left(-0.25 \cdot \left(\left(M \cdot \left(D \cdot M\right)\right) \cdot \left(0 - D\right)\right)\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 48.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{D \cdot \left(0.25 \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d}}{d}\\ t_1 := \frac{c0 \cdot d}{D}\\ \mathbf{if}\;w \leq -1.6 \cdot 10^{+117}:\\ \;\;\;\;\frac{\frac{h}{d} \cdot \left(-0.25 \cdot \left(\left(M \cdot \left(D \cdot M\right)\right) \cdot \left(0 - D\right)\right)\right)}{d}\\ \mathbf{elif}\;w \leq -1.7 \cdot 10^{+44}:\\ \;\;\;\;t\_1 \cdot \frac{c0 \cdot d}{w \cdot \left(\left(w \cdot h\right) \cdot D\right)}\\ \mathbf{elif}\;w \leq -6.8 \cdot 10^{-130}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;w \leq -1.26 \cdot 10^{-228}:\\ \;\;\;\;t\_1 \cdot \left(\frac{d}{w \cdot h} \cdot \frac{\frac{c0}{D}}{w}\right)\\ \mathbf{elif}\;w \leq 9 \cdot 10^{-190}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;w \leq 3.9 \cdot 10^{+126}:\\ \;\;\;\;t\_1 \cdot \frac{\frac{\frac{d}{D}}{w}}{\frac{w \cdot h}{c0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{h}{d} \cdot \frac{M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (/ (* D (* 0.25 (* D (* h (* M M))))) d) d))
        (t_1 (/ (* c0 d) D)))
   (if (<= w -1.6e+117)
     (/ (* (/ h d) (* -0.25 (* (* M (* D M)) (- 0.0 D)))) d)
     (if (<= w -1.7e+44)
       (* t_1 (/ (* c0 d) (* w (* (* w h) D))))
       (if (<= w -6.8e-130)
         t_0
         (if (<= w -1.26e-228)
           (* t_1 (* (/ d (* w h)) (/ (/ c0 D) w)))
           (if (<= w 9e-190)
             t_0
             (if (<= w 3.9e+126)
               (* t_1 (/ (/ (/ d D) w) (/ (* w h) c0)))
               (* (/ h d) (/ (* M (* 0.25 (* D (* D M)))) d))))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((D * (0.25 * (D * (h * (M * M))))) / d) / d;
	double t_1 = (c0 * d) / D;
	double tmp;
	if (w <= -1.6e+117) {
		tmp = ((h / d) * (-0.25 * ((M * (D * M)) * (0.0 - D)))) / d;
	} else if (w <= -1.7e+44) {
		tmp = t_1 * ((c0 * d) / (w * ((w * h) * D)));
	} else if (w <= -6.8e-130) {
		tmp = t_0;
	} else if (w <= -1.26e-228) {
		tmp = t_1 * ((d / (w * h)) * ((c0 / D) / w));
	} else if (w <= 9e-190) {
		tmp = t_0;
	} else if (w <= 3.9e+126) {
		tmp = t_1 * (((d / D) / w) / ((w * h) / c0));
	} else {
		tmp = (h / d) * ((M * (0.25 * (D * (D * M)))) / 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 = ((d * (0.25d0 * (d * (h * (m * m))))) / d_1) / d_1
    t_1 = (c0 * d_1) / d
    if (w <= (-1.6d+117)) then
        tmp = ((h / d_1) * ((-0.25d0) * ((m * (d * m)) * (0.0d0 - d)))) / d_1
    else if (w <= (-1.7d+44)) then
        tmp = t_1 * ((c0 * d_1) / (w * ((w * h) * d)))
    else if (w <= (-6.8d-130)) then
        tmp = t_0
    else if (w <= (-1.26d-228)) then
        tmp = t_1 * ((d_1 / (w * h)) * ((c0 / d) / w))
    else if (w <= 9d-190) then
        tmp = t_0
    else if (w <= 3.9d+126) then
        tmp = t_1 * (((d_1 / d) / w) / ((w * h) / c0))
    else
        tmp = (h / d_1) * ((m * (0.25d0 * (d * (d * m)))) / 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 * (0.25 * (D * (h * (M * M))))) / d) / d;
	double t_1 = (c0 * d) / D;
	double tmp;
	if (w <= -1.6e+117) {
		tmp = ((h / d) * (-0.25 * ((M * (D * M)) * (0.0 - D)))) / d;
	} else if (w <= -1.7e+44) {
		tmp = t_1 * ((c0 * d) / (w * ((w * h) * D)));
	} else if (w <= -6.8e-130) {
		tmp = t_0;
	} else if (w <= -1.26e-228) {
		tmp = t_1 * ((d / (w * h)) * ((c0 / D) / w));
	} else if (w <= 9e-190) {
		tmp = t_0;
	} else if (w <= 3.9e+126) {
		tmp = t_1 * (((d / D) / w) / ((w * h) / c0));
	} else {
		tmp = (h / d) * ((M * (0.25 * (D * (D * M)))) / d);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = ((D * (0.25 * (D * (h * (M * M))))) / d) / d
	t_1 = (c0 * d) / D
	tmp = 0
	if w <= -1.6e+117:
		tmp = ((h / d) * (-0.25 * ((M * (D * M)) * (0.0 - D)))) / d
	elif w <= -1.7e+44:
		tmp = t_1 * ((c0 * d) / (w * ((w * h) * D)))
	elif w <= -6.8e-130:
		tmp = t_0
	elif w <= -1.26e-228:
		tmp = t_1 * ((d / (w * h)) * ((c0 / D) / w))
	elif w <= 9e-190:
		tmp = t_0
	elif w <= 3.9e+126:
		tmp = t_1 * (((d / D) / w) / ((w * h) / c0))
	else:
		tmp = (h / d) * ((M * (0.25 * (D * (D * M)))) / d)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(Float64(D * Float64(0.25 * Float64(D * Float64(h * Float64(M * M))))) / d) / d)
	t_1 = Float64(Float64(c0 * d) / D)
	tmp = 0.0
	if (w <= -1.6e+117)
		tmp = Float64(Float64(Float64(h / d) * Float64(-0.25 * Float64(Float64(M * Float64(D * M)) * Float64(0.0 - D)))) / d);
	elseif (w <= -1.7e+44)
		tmp = Float64(t_1 * Float64(Float64(c0 * d) / Float64(w * Float64(Float64(w * h) * D))));
	elseif (w <= -6.8e-130)
		tmp = t_0;
	elseif (w <= -1.26e-228)
		tmp = Float64(t_1 * Float64(Float64(d / Float64(w * h)) * Float64(Float64(c0 / D) / w)));
	elseif (w <= 9e-190)
		tmp = t_0;
	elseif (w <= 3.9e+126)
		tmp = Float64(t_1 * Float64(Float64(Float64(d / D) / w) / Float64(Float64(w * h) / c0)));
	else
		tmp = Float64(Float64(h / d) * Float64(Float64(M * Float64(0.25 * Float64(D * Float64(D * M)))) / d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = ((D * (0.25 * (D * (h * (M * M))))) / d) / d;
	t_1 = (c0 * d) / D;
	tmp = 0.0;
	if (w <= -1.6e+117)
		tmp = ((h / d) * (-0.25 * ((M * (D * M)) * (0.0 - D)))) / d;
	elseif (w <= -1.7e+44)
		tmp = t_1 * ((c0 * d) / (w * ((w * h) * D)));
	elseif (w <= -6.8e-130)
		tmp = t_0;
	elseif (w <= -1.26e-228)
		tmp = t_1 * ((d / (w * h)) * ((c0 / D) / w));
	elseif (w <= 9e-190)
		tmp = t_0;
	elseif (w <= 3.9e+126)
		tmp = t_1 * (((d / D) / w) / ((w * h) / c0));
	else
		tmp = (h / d) * ((M * (0.25 * (D * (D * M)))) / d);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(N[(D * N[(0.25 * N[(D * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * d), $MachinePrecision] / D), $MachinePrecision]}, If[LessEqual[w, -1.6e+117], N[(N[(N[(h / d), $MachinePrecision] * N[(-0.25 * N[(N[(M * N[(D * M), $MachinePrecision]), $MachinePrecision] * N[(0.0 - D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[w, -1.7e+44], N[(t$95$1 * N[(N[(c0 * d), $MachinePrecision] / N[(w * N[(N[(w * h), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, -6.8e-130], t$95$0, If[LessEqual[w, -1.26e-228], N[(t$95$1 * N[(N[(d / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[(N[(c0 / D), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 9e-190], t$95$0, If[LessEqual[w, 3.9e+126], N[(t$95$1 * N[(N[(N[(d / D), $MachinePrecision] / w), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(h / d), $MachinePrecision] * N[(N[(M * N[(0.25 * N[(D * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if w < -1.60000000000000002e117

   1. Initial program 3.7%

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

   3. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   4. 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. +-commutative N/A

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

      5. distribute-lft1-in N/A

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

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0}{w}\right)\right) \]

      8. div0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot 0\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + 0\right)\right) \]

      10. metadata-eval N/A

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

   5. Simplified 13.8%

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

   6. Taylor expanded in c0 around 0

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

      1. associate-*r/ N/A

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

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

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

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

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

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

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

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

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

      17. *-commutative N/A

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

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

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

      19. unpow2 N/A

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

      20. *-lowering-*.f64 57.1%

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

   8. Simplified 57.1%

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

      1. associate-/r* N/A

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

      2. frac-2neg N/A

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

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

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

   10. Applied egg-rr 70.3%

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

   if -1.60000000000000002e117 < w < -1.7e44

   1. Initial program 44.2%

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

   3. Taylor expanded in c0 around inf

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. unpow2 N/A

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 38.6%

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

   5. Simplified 38.6%

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

      1. unswap-sqr N/A

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

      2. associate-*l* N/A

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

      3. times-frac N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

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

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

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

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 75.4%

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

   7. Applied egg-rr 75.4%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

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

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

      5. associate-*l* N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 79.9%

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

   9. Applied egg-rr 79.9%

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

   if -1.7e44 < w < -6.8000000000000001e-130 or -1.25999999999999997e-228 < w < 9.00000000000000042e-190

   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. Add Preprocessing

   3. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   4. 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. +-commutative N/A

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

      5. distribute-lft1-in N/A

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

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0}{w}\right)\right) \]

      8. div0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot 0\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + 0\right)\right) \]

      10. metadata-eval N/A

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

   5. Simplified 17.6%

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

   6. Taylor expanded in c0 around 0

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

      1. associate-*r/ N/A

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

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

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

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

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

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

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

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

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

      17. *-commutative N/A

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

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

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

      19. unpow2 N/A

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

      20. *-lowering-*.f64 44.0%

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

   8. Simplified 44.0%

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

   9. Taylor expanded in h around 0

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

       1. associate-*r/ N/A

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

       2. unpow2 N/A

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

       3. associate-/r* N/A

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

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

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

   11. Simplified 62.0%

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

   if -6.8000000000000001e-130 < w < -1.25999999999999997e-228

   1. Initial program 52.5%

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

   3. Taylor expanded in c0 around inf

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. unpow2 N/A

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 52.2%

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

   5. Simplified 52.2%

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

      1. unswap-sqr N/A

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

      2. associate-*l* N/A

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

      3. times-frac N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

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

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

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

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 69.3%

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

   7. Applied egg-rr 69.3%

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

      1. associate-/r* N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. /-lowering-/.f64 76.6%

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

   9. Applied egg-rr 76.6%

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

   if 9.00000000000000042e-190 < w < 3.89999999999999993e126

   1. Initial program 21.1%

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

   3. Taylor expanded in c0 around inf

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. unpow2 N/A

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 26.0%

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

   5. Simplified 26.0%

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

      1. unswap-sqr N/A

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

      2. associate-*l* N/A

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

      3. times-frac N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

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

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

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

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 49.6%

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

   7. Applied egg-rr 49.6%

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

      1. associate-/r* N/A

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

      2. div-inv N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

      5. associate-*r/ N/A

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

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

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

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

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

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

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

      11. *-commutative N/A

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

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

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

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

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

      14. /-lowering-/.f64 52.8%

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

   9. Applied egg-rr 52.8%

      \[\leadsto \frac{d \cdot c0}{D} \cdot \color{blue}{\left(\frac{d}{\frac{w \cdot h}{c0}} \cdot \frac{\frac{1}{D}}{w}\right)} \]
   10. Step-by-step derivation

       1. associate-*l/ N/A

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

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

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

       3. associate-*r/ N/A

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

       4. div-inv N/A

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

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

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

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

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

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

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

       8. *-lowering-*.f64 52.9%

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

   11. Applied egg-rr 52.9%

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

   if 3.89999999999999993e126 < w

   1. Initial program 13.1%

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

   3. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   4. 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. +-commutative N/A

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

      5. distribute-lft1-in N/A

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

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0}{w}\right)\right) \]

      8. div0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot 0\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + 0\right)\right) \]

      10. metadata-eval N/A

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

   5. Simplified 29.7%

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

   6. Taylor expanded in c0 around 0

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

      1. associate-*r/ N/A

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

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

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

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

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

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

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

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

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

      17. *-commutative N/A

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

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

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

      19. unpow2 N/A

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

      20. *-lowering-*.f64 67.2%

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

   8. Simplified 67.2%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. /-lowering-/.f64 68.2%

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

   10. Applied egg-rr 68.2%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -1.6 \cdot 10^{+117}:\\ \;\;\;\;\frac{\frac{h}{d} \cdot \left(-0.25 \cdot \left(\left(M \cdot \left(D \cdot M\right)\right) \cdot \left(0 - D\right)\right)\right)}{d}\\ \mathbf{elif}\;w \leq -1.7 \cdot 10^{+44}:\\ \;\;\;\;\frac{c0 \cdot d}{D} \cdot \frac{c0 \cdot d}{w \cdot \left(\left(w \cdot h\right) \cdot D\right)}\\ \mathbf{elif}\;w \leq -6.8 \cdot 10^{-130}:\\ \;\;\;\;\frac{\frac{D \cdot \left(0.25 \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d}}{d}\\ \mathbf{elif}\;w \leq -1.26 \cdot 10^{-228}:\\ \;\;\;\;\frac{c0 \cdot d}{D} \cdot \left(\frac{d}{w \cdot h} \cdot \frac{\frac{c0}{D}}{w}\right)\\ \mathbf{elif}\;w \leq 9 \cdot 10^{-190}:\\ \;\;\;\;\frac{\frac{D \cdot \left(0.25 \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d}}{d}\\ \mathbf{elif}\;w \leq 3.9 \cdot 10^{+126}:\\ \;\;\;\;\frac{c0 \cdot d}{D} \cdot \frac{\frac{\frac{d}{D}}{w}}{\frac{w \cdot h}{c0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{h}{d} \cdot \frac{M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 48.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{D \cdot \left(0.25 \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d}}{d}\\ t_1 := \frac{c0 \cdot d}{D}\\ t_2 := \frac{M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}\\ \mathbf{if}\;w \leq -1.4 \cdot 10^{+117}:\\ \;\;\;\;h \cdot \frac{t\_2}{d}\\ \mathbf{elif}\;w \leq -7.7 \cdot 10^{+43}:\\ \;\;\;\;t\_1 \cdot \frac{c0 \cdot d}{w \cdot \left(\left(w \cdot h\right) \cdot D\right)}\\ \mathbf{elif}\;w \leq -1 \cdot 10^{-130}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;w \leq -3 \cdot 10^{-229}:\\ \;\;\;\;t\_1 \cdot \left(\frac{d}{w \cdot h} \cdot \frac{\frac{c0}{D}}{w}\right)\\ \mathbf{elif}\;w \leq 2.5 \cdot 10^{-189}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;w \leq 3.2 \cdot 10^{+125}:\\ \;\;\;\;t\_1 \cdot \frac{\frac{\frac{d}{D}}{w}}{\frac{w \cdot h}{c0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{h}{d} \cdot t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (/ (* D (* 0.25 (* D (* h (* M M))))) d) d))
        (t_1 (/ (* c0 d) D))
        (t_2 (/ (* M (* 0.25 (* D (* D M)))) d)))
   (if (<= w -1.4e+117)
     (* h (/ t_2 d))
     (if (<= w -7.7e+43)
       (* t_1 (/ (* c0 d) (* w (* (* w h) D))))
       (if (<= w -1e-130)
         t_0
         (if (<= w -3e-229)
           (* t_1 (* (/ d (* w h)) (/ (/ c0 D) w)))
           (if (<= w 2.5e-189)
             t_0
             (if (<= w 3.2e+125)
               (* t_1 (/ (/ (/ d D) w) (/ (* w h) c0)))
               (* (/ h d) t_2)))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((D * (0.25 * (D * (h * (M * M))))) / d) / d;
	double t_1 = (c0 * d) / D;
	double t_2 = (M * (0.25 * (D * (D * M)))) / d;
	double tmp;
	if (w <= -1.4e+117) {
		tmp = h * (t_2 / d);
	} else if (w <= -7.7e+43) {
		tmp = t_1 * ((c0 * d) / (w * ((w * h) * D)));
	} else if (w <= -1e-130) {
		tmp = t_0;
	} else if (w <= -3e-229) {
		tmp = t_1 * ((d / (w * h)) * ((c0 / D) / w));
	} else if (w <= 2.5e-189) {
		tmp = t_0;
	} else if (w <= 3.2e+125) {
		tmp = t_1 * (((d / D) / w) / ((w * h) / c0));
	} else {
		tmp = (h / d) * t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = ((d * (0.25d0 * (d * (h * (m * m))))) / d_1) / d_1
    t_1 = (c0 * d_1) / d
    t_2 = (m * (0.25d0 * (d * (d * m)))) / d_1
    if (w <= (-1.4d+117)) then
        tmp = h * (t_2 / d_1)
    else if (w <= (-7.7d+43)) then
        tmp = t_1 * ((c0 * d_1) / (w * ((w * h) * d)))
    else if (w <= (-1d-130)) then
        tmp = t_0
    else if (w <= (-3d-229)) then
        tmp = t_1 * ((d_1 / (w * h)) * ((c0 / d) / w))
    else if (w <= 2.5d-189) then
        tmp = t_0
    else if (w <= 3.2d+125) then
        tmp = t_1 * (((d_1 / d) / w) / ((w * h) / c0))
    else
        tmp = (h / d_1) * t_2
    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 * (0.25 * (D * (h * (M * M))))) / d) / d;
	double t_1 = (c0 * d) / D;
	double t_2 = (M * (0.25 * (D * (D * M)))) / d;
	double tmp;
	if (w <= -1.4e+117) {
		tmp = h * (t_2 / d);
	} else if (w <= -7.7e+43) {
		tmp = t_1 * ((c0 * d) / (w * ((w * h) * D)));
	} else if (w <= -1e-130) {
		tmp = t_0;
	} else if (w <= -3e-229) {
		tmp = t_1 * ((d / (w * h)) * ((c0 / D) / w));
	} else if (w <= 2.5e-189) {
		tmp = t_0;
	} else if (w <= 3.2e+125) {
		tmp = t_1 * (((d / D) / w) / ((w * h) / c0));
	} else {
		tmp = (h / d) * t_2;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = ((D * (0.25 * (D * (h * (M * M))))) / d) / d
	t_1 = (c0 * d) / D
	t_2 = (M * (0.25 * (D * (D * M)))) / d
	tmp = 0
	if w <= -1.4e+117:
		tmp = h * (t_2 / d)
	elif w <= -7.7e+43:
		tmp = t_1 * ((c0 * d) / (w * ((w * h) * D)))
	elif w <= -1e-130:
		tmp = t_0
	elif w <= -3e-229:
		tmp = t_1 * ((d / (w * h)) * ((c0 / D) / w))
	elif w <= 2.5e-189:
		tmp = t_0
	elif w <= 3.2e+125:
		tmp = t_1 * (((d / D) / w) / ((w * h) / c0))
	else:
		tmp = (h / d) * t_2
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(Float64(D * Float64(0.25 * Float64(D * Float64(h * Float64(M * M))))) / d) / d)
	t_1 = Float64(Float64(c0 * d) / D)
	t_2 = Float64(Float64(M * Float64(0.25 * Float64(D * Float64(D * M)))) / d)
	tmp = 0.0
	if (w <= -1.4e+117)
		tmp = Float64(h * Float64(t_2 / d));
	elseif (w <= -7.7e+43)
		tmp = Float64(t_1 * Float64(Float64(c0 * d) / Float64(w * Float64(Float64(w * h) * D))));
	elseif (w <= -1e-130)
		tmp = t_0;
	elseif (w <= -3e-229)
		tmp = Float64(t_1 * Float64(Float64(d / Float64(w * h)) * Float64(Float64(c0 / D) / w)));
	elseif (w <= 2.5e-189)
		tmp = t_0;
	elseif (w <= 3.2e+125)
		tmp = Float64(t_1 * Float64(Float64(Float64(d / D) / w) / Float64(Float64(w * h) / c0)));
	else
		tmp = Float64(Float64(h / d) * t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = ((D * (0.25 * (D * (h * (M * M))))) / d) / d;
	t_1 = (c0 * d) / D;
	t_2 = (M * (0.25 * (D * (D * M)))) / d;
	tmp = 0.0;
	if (w <= -1.4e+117)
		tmp = h * (t_2 / d);
	elseif (w <= -7.7e+43)
		tmp = t_1 * ((c0 * d) / (w * ((w * h) * D)));
	elseif (w <= -1e-130)
		tmp = t_0;
	elseif (w <= -3e-229)
		tmp = t_1 * ((d / (w * h)) * ((c0 / D) / w));
	elseif (w <= 2.5e-189)
		tmp = t_0;
	elseif (w <= 3.2e+125)
		tmp = t_1 * (((d / D) / w) / ((w * h) / c0));
	else
		tmp = (h / d) * t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(N[(D * N[(0.25 * N[(D * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * d), $MachinePrecision] / D), $MachinePrecision]}, Block[{t$95$2 = N[(N[(M * N[(0.25 * N[(D * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[w, -1.4e+117], N[(h * N[(t$95$2 / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, -7.7e+43], N[(t$95$1 * N[(N[(c0 * d), $MachinePrecision] / N[(w * N[(N[(w * h), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, -1e-130], t$95$0, If[LessEqual[w, -3e-229], N[(t$95$1 * N[(N[(d / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[(N[(c0 / D), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 2.5e-189], t$95$0, If[LessEqual[w, 3.2e+125], N[(t$95$1 * N[(N[(N[(d / D), $MachinePrecision] / w), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(h / d), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if w < -1.39999999999999999e117

   1. Initial program 3.7%

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

   3. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   4. 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. +-commutative N/A

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

      5. distribute-lft1-in N/A

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

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0}{w}\right)\right) \]

      8. div0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot 0\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + 0\right)\right) \]

      10. metadata-eval N/A

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

   5. Simplified 13.8%

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

   6. Taylor expanded in c0 around 0

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

      1. associate-*r/ N/A

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

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

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

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

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

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

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

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

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

      17. *-commutative N/A

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

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

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

      19. unpow2 N/A

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

      20. *-lowering-*.f64 57.1%

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

   8. Simplified 57.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. associate-/r* N/A

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

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

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

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

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

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

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

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

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

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

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

      10. *-lowering-*.f64 67.4%

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

   10. Applied egg-rr 67.4%

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

   if -1.39999999999999999e117 < w < -7.6999999999999996e43

   1. Initial program 44.2%

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

   3. Taylor expanded in c0 around inf

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. unpow2 N/A

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 38.6%

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

   5. Simplified 38.6%

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

      1. unswap-sqr N/A

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

      2. associate-*l* N/A

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

      3. times-frac N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

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

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

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

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 75.4%

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

   7. Applied egg-rr 75.4%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

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

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

      5. associate-*l* N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 79.9%

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

   9. Applied egg-rr 79.9%

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

   if -7.6999999999999996e43 < w < -1.0000000000000001e-130 or -3.00000000000000002e-229 < w < 2.4999999999999999e-189

   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. Add Preprocessing

   3. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   4. 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. +-commutative N/A

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

      5. distribute-lft1-in N/A

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

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0}{w}\right)\right) \]

      8. div0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot 0\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + 0\right)\right) \]

      10. metadata-eval N/A

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

   5. Simplified 17.6%

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

   6. Taylor expanded in c0 around 0

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

      1. associate-*r/ N/A

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

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

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

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

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

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

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

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

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

      17. *-commutative N/A

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

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

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

      19. unpow2 N/A

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

      20. *-lowering-*.f64 44.0%

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

   8. Simplified 44.0%

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

   9. Taylor expanded in h around 0

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

       1. associate-*r/ N/A

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

       2. unpow2 N/A

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

       3. associate-/r* N/A

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

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

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

   11. Simplified 62.0%

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

   if -1.0000000000000001e-130 < w < -3.00000000000000002e-229

   1. Initial program 52.5%

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

   3. Taylor expanded in c0 around inf

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. unpow2 N/A

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 52.2%

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

   5. Simplified 52.2%

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

      1. unswap-sqr N/A

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

      2. associate-*l* N/A

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

      3. times-frac N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

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

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

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

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 69.3%

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

   7. Applied egg-rr 69.3%

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

      1. associate-/r* N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. /-lowering-/.f64 76.6%

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

   9. Applied egg-rr 76.6%

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

   if 2.4999999999999999e-189 < w < 3.19999999999999983e125

   1. Initial program 21.1%

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

   3. Taylor expanded in c0 around inf

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. unpow2 N/A

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 26.0%

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

   5. Simplified 26.0%

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

      1. unswap-sqr N/A

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

      2. associate-*l* N/A

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

      3. times-frac N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

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

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

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

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 49.6%

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

   7. Applied egg-rr 49.6%

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

      1. associate-/r* N/A

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

      2. div-inv N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

      5. associate-*r/ N/A

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

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

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

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

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

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

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

      11. *-commutative N/A

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

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

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

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

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

      14. /-lowering-/.f64 52.8%

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

   9. Applied egg-rr 52.8%

      \[\leadsto \frac{d \cdot c0}{D} \cdot \color{blue}{\left(\frac{d}{\frac{w \cdot h}{c0}} \cdot \frac{\frac{1}{D}}{w}\right)} \]
   10. Step-by-step derivation

       1. associate-*l/ N/A

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

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

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

       3. associate-*r/ N/A

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

       4. div-inv N/A

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

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

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

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

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

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

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

       8. *-lowering-*.f64 52.9%

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

   11. Applied egg-rr 52.9%

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

   if 3.19999999999999983e125 < w

   1. Initial program 13.1%

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

   3. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   4. 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. +-commutative N/A

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

      5. distribute-lft1-in N/A

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

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0}{w}\right)\right) \]

      8. div0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot 0\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + 0\right)\right) \]

      10. metadata-eval N/A

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

   5. Simplified 29.7%

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

   6. Taylor expanded in c0 around 0

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

      1. associate-*r/ N/A

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

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

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

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

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

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

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

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

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

      17. *-commutative N/A

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

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

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

      19. unpow2 N/A

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

      20. *-lowering-*.f64 67.2%

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

   8. Simplified 67.2%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right), d\right), \left(\frac{h}{d}\right)\right) \]

      9. /-lowering-/.f64 68.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right), d\right), \mathsf{/.f64}\left(h, \color{blue}{d}\right)\right) \]

   10. Applied egg-rr 68.2%

       \[\leadsto \color{blue}{\frac{M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d} \cdot \frac{h}{d}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -1.4 \cdot 10^{+117}:\\ \;\;\;\;h \cdot \frac{\frac{M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}}{d}\\ \mathbf{elif}\;w \leq -7.7 \cdot 10^{+43}:\\ \;\;\;\;\frac{c0 \cdot d}{D} \cdot \frac{c0 \cdot d}{w \cdot \left(\left(w \cdot h\right) \cdot D\right)}\\ \mathbf{elif}\;w \leq -1 \cdot 10^{-130}:\\ \;\;\;\;\frac{\frac{D \cdot \left(0.25 \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d}}{d}\\ \mathbf{elif}\;w \leq -3 \cdot 10^{-229}:\\ \;\;\;\;\frac{c0 \cdot d}{D} \cdot \left(\frac{d}{w \cdot h} \cdot \frac{\frac{c0}{D}}{w}\right)\\ \mathbf{elif}\;w \leq 2.5 \cdot 10^{-189}:\\ \;\;\;\;\frac{\frac{D \cdot \left(0.25 \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d}}{d}\\ \mathbf{elif}\;w \leq 3.2 \cdot 10^{+125}:\\ \;\;\;\;\frac{c0 \cdot d}{D} \cdot \frac{\frac{\frac{d}{D}}{w}}{\frac{w \cdot h}{c0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{h}{d} \cdot \frac{M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 48.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{D \cdot \left(0.25 \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d}}{d}\\ t_1 := \frac{c0 \cdot d}{D}\\ t_2 := \frac{M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}\\ \mathbf{if}\;w \leq -2.6 \cdot 10^{+117}:\\ \;\;\;\;h \cdot \frac{t\_2}{d}\\ \mathbf{elif}\;w \leq -1.7 \cdot 10^{+44}:\\ \;\;\;\;t\_1 \cdot \frac{c0 \cdot d}{w \cdot \left(\left(w \cdot h\right) \cdot D\right)}\\ \mathbf{elif}\;w \leq -5.6 \cdot 10^{-131}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;w \leq -2 \cdot 10^{-228}:\\ \;\;\;\;t\_1 \cdot \left(\frac{d}{w \cdot h} \cdot \frac{\frac{c0}{D}}{w}\right)\\ \mathbf{elif}\;w \leq 8.5 \cdot 10^{-189}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;w \leq 10^{+127}:\\ \;\;\;\;t\_1 \cdot \frac{\frac{c0 \cdot \frac{d}{D}}{w \cdot h}}{w}\\ \mathbf{else}:\\ \;\;\;\;\frac{h}{d} \cdot t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (/ (* D (* 0.25 (* D (* h (* M M))))) d) d))
        (t_1 (/ (* c0 d) D))
        (t_2 (/ (* M (* 0.25 (* D (* D M)))) d)))
   (if (<= w -2.6e+117)
     (* h (/ t_2 d))
     (if (<= w -1.7e+44)
       (* t_1 (/ (* c0 d) (* w (* (* w h) D))))
       (if (<= w -5.6e-131)
         t_0
         (if (<= w -2e-228)
           (* t_1 (* (/ d (* w h)) (/ (/ c0 D) w)))
           (if (<= w 8.5e-189)
             t_0
             (if (<= w 1e+127)
               (* t_1 (/ (/ (* c0 (/ d D)) (* w h)) w))
               (* (/ h d) t_2)))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((D * (0.25 * (D * (h * (M * M))))) / d) / d;
	double t_1 = (c0 * d) / D;
	double t_2 = (M * (0.25 * (D * (D * M)))) / d;
	double tmp;
	if (w <= -2.6e+117) {
		tmp = h * (t_2 / d);
	} else if (w <= -1.7e+44) {
		tmp = t_1 * ((c0 * d) / (w * ((w * h) * D)));
	} else if (w <= -5.6e-131) {
		tmp = t_0;
	} else if (w <= -2e-228) {
		tmp = t_1 * ((d / (w * h)) * ((c0 / D) / w));
	} else if (w <= 8.5e-189) {
		tmp = t_0;
	} else if (w <= 1e+127) {
		tmp = t_1 * (((c0 * (d / D)) / (w * h)) / w);
	} else {
		tmp = (h / d) * t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = ((d * (0.25d0 * (d * (h * (m * m))))) / d_1) / d_1
    t_1 = (c0 * d_1) / d
    t_2 = (m * (0.25d0 * (d * (d * m)))) / d_1
    if (w <= (-2.6d+117)) then
        tmp = h * (t_2 / d_1)
    else if (w <= (-1.7d+44)) then
        tmp = t_1 * ((c0 * d_1) / (w * ((w * h) * d)))
    else if (w <= (-5.6d-131)) then
        tmp = t_0
    else if (w <= (-2d-228)) then
        tmp = t_1 * ((d_1 / (w * h)) * ((c0 / d) / w))
    else if (w <= 8.5d-189) then
        tmp = t_0
    else if (w <= 1d+127) then
        tmp = t_1 * (((c0 * (d_1 / d)) / (w * h)) / w)
    else
        tmp = (h / d_1) * t_2
    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 * (0.25 * (D * (h * (M * M))))) / d) / d;
	double t_1 = (c0 * d) / D;
	double t_2 = (M * (0.25 * (D * (D * M)))) / d;
	double tmp;
	if (w <= -2.6e+117) {
		tmp = h * (t_2 / d);
	} else if (w <= -1.7e+44) {
		tmp = t_1 * ((c0 * d) / (w * ((w * h) * D)));
	} else if (w <= -5.6e-131) {
		tmp = t_0;
	} else if (w <= -2e-228) {
		tmp = t_1 * ((d / (w * h)) * ((c0 / D) / w));
	} else if (w <= 8.5e-189) {
		tmp = t_0;
	} else if (w <= 1e+127) {
		tmp = t_1 * (((c0 * (d / D)) / (w * h)) / w);
	} else {
		tmp = (h / d) * t_2;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = ((D * (0.25 * (D * (h * (M * M))))) / d) / d
	t_1 = (c0 * d) / D
	t_2 = (M * (0.25 * (D * (D * M)))) / d
	tmp = 0
	if w <= -2.6e+117:
		tmp = h * (t_2 / d)
	elif w <= -1.7e+44:
		tmp = t_1 * ((c0 * d) / (w * ((w * h) * D)))
	elif w <= -5.6e-131:
		tmp = t_0
	elif w <= -2e-228:
		tmp = t_1 * ((d / (w * h)) * ((c0 / D) / w))
	elif w <= 8.5e-189:
		tmp = t_0
	elif w <= 1e+127:
		tmp = t_1 * (((c0 * (d / D)) / (w * h)) / w)
	else:
		tmp = (h / d) * t_2
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(Float64(D * Float64(0.25 * Float64(D * Float64(h * Float64(M * M))))) / d) / d)
	t_1 = Float64(Float64(c0 * d) / D)
	t_2 = Float64(Float64(M * Float64(0.25 * Float64(D * Float64(D * M)))) / d)
	tmp = 0.0
	if (w <= -2.6e+117)
		tmp = Float64(h * Float64(t_2 / d));
	elseif (w <= -1.7e+44)
		tmp = Float64(t_1 * Float64(Float64(c0 * d) / Float64(w * Float64(Float64(w * h) * D))));
	elseif (w <= -5.6e-131)
		tmp = t_0;
	elseif (w <= -2e-228)
		tmp = Float64(t_1 * Float64(Float64(d / Float64(w * h)) * Float64(Float64(c0 / D) / w)));
	elseif (w <= 8.5e-189)
		tmp = t_0;
	elseif (w <= 1e+127)
		tmp = Float64(t_1 * Float64(Float64(Float64(c0 * Float64(d / D)) / Float64(w * h)) / w));
	else
		tmp = Float64(Float64(h / d) * t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = ((D * (0.25 * (D * (h * (M * M))))) / d) / d;
	t_1 = (c0 * d) / D;
	t_2 = (M * (0.25 * (D * (D * M)))) / d;
	tmp = 0.0;
	if (w <= -2.6e+117)
		tmp = h * (t_2 / d);
	elseif (w <= -1.7e+44)
		tmp = t_1 * ((c0 * d) / (w * ((w * h) * D)));
	elseif (w <= -5.6e-131)
		tmp = t_0;
	elseif (w <= -2e-228)
		tmp = t_1 * ((d / (w * h)) * ((c0 / D) / w));
	elseif (w <= 8.5e-189)
		tmp = t_0;
	elseif (w <= 1e+127)
		tmp = t_1 * (((c0 * (d / D)) / (w * h)) / w);
	else
		tmp = (h / d) * t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(N[(D * N[(0.25 * N[(D * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * d), $MachinePrecision] / D), $MachinePrecision]}, Block[{t$95$2 = N[(N[(M * N[(0.25 * N[(D * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[w, -2.6e+117], N[(h * N[(t$95$2 / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, -1.7e+44], N[(t$95$1 * N[(N[(c0 * d), $MachinePrecision] / N[(w * N[(N[(w * h), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, -5.6e-131], t$95$0, If[LessEqual[w, -2e-228], N[(t$95$1 * N[(N[(d / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[(N[(c0 / D), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 8.5e-189], t$95$0, If[LessEqual[w, 1e+127], N[(t$95$1 * N[(N[(N[(c0 * N[(d / D), $MachinePrecision]), $MachinePrecision] / N[(w * h), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision], N[(N[(h / d), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if w < -2.5999999999999999e117

   1. Initial program 3.7%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   4. 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. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}}\right)\right) \]

      5. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0}{w}\right)\right) \]

      8. div0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot 0\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + 0\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \left(-1 + \color{blue}{1}\right)\right)\right) \]

   5. Simplified 13.8%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \left(0.25 \cdot \frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}}{c0 \cdot c0} + 0\right)} \]

   6. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \color{blue}{\left({d}^{2}\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)\right), \left({d}^{2}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right) \cdot h\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(h \cdot \left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot \left(M \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left(\left({D}^{2} \cdot M\right) \cdot M\right)\right)\right), \left({d}^{2}\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right) \cdot M\right)\right), \left({d}^{2}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(M \cdot \left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(\left(D \cdot D\right) \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \left(D \cdot M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \left(M \cdot D\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(M \cdot D\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(D \cdot M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

      20. *-lowering-*.f64 57.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   8. Simplified 57.1%

      \[\leadsto \color{blue}{\frac{h \cdot \left(M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right)}{d \cdot d}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto h \cdot \color{blue}{\frac{M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d \cdot d}} \]

      2. *-commutative N/A

         \[\leadsto \frac{M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d \cdot d} \cdot \color{blue}{h} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d \cdot d}\right), \color{blue}{h}\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}}{d}\right), h\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}\right), d\right), h\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right), d\right), d\right), h\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right), d\right), d\right), h\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \left(D \cdot M\right)\right)\right)\right), d\right), d\right), h\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(D \cdot M\right)\right)\right)\right), d\right), d\right), h\right) \]

      10. *-lowering-*.f64 67.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right), d\right), d\right), h\right) \]

   10. Applied egg-rr 67.4%

       \[\leadsto \color{blue}{\frac{\frac{M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}}{d} \cdot h} \]

   if -2.5999999999999999e117 < w < -1.7e44

   1. Initial program 44.2%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({c0}^{2} \cdot {d}^{2}\right), \color{blue}{\left({D}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left({d}^{2} \cdot {c0}^{2}\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({d}^{2}\right), \left({c0}^{2}\right)\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(d \cdot d\right), \left({c0}^{2}\right)\right), \left({\color{blue}{D}}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \left({c0}^{2}\right)\right), \left({\color{blue}{D}}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \left(c0 \cdot c0\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(h \cdot {w}^{2}\right)}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{h} \cdot {w}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{h} \cdot {w}^{2}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \color{blue}{\left({w}^{2}\right)}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(w \cdot \color{blue}{w}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 38.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w, \color{blue}{w}\right)\right)\right)\right) \]

   5. Simplified 38.6%

      \[\leadsto \color{blue}{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}} \]
   6. Step-by-step derivation

      1. unswap-sqr N/A

         \[\leadsto \frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{\color{blue}{\left(D \cdot D\right)} \cdot \left(h \cdot \left(w \cdot w\right)\right)} \]

      2. associate-*l* N/A

         \[\leadsto \frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{D \cdot \color{blue}{\left(D \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)}} \]

      3. times-frac N/A

         \[\leadsto \frac{d \cdot c0}{D} \cdot \color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{d \cdot c0}{D}\right), \color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{c0 \cdot d}{D}\right), \left(\frac{\color{blue}{d} \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(c0 \cdot d\right), D\right), \left(\frac{\color{blue}{d \cdot c0}}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(d \cdot c0\right), D\right), \left(\frac{\color{blue}{d} \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \left(\frac{\color{blue}{d} \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \left(\frac{c0 \cdot d}{\color{blue}{D} \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\left(c0 \cdot d\right), \color{blue}{\left(D \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\left(d \cdot c0\right), \left(\color{blue}{D} \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \left(\color{blue}{D} \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \color{blue}{\left(h \cdot \left(w \cdot w\right)\right)}\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \left(\left(w \cdot w\right) \cdot \color{blue}{h}\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \left(w \cdot \color{blue}{\left(w \cdot h\right)}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, \color{blue}{\left(w \cdot h\right)}\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, \left(h \cdot \color{blue}{w}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 75.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(h, \color{blue}{w}\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 75.4%

      \[\leadsto \color{blue}{\frac{d \cdot c0}{D} \cdot \frac{d \cdot c0}{D \cdot \left(w \cdot \left(h \cdot w\right)\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \left(D \cdot \left(\left(h \cdot w\right) \cdot \color{blue}{w}\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \left(\left(D \cdot \left(h \cdot w\right)\right) \cdot \color{blue}{w}\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \left(\left(\left(D \cdot h\right) \cdot w\right) \cdot w\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(\left(\left(D \cdot h\right) \cdot w\right), \color{blue}{w}\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(\left(D \cdot \left(h \cdot w\right)\right), w\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \left(h \cdot w\right)\right), w\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \left(w \cdot h\right)\right), w\right)\right)\right) \]

      8. *-lowering-*.f64 79.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, h\right)\right), w\right)\right)\right) \]

   9. Applied egg-rr 79.9%

      \[\leadsto \frac{d \cdot c0}{D} \cdot \frac{d \cdot c0}{\color{blue}{\left(D \cdot \left(w \cdot h\right)\right) \cdot w}} \]

   if -1.7e44 < w < -5.5999999999999999e-131 or -2.00000000000000007e-228 < w < 8.50000000000000068e-189

   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. Add Preprocessing

   3. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   4. 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. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}}\right)\right) \]

      5. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0}{w}\right)\right) \]

      8. div0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot 0\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + 0\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \left(-1 + \color{blue}{1}\right)\right)\right) \]

   5. Simplified 17.6%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \left(0.25 \cdot \frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}}{c0 \cdot c0} + 0\right)} \]

   6. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \color{blue}{\left({d}^{2}\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)\right), \left({d}^{2}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right) \cdot h\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(h \cdot \left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot \left(M \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left(\left({D}^{2} \cdot M\right) \cdot M\right)\right)\right), \left({d}^{2}\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right) \cdot M\right)\right), \left({d}^{2}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(M \cdot \left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(\left(D \cdot D\right) \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \left(D \cdot M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \left(M \cdot D\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(M \cdot D\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(D \cdot M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

      20. *-lowering-*.f64 44.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   8. Simplified 44.0%

      \[\leadsto \color{blue}{\frac{h \cdot \left(M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right)}{d \cdot d}} \]

   9. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2}}} \]

       2. unpow2 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{d \cdot \color{blue}{d}} \]

       3. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{d}}{\color{blue}{d}} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{d}\right), \color{blue}{d}\right) \]

   11. Simplified 62.0%

       \[\leadsto \color{blue}{\frac{\frac{D \cdot \left(0.25 \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d}}{d}} \]

   if -5.5999999999999999e-131 < w < -2.00000000000000007e-228

   1. Initial program 52.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({c0}^{2} \cdot {d}^{2}\right), \color{blue}{\left({D}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left({d}^{2} \cdot {c0}^{2}\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({d}^{2}\right), \left({c0}^{2}\right)\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(d \cdot d\right), \left({c0}^{2}\right)\right), \left({\color{blue}{D}}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \left({c0}^{2}\right)\right), \left({\color{blue}{D}}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \left(c0 \cdot c0\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(h \cdot {w}^{2}\right)}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{h} \cdot {w}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{h} \cdot {w}^{2}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \color{blue}{\left({w}^{2}\right)}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(w \cdot \color{blue}{w}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 52.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w, \color{blue}{w}\right)\right)\right)\right) \]

   5. Simplified 52.2%

      \[\leadsto \color{blue}{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}} \]
   6. Step-by-step derivation

      1. unswap-sqr N/A

         \[\leadsto \frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{\color{blue}{\left(D \cdot D\right)} \cdot \left(h \cdot \left(w \cdot w\right)\right)} \]

      2. associate-*l* N/A

         \[\leadsto \frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{D \cdot \color{blue}{\left(D \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)}} \]

      3. times-frac N/A

         \[\leadsto \frac{d \cdot c0}{D} \cdot \color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{d \cdot c0}{D}\right), \color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{c0 \cdot d}{D}\right), \left(\frac{\color{blue}{d} \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(c0 \cdot d\right), D\right), \left(\frac{\color{blue}{d \cdot c0}}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(d \cdot c0\right), D\right), \left(\frac{\color{blue}{d} \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \left(\frac{\color{blue}{d} \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \left(\frac{c0 \cdot d}{\color{blue}{D} \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\left(c0 \cdot d\right), \color{blue}{\left(D \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\left(d \cdot c0\right), \left(\color{blue}{D} \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \left(\color{blue}{D} \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \color{blue}{\left(h \cdot \left(w \cdot w\right)\right)}\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \left(\left(w \cdot w\right) \cdot \color{blue}{h}\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \left(w \cdot \color{blue}{\left(w \cdot h\right)}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, \color{blue}{\left(w \cdot h\right)}\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, \left(h \cdot \color{blue}{w}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 69.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(h, \color{blue}{w}\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 69.3%

      \[\leadsto \color{blue}{\frac{d \cdot c0}{D} \cdot \frac{d \cdot c0}{D \cdot \left(w \cdot \left(h \cdot w\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \left(\frac{\frac{d \cdot c0}{D}}{\color{blue}{w \cdot \left(h \cdot w\right)}}\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \left(\frac{d \cdot \frac{c0}{D}}{\color{blue}{w} \cdot \left(h \cdot w\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \left(\frac{d \cdot \frac{c0}{D}}{\left(h \cdot w\right) \cdot \color{blue}{w}}\right)\right) \]

      4. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \left(\frac{d}{h \cdot w} \cdot \color{blue}{\frac{\frac{c0}{D}}{w}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{*.f64}\left(\left(\frac{d}{h \cdot w}\right), \color{blue}{\left(\frac{\frac{c0}{D}}{w}\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(h \cdot w\right)\right), \left(\frac{\color{blue}{\frac{c0}{D}}}{w}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(w \cdot h\right)\right), \left(\frac{\frac{c0}{\color{blue}{D}}}{w}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(w, h\right)\right), \left(\frac{\frac{c0}{\color{blue}{D}}}{w}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(\left(\frac{c0}{D}\right), \color{blue}{w}\right)\right)\right) \]

      10. /-lowering-/.f64 76.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, D\right), w\right)\right)\right) \]

   9. Applied egg-rr 76.6%

      \[\leadsto \frac{d \cdot c0}{D} \cdot \color{blue}{\left(\frac{d}{w \cdot h} \cdot \frac{\frac{c0}{D}}{w}\right)} \]

   if 8.50000000000000068e-189 < w < 9.99999999999999955e126

   1. Initial program 21.1%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({c0}^{2} \cdot {d}^{2}\right), \color{blue}{\left({D}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left({d}^{2} \cdot {c0}^{2}\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({d}^{2}\right), \left({c0}^{2}\right)\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(d \cdot d\right), \left({c0}^{2}\right)\right), \left({\color{blue}{D}}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \left({c0}^{2}\right)\right), \left({\color{blue}{D}}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \left(c0 \cdot c0\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(h \cdot {w}^{2}\right)}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{h} \cdot {w}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{h} \cdot {w}^{2}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \color{blue}{\left({w}^{2}\right)}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(w \cdot \color{blue}{w}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 26.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w, \color{blue}{w}\right)\right)\right)\right) \]

   5. Simplified 26.0%

      \[\leadsto \color{blue}{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}} \]
   6. Step-by-step derivation

      1. unswap-sqr N/A

         \[\leadsto \frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{\color{blue}{\left(D \cdot D\right)} \cdot \left(h \cdot \left(w \cdot w\right)\right)} \]

      2. associate-*l* N/A

         \[\leadsto \frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{D \cdot \color{blue}{\left(D \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)}} \]

      3. times-frac N/A

         \[\leadsto \frac{d \cdot c0}{D} \cdot \color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{d \cdot c0}{D}\right), \color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{c0 \cdot d}{D}\right), \left(\frac{\color{blue}{d} \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(c0 \cdot d\right), D\right), \left(\frac{\color{blue}{d \cdot c0}}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(d \cdot c0\right), D\right), \left(\frac{\color{blue}{d} \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \left(\frac{\color{blue}{d} \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \left(\frac{c0 \cdot d}{\color{blue}{D} \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\left(c0 \cdot d\right), \color{blue}{\left(D \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\left(d \cdot c0\right), \left(\color{blue}{D} \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \left(\color{blue}{D} \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \color{blue}{\left(h \cdot \left(w \cdot w\right)\right)}\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \left(\left(w \cdot w\right) \cdot \color{blue}{h}\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \left(w \cdot \color{blue}{\left(w \cdot h\right)}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, \color{blue}{\left(w \cdot h\right)}\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, \left(h \cdot \color{blue}{w}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 49.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(h, \color{blue}{w}\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 49.6%

      \[\leadsto \color{blue}{\frac{d \cdot c0}{D} \cdot \frac{d \cdot c0}{D \cdot \left(w \cdot \left(h \cdot w\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \left(\frac{\frac{d \cdot c0}{D}}{\color{blue}{w \cdot \left(h \cdot w\right)}}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \left(\frac{\frac{d \cdot c0}{D}}{\left(h \cdot w\right) \cdot \color{blue}{w}}\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \left(\frac{\frac{\frac{d \cdot c0}{D}}{h \cdot w}}{\color{blue}{w}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\left(\frac{\frac{d \cdot c0}{D}}{h \cdot w}\right), \color{blue}{w}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{d \cdot c0}{D}\right), \left(h \cdot w\right)\right), w\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{c0 \cdot d}{D}\right), \left(h \cdot w\right)\right), w\right)\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(c0 \cdot \frac{d}{D}\right), \left(h \cdot w\right)\right), w\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{d}{D}\right)\right), \left(h \cdot w\right)\right), w\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, D\right)\right), \left(h \cdot w\right)\right), w\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, D\right)\right), \left(w \cdot h\right)\right), w\right)\right) \]

      11. *-lowering-*.f64 51.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, D\right)\right), \mathsf{*.f64}\left(w, h\right)\right), w\right)\right) \]

   9. Applied egg-rr 51.3%

      \[\leadsto \frac{d \cdot c0}{D} \cdot \color{blue}{\frac{\frac{c0 \cdot \frac{d}{D}}{w \cdot h}}{w}} \]

   if 9.99999999999999955e126 < w

   1. Initial program 13.1%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   4. 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. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}}\right)\right) \]

      5. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0}{w}\right)\right) \]

      8. div0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot 0\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + 0\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \left(-1 + \color{blue}{1}\right)\right)\right) \]

   5. Simplified 29.7%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \left(0.25 \cdot \frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}}{c0 \cdot c0} + 0\right)} \]

   6. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \color{blue}{\left({d}^{2}\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)\right), \left({d}^{2}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right) \cdot h\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(h \cdot \left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot \left(M \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left(\left({D}^{2} \cdot M\right) \cdot M\right)\right)\right), \left({d}^{2}\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right) \cdot M\right)\right), \left({d}^{2}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(M \cdot \left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(\left(D \cdot D\right) \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \left(D \cdot M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \left(M \cdot D\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(M \cdot D\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(D \cdot M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

      20. *-lowering-*.f64 67.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   8. Simplified 67.2%

      \[\leadsto \color{blue}{\frac{h \cdot \left(M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right)}{d \cdot d}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right) \cdot h}{\color{blue}{d} \cdot d} \]

      2. times-frac N/A

         \[\leadsto \frac{M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d} \cdot \color{blue}{\frac{h}{d}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}\right), \color{blue}{\left(\frac{h}{d}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right), d\right), \left(\frac{\color{blue}{h}}{d}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right), d\right), \left(\frac{h}{d}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \left(D \cdot M\right)\right)\right)\right), d\right), \left(\frac{h}{d}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(D \cdot M\right)\right)\right)\right), d\right), \left(\frac{h}{d}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right), d\right), \left(\frac{h}{d}\right)\right) \]

      9. /-lowering-/.f64 68.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right), d\right), \mathsf{/.f64}\left(h, \color{blue}{d}\right)\right) \]

   10. Applied egg-rr 68.2%

       \[\leadsto \color{blue}{\frac{M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d} \cdot \frac{h}{d}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -2.6 \cdot 10^{+117}:\\ \;\;\;\;h \cdot \frac{\frac{M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}}{d}\\ \mathbf{elif}\;w \leq -1.7 \cdot 10^{+44}:\\ \;\;\;\;\frac{c0 \cdot d}{D} \cdot \frac{c0 \cdot d}{w \cdot \left(\left(w \cdot h\right) \cdot D\right)}\\ \mathbf{elif}\;w \leq -5.6 \cdot 10^{-131}:\\ \;\;\;\;\frac{\frac{D \cdot \left(0.25 \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d}}{d}\\ \mathbf{elif}\;w \leq -2 \cdot 10^{-228}:\\ \;\;\;\;\frac{c0 \cdot d}{D} \cdot \left(\frac{d}{w \cdot h} \cdot \frac{\frac{c0}{D}}{w}\right)\\ \mathbf{elif}\;w \leq 8.5 \cdot 10^{-189}:\\ \;\;\;\;\frac{\frac{D \cdot \left(0.25 \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d}}{d}\\ \mathbf{elif}\;w \leq 10^{+127}:\\ \;\;\;\;\frac{c0 \cdot d}{D} \cdot \frac{\frac{c0 \cdot \frac{d}{D}}{w \cdot h}}{w}\\ \mathbf{else}:\\ \;\;\;\;\frac{h}{d} \cdot \frac{M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 48.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{D \cdot \left(0.25 \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d}}{d}\\ t_1 := \frac{c0 \cdot d}{D}\\ t_2 := \frac{M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}\\ \mathbf{if}\;w \leq -2.1 \cdot 10^{+117}:\\ \;\;\;\;h \cdot \frac{t\_2}{d}\\ \mathbf{elif}\;w \leq -1.48 \cdot 10^{+44}:\\ \;\;\;\;t\_1 \cdot \frac{c0 \cdot d}{w \cdot \left(\left(w \cdot h\right) \cdot D\right)}\\ \mathbf{elif}\;w \leq -6.2 \cdot 10^{-132}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;w \leq -4.1 \cdot 10^{-230}:\\ \;\;\;\;t\_1 \cdot \left(\frac{d}{w \cdot h} \cdot \frac{\frac{c0}{D}}{w}\right)\\ \mathbf{elif}\;w \leq 4.1 \cdot 10^{-191}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;w \leq 1.8 \cdot 10^{+126}:\\ \;\;\;\;t\_1 \cdot \frac{c0 \cdot d}{D \cdot \left(w \cdot \left(w \cdot h\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{h}{d} \cdot t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (/ (* D (* 0.25 (* D (* h (* M M))))) d) d))
        (t_1 (/ (* c0 d) D))
        (t_2 (/ (* M (* 0.25 (* D (* D M)))) d)))
   (if (<= w -2.1e+117)
     (* h (/ t_2 d))
     (if (<= w -1.48e+44)
       (* t_1 (/ (* c0 d) (* w (* (* w h) D))))
       (if (<= w -6.2e-132)
         t_0
         (if (<= w -4.1e-230)
           (* t_1 (* (/ d (* w h)) (/ (/ c0 D) w)))
           (if (<= w 4.1e-191)
             t_0
             (if (<= w 1.8e+126)
               (* t_1 (/ (* c0 d) (* D (* w (* w h)))))
               (* (/ h d) t_2)))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((D * (0.25 * (D * (h * (M * M))))) / d) / d;
	double t_1 = (c0 * d) / D;
	double t_2 = (M * (0.25 * (D * (D * M)))) / d;
	double tmp;
	if (w <= -2.1e+117) {
		tmp = h * (t_2 / d);
	} else if (w <= -1.48e+44) {
		tmp = t_1 * ((c0 * d) / (w * ((w * h) * D)));
	} else if (w <= -6.2e-132) {
		tmp = t_0;
	} else if (w <= -4.1e-230) {
		tmp = t_1 * ((d / (w * h)) * ((c0 / D) / w));
	} else if (w <= 4.1e-191) {
		tmp = t_0;
	} else if (w <= 1.8e+126) {
		tmp = t_1 * ((c0 * d) / (D * (w * (w * h))));
	} else {
		tmp = (h / d) * t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = ((d * (0.25d0 * (d * (h * (m * m))))) / d_1) / d_1
    t_1 = (c0 * d_1) / d
    t_2 = (m * (0.25d0 * (d * (d * m)))) / d_1
    if (w <= (-2.1d+117)) then
        tmp = h * (t_2 / d_1)
    else if (w <= (-1.48d+44)) then
        tmp = t_1 * ((c0 * d_1) / (w * ((w * h) * d)))
    else if (w <= (-6.2d-132)) then
        tmp = t_0
    else if (w <= (-4.1d-230)) then
        tmp = t_1 * ((d_1 / (w * h)) * ((c0 / d) / w))
    else if (w <= 4.1d-191) then
        tmp = t_0
    else if (w <= 1.8d+126) then
        tmp = t_1 * ((c0 * d_1) / (d * (w * (w * h))))
    else
        tmp = (h / d_1) * t_2
    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 * (0.25 * (D * (h * (M * M))))) / d) / d;
	double t_1 = (c0 * d) / D;
	double t_2 = (M * (0.25 * (D * (D * M)))) / d;
	double tmp;
	if (w <= -2.1e+117) {
		tmp = h * (t_2 / d);
	} else if (w <= -1.48e+44) {
		tmp = t_1 * ((c0 * d) / (w * ((w * h) * D)));
	} else if (w <= -6.2e-132) {
		tmp = t_0;
	} else if (w <= -4.1e-230) {
		tmp = t_1 * ((d / (w * h)) * ((c0 / D) / w));
	} else if (w <= 4.1e-191) {
		tmp = t_0;
	} else if (w <= 1.8e+126) {
		tmp = t_1 * ((c0 * d) / (D * (w * (w * h))));
	} else {
		tmp = (h / d) * t_2;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = ((D * (0.25 * (D * (h * (M * M))))) / d) / d
	t_1 = (c0 * d) / D
	t_2 = (M * (0.25 * (D * (D * M)))) / d
	tmp = 0
	if w <= -2.1e+117:
		tmp = h * (t_2 / d)
	elif w <= -1.48e+44:
		tmp = t_1 * ((c0 * d) / (w * ((w * h) * D)))
	elif w <= -6.2e-132:
		tmp = t_0
	elif w <= -4.1e-230:
		tmp = t_1 * ((d / (w * h)) * ((c0 / D) / w))
	elif w <= 4.1e-191:
		tmp = t_0
	elif w <= 1.8e+126:
		tmp = t_1 * ((c0 * d) / (D * (w * (w * h))))
	else:
		tmp = (h / d) * t_2
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(Float64(D * Float64(0.25 * Float64(D * Float64(h * Float64(M * M))))) / d) / d)
	t_1 = Float64(Float64(c0 * d) / D)
	t_2 = Float64(Float64(M * Float64(0.25 * Float64(D * Float64(D * M)))) / d)
	tmp = 0.0
	if (w <= -2.1e+117)
		tmp = Float64(h * Float64(t_2 / d));
	elseif (w <= -1.48e+44)
		tmp = Float64(t_1 * Float64(Float64(c0 * d) / Float64(w * Float64(Float64(w * h) * D))));
	elseif (w <= -6.2e-132)
		tmp = t_0;
	elseif (w <= -4.1e-230)
		tmp = Float64(t_1 * Float64(Float64(d / Float64(w * h)) * Float64(Float64(c0 / D) / w)));
	elseif (w <= 4.1e-191)
		tmp = t_0;
	elseif (w <= 1.8e+126)
		tmp = Float64(t_1 * Float64(Float64(c0 * d) / Float64(D * Float64(w * Float64(w * h)))));
	else
		tmp = Float64(Float64(h / d) * t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = ((D * (0.25 * (D * (h * (M * M))))) / d) / d;
	t_1 = (c0 * d) / D;
	t_2 = (M * (0.25 * (D * (D * M)))) / d;
	tmp = 0.0;
	if (w <= -2.1e+117)
		tmp = h * (t_2 / d);
	elseif (w <= -1.48e+44)
		tmp = t_1 * ((c0 * d) / (w * ((w * h) * D)));
	elseif (w <= -6.2e-132)
		tmp = t_0;
	elseif (w <= -4.1e-230)
		tmp = t_1 * ((d / (w * h)) * ((c0 / D) / w));
	elseif (w <= 4.1e-191)
		tmp = t_0;
	elseif (w <= 1.8e+126)
		tmp = t_1 * ((c0 * d) / (D * (w * (w * h))));
	else
		tmp = (h / d) * t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(N[(D * N[(0.25 * N[(D * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * d), $MachinePrecision] / D), $MachinePrecision]}, Block[{t$95$2 = N[(N[(M * N[(0.25 * N[(D * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[w, -2.1e+117], N[(h * N[(t$95$2 / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, -1.48e+44], N[(t$95$1 * N[(N[(c0 * d), $MachinePrecision] / N[(w * N[(N[(w * h), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, -6.2e-132], t$95$0, If[LessEqual[w, -4.1e-230], N[(t$95$1 * N[(N[(d / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[(N[(c0 / D), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 4.1e-191], t$95$0, If[LessEqual[w, 1.8e+126], N[(t$95$1 * N[(N[(c0 * d), $MachinePrecision] / N[(D * N[(w * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(h / d), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if w < -2.1000000000000001e117

   1. Initial program 3.7%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   4. 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. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}}\right)\right) \]

      5. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0}{w}\right)\right) \]

      8. div0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot 0\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + 0\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \left(-1 + \color{blue}{1}\right)\right)\right) \]

   5. Simplified 13.8%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \left(0.25 \cdot \frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}}{c0 \cdot c0} + 0\right)} \]

   6. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \color{blue}{\left({d}^{2}\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)\right), \left({d}^{2}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right) \cdot h\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(h \cdot \left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot \left(M \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left(\left({D}^{2} \cdot M\right) \cdot M\right)\right)\right), \left({d}^{2}\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right) \cdot M\right)\right), \left({d}^{2}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(M \cdot \left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(\left(D \cdot D\right) \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \left(D \cdot M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \left(M \cdot D\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(M \cdot D\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(D \cdot M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

      20. *-lowering-*.f64 57.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   8. Simplified 57.1%

      \[\leadsto \color{blue}{\frac{h \cdot \left(M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right)}{d \cdot d}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto h \cdot \color{blue}{\frac{M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d \cdot d}} \]

      2. *-commutative N/A

         \[\leadsto \frac{M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d \cdot d} \cdot \color{blue}{h} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d \cdot d}\right), \color{blue}{h}\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}}{d}\right), h\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}\right), d\right), h\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right), d\right), d\right), h\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right), d\right), d\right), h\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \left(D \cdot M\right)\right)\right)\right), d\right), d\right), h\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(D \cdot M\right)\right)\right)\right), d\right), d\right), h\right) \]

      10. *-lowering-*.f64 67.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right), d\right), d\right), h\right) \]

   10. Applied egg-rr 67.4%

       \[\leadsto \color{blue}{\frac{\frac{M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}}{d} \cdot h} \]

   if -2.1000000000000001e117 < w < -1.48e44

   1. Initial program 44.2%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({c0}^{2} \cdot {d}^{2}\right), \color{blue}{\left({D}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left({d}^{2} \cdot {c0}^{2}\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({d}^{2}\right), \left({c0}^{2}\right)\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(d \cdot d\right), \left({c0}^{2}\right)\right), \left({\color{blue}{D}}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \left({c0}^{2}\right)\right), \left({\color{blue}{D}}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \left(c0 \cdot c0\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(h \cdot {w}^{2}\right)}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{h} \cdot {w}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{h} \cdot {w}^{2}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \color{blue}{\left({w}^{2}\right)}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(w \cdot \color{blue}{w}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 38.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w, \color{blue}{w}\right)\right)\right)\right) \]

   5. Simplified 38.6%

      \[\leadsto \color{blue}{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}} \]
   6. Step-by-step derivation

      1. unswap-sqr N/A

         \[\leadsto \frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{\color{blue}{\left(D \cdot D\right)} \cdot \left(h \cdot \left(w \cdot w\right)\right)} \]

      2. associate-*l* N/A

         \[\leadsto \frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{D \cdot \color{blue}{\left(D \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)}} \]

      3. times-frac N/A

         \[\leadsto \frac{d \cdot c0}{D} \cdot \color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{d \cdot c0}{D}\right), \color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{c0 \cdot d}{D}\right), \left(\frac{\color{blue}{d} \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(c0 \cdot d\right), D\right), \left(\frac{\color{blue}{d \cdot c0}}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(d \cdot c0\right), D\right), \left(\frac{\color{blue}{d} \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \left(\frac{\color{blue}{d} \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \left(\frac{c0 \cdot d}{\color{blue}{D} \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\left(c0 \cdot d\right), \color{blue}{\left(D \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\left(d \cdot c0\right), \left(\color{blue}{D} \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \left(\color{blue}{D} \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \color{blue}{\left(h \cdot \left(w \cdot w\right)\right)}\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \left(\left(w \cdot w\right) \cdot \color{blue}{h}\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \left(w \cdot \color{blue}{\left(w \cdot h\right)}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, \color{blue}{\left(w \cdot h\right)}\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, \left(h \cdot \color{blue}{w}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 75.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(h, \color{blue}{w}\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 75.4%

      \[\leadsto \color{blue}{\frac{d \cdot c0}{D} \cdot \frac{d \cdot c0}{D \cdot \left(w \cdot \left(h \cdot w\right)\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \left(D \cdot \left(\left(h \cdot w\right) \cdot \color{blue}{w}\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \left(\left(D \cdot \left(h \cdot w\right)\right) \cdot \color{blue}{w}\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \left(\left(\left(D \cdot h\right) \cdot w\right) \cdot w\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(\left(\left(D \cdot h\right) \cdot w\right), \color{blue}{w}\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(\left(D \cdot \left(h \cdot w\right)\right), w\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \left(h \cdot w\right)\right), w\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \left(w \cdot h\right)\right), w\right)\right)\right) \]

      8. *-lowering-*.f64 79.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, h\right)\right), w\right)\right)\right) \]

   9. Applied egg-rr 79.9%

      \[\leadsto \frac{d \cdot c0}{D} \cdot \frac{d \cdot c0}{\color{blue}{\left(D \cdot \left(w \cdot h\right)\right) \cdot w}} \]

   if -1.48e44 < w < -6.20000000000000016e-132 or -4.1000000000000002e-230 < w < 4.1000000000000002e-191

   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. Add Preprocessing

   3. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   4. 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. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}}\right)\right) \]

      5. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0}{w}\right)\right) \]

      8. div0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot 0\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + 0\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \left(-1 + \color{blue}{1}\right)\right)\right) \]

   5. Simplified 17.6%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \left(0.25 \cdot \frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}}{c0 \cdot c0} + 0\right)} \]

   6. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \color{blue}{\left({d}^{2}\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)\right), \left({d}^{2}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right) \cdot h\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(h \cdot \left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot \left(M \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left(\left({D}^{2} \cdot M\right) \cdot M\right)\right)\right), \left({d}^{2}\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right) \cdot M\right)\right), \left({d}^{2}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(M \cdot \left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(\left(D \cdot D\right) \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \left(D \cdot M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \left(M \cdot D\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(M \cdot D\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(D \cdot M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

      20. *-lowering-*.f64 44.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   8. Simplified 44.0%

      \[\leadsto \color{blue}{\frac{h \cdot \left(M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right)}{d \cdot d}} \]

   9. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2}}} \]

       2. unpow2 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{d \cdot \color{blue}{d}} \]

       3. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{d}}{\color{blue}{d}} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{d}\right), \color{blue}{d}\right) \]

   11. Simplified 62.0%

       \[\leadsto \color{blue}{\frac{\frac{D \cdot \left(0.25 \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d}}{d}} \]

   if -6.20000000000000016e-132 < w < -4.1000000000000002e-230

   1. Initial program 52.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({c0}^{2} \cdot {d}^{2}\right), \color{blue}{\left({D}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left({d}^{2} \cdot {c0}^{2}\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({d}^{2}\right), \left({c0}^{2}\right)\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(d \cdot d\right), \left({c0}^{2}\right)\right), \left({\color{blue}{D}}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \left({c0}^{2}\right)\right), \left({\color{blue}{D}}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \left(c0 \cdot c0\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(h \cdot {w}^{2}\right)}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{h} \cdot {w}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{h} \cdot {w}^{2}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \color{blue}{\left({w}^{2}\right)}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(w \cdot \color{blue}{w}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 52.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w, \color{blue}{w}\right)\right)\right)\right) \]

   5. Simplified 52.2%

      \[\leadsto \color{blue}{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}} \]
   6. Step-by-step derivation

      1. unswap-sqr N/A

         \[\leadsto \frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{\color{blue}{\left(D \cdot D\right)} \cdot \left(h \cdot \left(w \cdot w\right)\right)} \]

      2. associate-*l* N/A

         \[\leadsto \frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{D \cdot \color{blue}{\left(D \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)}} \]

      3. times-frac N/A

         \[\leadsto \frac{d \cdot c0}{D} \cdot \color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{d \cdot c0}{D}\right), \color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{c0 \cdot d}{D}\right), \left(\frac{\color{blue}{d} \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(c0 \cdot d\right), D\right), \left(\frac{\color{blue}{d \cdot c0}}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(d \cdot c0\right), D\right), \left(\frac{\color{blue}{d} \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \left(\frac{\color{blue}{d} \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \left(\frac{c0 \cdot d}{\color{blue}{D} \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\left(c0 \cdot d\right), \color{blue}{\left(D \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\left(d \cdot c0\right), \left(\color{blue}{D} \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \left(\color{blue}{D} \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \color{blue}{\left(h \cdot \left(w \cdot w\right)\right)}\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \left(\left(w \cdot w\right) \cdot \color{blue}{h}\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \left(w \cdot \color{blue}{\left(w \cdot h\right)}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, \color{blue}{\left(w \cdot h\right)}\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, \left(h \cdot \color{blue}{w}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 69.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(h, \color{blue}{w}\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 69.3%

      \[\leadsto \color{blue}{\frac{d \cdot c0}{D} \cdot \frac{d \cdot c0}{D \cdot \left(w \cdot \left(h \cdot w\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \left(\frac{\frac{d \cdot c0}{D}}{\color{blue}{w \cdot \left(h \cdot w\right)}}\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \left(\frac{d \cdot \frac{c0}{D}}{\color{blue}{w} \cdot \left(h \cdot w\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \left(\frac{d \cdot \frac{c0}{D}}{\left(h \cdot w\right) \cdot \color{blue}{w}}\right)\right) \]

      4. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \left(\frac{d}{h \cdot w} \cdot \color{blue}{\frac{\frac{c0}{D}}{w}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{*.f64}\left(\left(\frac{d}{h \cdot w}\right), \color{blue}{\left(\frac{\frac{c0}{D}}{w}\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(h \cdot w\right)\right), \left(\frac{\color{blue}{\frac{c0}{D}}}{w}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(w \cdot h\right)\right), \left(\frac{\frac{c0}{\color{blue}{D}}}{w}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(w, h\right)\right), \left(\frac{\frac{c0}{\color{blue}{D}}}{w}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(\left(\frac{c0}{D}\right), \color{blue}{w}\right)\right)\right) \]

      10. /-lowering-/.f64 76.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, D\right), w\right)\right)\right) \]

   9. Applied egg-rr 76.6%

      \[\leadsto \frac{d \cdot c0}{D} \cdot \color{blue}{\left(\frac{d}{w \cdot h} \cdot \frac{\frac{c0}{D}}{w}\right)} \]

   if 4.1000000000000002e-191 < w < 1.8e126

   1. Initial program 21.1%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({c0}^{2} \cdot {d}^{2}\right), \color{blue}{\left({D}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left({d}^{2} \cdot {c0}^{2}\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({d}^{2}\right), \left({c0}^{2}\right)\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(d \cdot d\right), \left({c0}^{2}\right)\right), \left({\color{blue}{D}}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \left({c0}^{2}\right)\right), \left({\color{blue}{D}}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \left(c0 \cdot c0\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(h \cdot {w}^{2}\right)}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{h} \cdot {w}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{h} \cdot {w}^{2}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \color{blue}{\left({w}^{2}\right)}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(w \cdot \color{blue}{w}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 26.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w, \color{blue}{w}\right)\right)\right)\right) \]

   5. Simplified 26.0%

      \[\leadsto \color{blue}{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}} \]
   6. Step-by-step derivation

      1. unswap-sqr N/A

         \[\leadsto \frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{\color{blue}{\left(D \cdot D\right)} \cdot \left(h \cdot \left(w \cdot w\right)\right)} \]

      2. associate-*l* N/A

         \[\leadsto \frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{D \cdot \color{blue}{\left(D \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)}} \]

      3. times-frac N/A

         \[\leadsto \frac{d \cdot c0}{D} \cdot \color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{d \cdot c0}{D}\right), \color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{c0 \cdot d}{D}\right), \left(\frac{\color{blue}{d} \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(c0 \cdot d\right), D\right), \left(\frac{\color{blue}{d \cdot c0}}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(d \cdot c0\right), D\right), \left(\frac{\color{blue}{d} \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \left(\frac{\color{blue}{d} \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \left(\frac{c0 \cdot d}{\color{blue}{D} \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\left(c0 \cdot d\right), \color{blue}{\left(D \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\left(d \cdot c0\right), \left(\color{blue}{D} \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \left(\color{blue}{D} \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \color{blue}{\left(h \cdot \left(w \cdot w\right)\right)}\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \left(\left(w \cdot w\right) \cdot \color{blue}{h}\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \left(w \cdot \color{blue}{\left(w \cdot h\right)}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, \color{blue}{\left(w \cdot h\right)}\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, \left(h \cdot \color{blue}{w}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 49.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(h, \color{blue}{w}\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 49.6%

      \[\leadsto \color{blue}{\frac{d \cdot c0}{D} \cdot \frac{d \cdot c0}{D \cdot \left(w \cdot \left(h \cdot w\right)\right)}} \]

   if 1.8e126 < w

   1. Initial program 13.1%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   4. 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. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}}\right)\right) \]

      5. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0}{w}\right)\right) \]

      8. div0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot 0\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + 0\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \left(-1 + \color{blue}{1}\right)\right)\right) \]

   5. Simplified 29.7%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \left(0.25 \cdot \frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}}{c0 \cdot c0} + 0\right)} \]

   6. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \color{blue}{\left({d}^{2}\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)\right), \left({d}^{2}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right) \cdot h\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(h \cdot \left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot \left(M \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left(\left({D}^{2} \cdot M\right) \cdot M\right)\right)\right), \left({d}^{2}\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right) \cdot M\right)\right), \left({d}^{2}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(M \cdot \left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(\left(D \cdot D\right) \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \left(D \cdot M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \left(M \cdot D\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(M \cdot D\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(D \cdot M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

      20. *-lowering-*.f64 67.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   8. Simplified 67.2%

      \[\leadsto \color{blue}{\frac{h \cdot \left(M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right)}{d \cdot d}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right) \cdot h}{\color{blue}{d} \cdot d} \]

      2. times-frac N/A

         \[\leadsto \frac{M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d} \cdot \color{blue}{\frac{h}{d}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}\right), \color{blue}{\left(\frac{h}{d}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right), d\right), \left(\frac{\color{blue}{h}}{d}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right), d\right), \left(\frac{h}{d}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \left(D \cdot M\right)\right)\right)\right), d\right), \left(\frac{h}{d}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(D \cdot M\right)\right)\right)\right), d\right), \left(\frac{h}{d}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right), d\right), \left(\frac{h}{d}\right)\right) \]

      9. /-lowering-/.f64 68.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right), d\right), \mathsf{/.f64}\left(h, \color{blue}{d}\right)\right) \]

   10. Applied egg-rr 68.2%

       \[\leadsto \color{blue}{\frac{M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d} \cdot \frac{h}{d}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -2.1 \cdot 10^{+117}:\\ \;\;\;\;h \cdot \frac{\frac{M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}}{d}\\ \mathbf{elif}\;w \leq -1.48 \cdot 10^{+44}:\\ \;\;\;\;\frac{c0 \cdot d}{D} \cdot \frac{c0 \cdot d}{w \cdot \left(\left(w \cdot h\right) \cdot D\right)}\\ \mathbf{elif}\;w \leq -6.2 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{D \cdot \left(0.25 \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d}}{d}\\ \mathbf{elif}\;w \leq -4.1 \cdot 10^{-230}:\\ \;\;\;\;\frac{c0 \cdot d}{D} \cdot \left(\frac{d}{w \cdot h} \cdot \frac{\frac{c0}{D}}{w}\right)\\ \mathbf{elif}\;w \leq 4.1 \cdot 10^{-191}:\\ \;\;\;\;\frac{\frac{D \cdot \left(0.25 \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d}}{d}\\ \mathbf{elif}\;w \leq 1.8 \cdot 10^{+126}:\\ \;\;\;\;\frac{c0 \cdot d}{D} \cdot \frac{c0 \cdot d}{D \cdot \left(w \cdot \left(w \cdot h\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{h}{d} \cdot \frac{M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 48.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{D \cdot \left(0.25 \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d}}{d}\\ t_1 := \frac{c0 \cdot d}{D}\\ t_2 := t\_1 \cdot \frac{c0 \cdot d}{D \cdot \left(w \cdot \left(w \cdot h\right)\right)}\\ t_3 := \frac{M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}\\ \mathbf{if}\;w \leq -1.18 \cdot 10^{+111}:\\ \;\;\;\;h \cdot \frac{t\_3}{d}\\ \mathbf{elif}\;w \leq -2.15 \cdot 10^{+43}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;w \leq -3.4 \cdot 10^{-130}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;w \leq -3 \cdot 10^{-230}:\\ \;\;\;\;t\_1 \cdot \left(\frac{d}{w \cdot h} \cdot \frac{\frac{c0}{D}}{w}\right)\\ \mathbf{elif}\;w \leq 9.8 \cdot 10^{-189}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;w \leq 1.9 \cdot 10^{+126}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{h}{d} \cdot t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (/ (* D (* 0.25 (* D (* h (* M M))))) d) d))
        (t_1 (/ (* c0 d) D))
        (t_2 (* t_1 (/ (* c0 d) (* D (* w (* w h))))))
        (t_3 (/ (* M (* 0.25 (* D (* D M)))) d)))
   (if (<= w -1.18e+111)
     (* h (/ t_3 d))
     (if (<= w -2.15e+43)
       t_2
       (if (<= w -3.4e-130)
         t_0
         (if (<= w -3e-230)
           (* t_1 (* (/ d (* w h)) (/ (/ c0 D) w)))
           (if (<= w 9.8e-189)
             t_0
             (if (<= w 1.9e+126) t_2 (* (/ h d) t_3)))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((D * (0.25 * (D * (h * (M * M))))) / d) / d;
	double t_1 = (c0 * d) / D;
	double t_2 = t_1 * ((c0 * d) / (D * (w * (w * h))));
	double t_3 = (M * (0.25 * (D * (D * M)))) / d;
	double tmp;
	if (w <= -1.18e+111) {
		tmp = h * (t_3 / d);
	} else if (w <= -2.15e+43) {
		tmp = t_2;
	} else if (w <= -3.4e-130) {
		tmp = t_0;
	} else if (w <= -3e-230) {
		tmp = t_1 * ((d / (w * h)) * ((c0 / D) / w));
	} else if (w <= 9.8e-189) {
		tmp = t_0;
	} else if (w <= 1.9e+126) {
		tmp = t_2;
	} else {
		tmp = (h / d) * t_3;
	}
	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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = ((d * (0.25d0 * (d * (h * (m * m))))) / d_1) / d_1
    t_1 = (c0 * d_1) / d
    t_2 = t_1 * ((c0 * d_1) / (d * (w * (w * h))))
    t_3 = (m * (0.25d0 * (d * (d * m)))) / d_1
    if (w <= (-1.18d+111)) then
        tmp = h * (t_3 / d_1)
    else if (w <= (-2.15d+43)) then
        tmp = t_2
    else if (w <= (-3.4d-130)) then
        tmp = t_0
    else if (w <= (-3d-230)) then
        tmp = t_1 * ((d_1 / (w * h)) * ((c0 / d) / w))
    else if (w <= 9.8d-189) then
        tmp = t_0
    else if (w <= 1.9d+126) then
        tmp = t_2
    else
        tmp = (h / d_1) * t_3
    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 * (0.25 * (D * (h * (M * M))))) / d) / d;
	double t_1 = (c0 * d) / D;
	double t_2 = t_1 * ((c0 * d) / (D * (w * (w * h))));
	double t_3 = (M * (0.25 * (D * (D * M)))) / d;
	double tmp;
	if (w <= -1.18e+111) {
		tmp = h * (t_3 / d);
	} else if (w <= -2.15e+43) {
		tmp = t_2;
	} else if (w <= -3.4e-130) {
		tmp = t_0;
	} else if (w <= -3e-230) {
		tmp = t_1 * ((d / (w * h)) * ((c0 / D) / w));
	} else if (w <= 9.8e-189) {
		tmp = t_0;
	} else if (w <= 1.9e+126) {
		tmp = t_2;
	} else {
		tmp = (h / d) * t_3;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = ((D * (0.25 * (D * (h * (M * M))))) / d) / d
	t_1 = (c0 * d) / D
	t_2 = t_1 * ((c0 * d) / (D * (w * (w * h))))
	t_3 = (M * (0.25 * (D * (D * M)))) / d
	tmp = 0
	if w <= -1.18e+111:
		tmp = h * (t_3 / d)
	elif w <= -2.15e+43:
		tmp = t_2
	elif w <= -3.4e-130:
		tmp = t_0
	elif w <= -3e-230:
		tmp = t_1 * ((d / (w * h)) * ((c0 / D) / w))
	elif w <= 9.8e-189:
		tmp = t_0
	elif w <= 1.9e+126:
		tmp = t_2
	else:
		tmp = (h / d) * t_3
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(Float64(D * Float64(0.25 * Float64(D * Float64(h * Float64(M * M))))) / d) / d)
	t_1 = Float64(Float64(c0 * d) / D)
	t_2 = Float64(t_1 * Float64(Float64(c0 * d) / Float64(D * Float64(w * Float64(w * h)))))
	t_3 = Float64(Float64(M * Float64(0.25 * Float64(D * Float64(D * M)))) / d)
	tmp = 0.0
	if (w <= -1.18e+111)
		tmp = Float64(h * Float64(t_3 / d));
	elseif (w <= -2.15e+43)
		tmp = t_2;
	elseif (w <= -3.4e-130)
		tmp = t_0;
	elseif (w <= -3e-230)
		tmp = Float64(t_1 * Float64(Float64(d / Float64(w * h)) * Float64(Float64(c0 / D) / w)));
	elseif (w <= 9.8e-189)
		tmp = t_0;
	elseif (w <= 1.9e+126)
		tmp = t_2;
	else
		tmp = Float64(Float64(h / d) * t_3);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = ((D * (0.25 * (D * (h * (M * M))))) / d) / d;
	t_1 = (c0 * d) / D;
	t_2 = t_1 * ((c0 * d) / (D * (w * (w * h))));
	t_3 = (M * (0.25 * (D * (D * M)))) / d;
	tmp = 0.0;
	if (w <= -1.18e+111)
		tmp = h * (t_3 / d);
	elseif (w <= -2.15e+43)
		tmp = t_2;
	elseif (w <= -3.4e-130)
		tmp = t_0;
	elseif (w <= -3e-230)
		tmp = t_1 * ((d / (w * h)) * ((c0 / D) / w));
	elseif (w <= 9.8e-189)
		tmp = t_0;
	elseif (w <= 1.9e+126)
		tmp = t_2;
	else
		tmp = (h / d) * t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(N[(D * N[(0.25 * N[(D * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * d), $MachinePrecision] / D), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(c0 * d), $MachinePrecision] / N[(D * N[(w * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(M * N[(0.25 * N[(D * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[w, -1.18e+111], N[(h * N[(t$95$3 / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, -2.15e+43], t$95$2, If[LessEqual[w, -3.4e-130], t$95$0, If[LessEqual[w, -3e-230], N[(t$95$1 * N[(N[(d / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[(N[(c0 / D), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 9.8e-189], t$95$0, If[LessEqual[w, 1.9e+126], t$95$2, N[(N[(h / d), $MachinePrecision] * t$95$3), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if w < -1.1799999999999999e111

   1. Initial program 6.3%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   4. 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. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}}\right)\right) \]

      5. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0}{w}\right)\right) \]

      8. div0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot 0\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + 0\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \left(-1 + \color{blue}{1}\right)\right)\right) \]

   5. Simplified 12.5%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \left(0.25 \cdot \frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}}{c0 \cdot c0} + 0\right)} \]

   6. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \color{blue}{\left({d}^{2}\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)\right), \left({d}^{2}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right) \cdot h\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(h \cdot \left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot \left(M \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left(\left({D}^{2} \cdot M\right) \cdot M\right)\right)\right), \left({d}^{2}\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right) \cdot M\right)\right), \left({d}^{2}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(M \cdot \left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(\left(D \cdot D\right) \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \left(D \cdot M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \left(M \cdot D\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(M \cdot D\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(D \cdot M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

      20. *-lowering-*.f64 55.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   8. Simplified 55.1%

      \[\leadsto \color{blue}{\frac{h \cdot \left(M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right)}{d \cdot d}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto h \cdot \color{blue}{\frac{M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d \cdot d}} \]

      2. *-commutative N/A

         \[\leadsto \frac{M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d \cdot d} \cdot \color{blue}{h} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d \cdot d}\right), \color{blue}{h}\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}}{d}\right), h\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}\right), d\right), h\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right), d\right), d\right), h\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right), d\right), d\right), h\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \left(D \cdot M\right)\right)\right)\right), d\right), d\right), h\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(D \cdot M\right)\right)\right)\right), d\right), d\right), h\right) \]

      10. *-lowering-*.f64 64.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right), d\right), d\right), h\right) \]

   10. Applied egg-rr 64.5%

       \[\leadsto \color{blue}{\frac{\frac{M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}}{d} \cdot h} \]

   if -1.1799999999999999e111 < w < -2.15e43 or 9.7999999999999994e-189 < w < 1.90000000000000008e126

   1. Initial program 25.8%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({c0}^{2} \cdot {d}^{2}\right), \color{blue}{\left({D}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left({d}^{2} \cdot {c0}^{2}\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({d}^{2}\right), \left({c0}^{2}\right)\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(d \cdot d\right), \left({c0}^{2}\right)\right), \left({\color{blue}{D}}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \left({c0}^{2}\right)\right), \left({\color{blue}{D}}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \left(c0 \cdot c0\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(h \cdot {w}^{2}\right)}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{h} \cdot {w}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{h} \cdot {w}^{2}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \color{blue}{\left({w}^{2}\right)}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(w \cdot \color{blue}{w}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 29.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w, \color{blue}{w}\right)\right)\right)\right) \]

   5. Simplified 29.8%

      \[\leadsto \color{blue}{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}} \]
   6. Step-by-step derivation

      1. unswap-sqr N/A

         \[\leadsto \frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{\color{blue}{\left(D \cdot D\right)} \cdot \left(h \cdot \left(w \cdot w\right)\right)} \]

      2. associate-*l* N/A

         \[\leadsto \frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{D \cdot \color{blue}{\left(D \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)}} \]

      3. times-frac N/A

         \[\leadsto \frac{d \cdot c0}{D} \cdot \color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{d \cdot c0}{D}\right), \color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{c0 \cdot d}{D}\right), \left(\frac{\color{blue}{d} \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(c0 \cdot d\right), D\right), \left(\frac{\color{blue}{d \cdot c0}}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(d \cdot c0\right), D\right), \left(\frac{\color{blue}{d} \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \left(\frac{\color{blue}{d} \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \left(\frac{c0 \cdot d}{\color{blue}{D} \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\left(c0 \cdot d\right), \color{blue}{\left(D \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\left(d \cdot c0\right), \left(\color{blue}{D} \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \left(\color{blue}{D} \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \color{blue}{\left(h \cdot \left(w \cdot w\right)\right)}\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \left(\left(w \cdot w\right) \cdot \color{blue}{h}\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \left(w \cdot \color{blue}{\left(w \cdot h\right)}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, \color{blue}{\left(w \cdot h\right)}\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, \left(h \cdot \color{blue}{w}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 56.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(h, \color{blue}{w}\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 56.0%

      \[\leadsto \color{blue}{\frac{d \cdot c0}{D} \cdot \frac{d \cdot c0}{D \cdot \left(w \cdot \left(h \cdot w\right)\right)}} \]

   if -2.15e43 < w < -3.40000000000000005e-130 or -3e-230 < w < 9.7999999999999994e-189

   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. Add Preprocessing

   3. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   4. 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. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}}\right)\right) \]

      5. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0}{w}\right)\right) \]

      8. div0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot 0\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + 0\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \left(-1 + \color{blue}{1}\right)\right)\right) \]

   5. Simplified 17.6%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \left(0.25 \cdot \frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}}{c0 \cdot c0} + 0\right)} \]

   6. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \color{blue}{\left({d}^{2}\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)\right), \left({d}^{2}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right) \cdot h\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(h \cdot \left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot \left(M \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left(\left({D}^{2} \cdot M\right) \cdot M\right)\right)\right), \left({d}^{2}\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right) \cdot M\right)\right), \left({d}^{2}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(M \cdot \left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(\left(D \cdot D\right) \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \left(D \cdot M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \left(M \cdot D\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(M \cdot D\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(D \cdot M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

      20. *-lowering-*.f64 44.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   8. Simplified 44.0%

      \[\leadsto \color{blue}{\frac{h \cdot \left(M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right)}{d \cdot d}} \]

   9. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2}}} \]

       2. unpow2 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{d \cdot \color{blue}{d}} \]

       3. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{d}}{\color{blue}{d}} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{d}\right), \color{blue}{d}\right) \]

   11. Simplified 62.0%

       \[\leadsto \color{blue}{\frac{\frac{D \cdot \left(0.25 \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d}}{d}} \]

   if -3.40000000000000005e-130 < w < -3e-230

   1. Initial program 52.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({c0}^{2} \cdot {d}^{2}\right), \color{blue}{\left({D}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left({d}^{2} \cdot {c0}^{2}\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({d}^{2}\right), \left({c0}^{2}\right)\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(d \cdot d\right), \left({c0}^{2}\right)\right), \left({\color{blue}{D}}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \left({c0}^{2}\right)\right), \left({\color{blue}{D}}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \left(c0 \cdot c0\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(h \cdot {w}^{2}\right)}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{h} \cdot {w}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{h} \cdot {w}^{2}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \color{blue}{\left({w}^{2}\right)}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(w \cdot \color{blue}{w}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 52.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w, \color{blue}{w}\right)\right)\right)\right) \]

   5. Simplified 52.2%

      \[\leadsto \color{blue}{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}} \]
   6. Step-by-step derivation

      1. unswap-sqr N/A

         \[\leadsto \frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{\color{blue}{\left(D \cdot D\right)} \cdot \left(h \cdot \left(w \cdot w\right)\right)} \]

      2. associate-*l* N/A

         \[\leadsto \frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{D \cdot \color{blue}{\left(D \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)}} \]

      3. times-frac N/A

         \[\leadsto \frac{d \cdot c0}{D} \cdot \color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{d \cdot c0}{D}\right), \color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{c0 \cdot d}{D}\right), \left(\frac{\color{blue}{d} \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(c0 \cdot d\right), D\right), \left(\frac{\color{blue}{d \cdot c0}}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(d \cdot c0\right), D\right), \left(\frac{\color{blue}{d} \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \left(\frac{\color{blue}{d} \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \left(\frac{c0 \cdot d}{\color{blue}{D} \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\left(c0 \cdot d\right), \color{blue}{\left(D \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\left(d \cdot c0\right), \left(\color{blue}{D} \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \left(\color{blue}{D} \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \color{blue}{\left(h \cdot \left(w \cdot w\right)\right)}\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \left(\left(w \cdot w\right) \cdot \color{blue}{h}\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \left(w \cdot \color{blue}{\left(w \cdot h\right)}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, \color{blue}{\left(w \cdot h\right)}\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, \left(h \cdot \color{blue}{w}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 69.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(h, \color{blue}{w}\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 69.3%

      \[\leadsto \color{blue}{\frac{d \cdot c0}{D} \cdot \frac{d \cdot c0}{D \cdot \left(w \cdot \left(h \cdot w\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \left(\frac{\frac{d \cdot c0}{D}}{\color{blue}{w \cdot \left(h \cdot w\right)}}\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \left(\frac{d \cdot \frac{c0}{D}}{\color{blue}{w} \cdot \left(h \cdot w\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \left(\frac{d \cdot \frac{c0}{D}}{\left(h \cdot w\right) \cdot \color{blue}{w}}\right)\right) \]

      4. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \left(\frac{d}{h \cdot w} \cdot \color{blue}{\frac{\frac{c0}{D}}{w}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{*.f64}\left(\left(\frac{d}{h \cdot w}\right), \color{blue}{\left(\frac{\frac{c0}{D}}{w}\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(h \cdot w\right)\right), \left(\frac{\color{blue}{\frac{c0}{D}}}{w}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(w \cdot h\right)\right), \left(\frac{\frac{c0}{\color{blue}{D}}}{w}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(w, h\right)\right), \left(\frac{\frac{c0}{\color{blue}{D}}}{w}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(\left(\frac{c0}{D}\right), \color{blue}{w}\right)\right)\right) \]

      10. /-lowering-/.f64 76.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, D\right), w\right)\right)\right) \]

   9. Applied egg-rr 76.6%

      \[\leadsto \frac{d \cdot c0}{D} \cdot \color{blue}{\left(\frac{d}{w \cdot h} \cdot \frac{\frac{c0}{D}}{w}\right)} \]

   if 1.90000000000000008e126 < w

   1. Initial program 13.1%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   4. 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. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}}\right)\right) \]

      5. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0}{w}\right)\right) \]

      8. div0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot 0\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + 0\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \left(-1 + \color{blue}{1}\right)\right)\right) \]

   5. Simplified 29.7%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \left(0.25 \cdot \frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}}{c0 \cdot c0} + 0\right)} \]

   6. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \color{blue}{\left({d}^{2}\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)\right), \left({d}^{2}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right) \cdot h\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(h \cdot \left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot \left(M \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left(\left({D}^{2} \cdot M\right) \cdot M\right)\right)\right), \left({d}^{2}\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right) \cdot M\right)\right), \left({d}^{2}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(M \cdot \left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(\left(D \cdot D\right) \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \left(D \cdot M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \left(M \cdot D\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(M \cdot D\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(D \cdot M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

      20. *-lowering-*.f64 67.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   8. Simplified 67.2%

      \[\leadsto \color{blue}{\frac{h \cdot \left(M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right)}{d \cdot d}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right) \cdot h}{\color{blue}{d} \cdot d} \]

      2. times-frac N/A

         \[\leadsto \frac{M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d} \cdot \color{blue}{\frac{h}{d}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}\right), \color{blue}{\left(\frac{h}{d}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right), d\right), \left(\frac{\color{blue}{h}}{d}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right), d\right), \left(\frac{h}{d}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \left(D \cdot M\right)\right)\right)\right), d\right), \left(\frac{h}{d}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(D \cdot M\right)\right)\right)\right), d\right), \left(\frac{h}{d}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right), d\right), \left(\frac{h}{d}\right)\right) \]

      9. /-lowering-/.f64 68.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right), d\right), \mathsf{/.f64}\left(h, \color{blue}{d}\right)\right) \]

   10. Applied egg-rr 68.2%

       \[\leadsto \color{blue}{\frac{M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d} \cdot \frac{h}{d}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -1.18 \cdot 10^{+111}:\\ \;\;\;\;h \cdot \frac{\frac{M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}}{d}\\ \mathbf{elif}\;w \leq -2.15 \cdot 10^{+43}:\\ \;\;\;\;\frac{c0 \cdot d}{D} \cdot \frac{c0 \cdot d}{D \cdot \left(w \cdot \left(w \cdot h\right)\right)}\\ \mathbf{elif}\;w \leq -3.4 \cdot 10^{-130}:\\ \;\;\;\;\frac{\frac{D \cdot \left(0.25 \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d}}{d}\\ \mathbf{elif}\;w \leq -3 \cdot 10^{-230}:\\ \;\;\;\;\frac{c0 \cdot d}{D} \cdot \left(\frac{d}{w \cdot h} \cdot \frac{\frac{c0}{D}}{w}\right)\\ \mathbf{elif}\;w \leq 9.8 \cdot 10^{-189}:\\ \;\;\;\;\frac{\frac{D \cdot \left(0.25 \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d}}{d}\\ \mathbf{elif}\;w \leq 1.9 \cdot 10^{+126}:\\ \;\;\;\;\frac{c0 \cdot d}{D} \cdot \frac{c0 \cdot d}{D \cdot \left(w \cdot \left(w \cdot h\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{h}{d} \cdot \frac{M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 48.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{D \cdot \left(0.25 \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d}}{d}\\ t_1 := \frac{c0 \cdot d}{D} \cdot \left(\frac{d}{w \cdot h} \cdot \frac{\frac{c0}{D}}{w}\right)\\ t_2 := \frac{M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}\\ \mathbf{if}\;w \leq -3.4 \cdot 10^{+109}:\\ \;\;\;\;h \cdot \frac{t\_2}{d}\\ \mathbf{elif}\;w \leq -2.5 \cdot 10^{+44}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;w \leq -5 \cdot 10^{-131}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;w \leq -8.2 \cdot 10^{-231}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;w \leq 3.1 \cdot 10^{-189}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;w \leq 2.45 \cdot 10^{+126}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{h}{d} \cdot t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (/ (* D (* 0.25 (* D (* h (* M M))))) d) d))
        (t_1 (* (/ (* c0 d) D) (* (/ d (* w h)) (/ (/ c0 D) w))))
        (t_2 (/ (* M (* 0.25 (* D (* D M)))) d)))
   (if (<= w -3.4e+109)
     (* h (/ t_2 d))
     (if (<= w -2.5e+44)
       t_1
       (if (<= w -5e-131)
         t_0
         (if (<= w -8.2e-231)
           t_1
           (if (<= w 3.1e-189)
             t_0
             (if (<= w 2.45e+126) t_1 (* (/ h d) t_2)))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((D * (0.25 * (D * (h * (M * M))))) / d) / d;
	double t_1 = ((c0 * d) / D) * ((d / (w * h)) * ((c0 / D) / w));
	double t_2 = (M * (0.25 * (D * (D * M)))) / d;
	double tmp;
	if (w <= -3.4e+109) {
		tmp = h * (t_2 / d);
	} else if (w <= -2.5e+44) {
		tmp = t_1;
	} else if (w <= -5e-131) {
		tmp = t_0;
	} else if (w <= -8.2e-231) {
		tmp = t_1;
	} else if (w <= 3.1e-189) {
		tmp = t_0;
	} else if (w <= 2.45e+126) {
		tmp = t_1;
	} else {
		tmp = (h / d) * t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = ((d * (0.25d0 * (d * (h * (m * m))))) / d_1) / d_1
    t_1 = ((c0 * d_1) / d) * ((d_1 / (w * h)) * ((c0 / d) / w))
    t_2 = (m * (0.25d0 * (d * (d * m)))) / d_1
    if (w <= (-3.4d+109)) then
        tmp = h * (t_2 / d_1)
    else if (w <= (-2.5d+44)) then
        tmp = t_1
    else if (w <= (-5d-131)) then
        tmp = t_0
    else if (w <= (-8.2d-231)) then
        tmp = t_1
    else if (w <= 3.1d-189) then
        tmp = t_0
    else if (w <= 2.45d+126) then
        tmp = t_1
    else
        tmp = (h / d_1) * t_2
    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 * (0.25 * (D * (h * (M * M))))) / d) / d;
	double t_1 = ((c0 * d) / D) * ((d / (w * h)) * ((c0 / D) / w));
	double t_2 = (M * (0.25 * (D * (D * M)))) / d;
	double tmp;
	if (w <= -3.4e+109) {
		tmp = h * (t_2 / d);
	} else if (w <= -2.5e+44) {
		tmp = t_1;
	} else if (w <= -5e-131) {
		tmp = t_0;
	} else if (w <= -8.2e-231) {
		tmp = t_1;
	} else if (w <= 3.1e-189) {
		tmp = t_0;
	} else if (w <= 2.45e+126) {
		tmp = t_1;
	} else {
		tmp = (h / d) * t_2;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = ((D * (0.25 * (D * (h * (M * M))))) / d) / d
	t_1 = ((c0 * d) / D) * ((d / (w * h)) * ((c0 / D) / w))
	t_2 = (M * (0.25 * (D * (D * M)))) / d
	tmp = 0
	if w <= -3.4e+109:
		tmp = h * (t_2 / d)
	elif w <= -2.5e+44:
		tmp = t_1
	elif w <= -5e-131:
		tmp = t_0
	elif w <= -8.2e-231:
		tmp = t_1
	elif w <= 3.1e-189:
		tmp = t_0
	elif w <= 2.45e+126:
		tmp = t_1
	else:
		tmp = (h / d) * t_2
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(Float64(D * Float64(0.25 * Float64(D * Float64(h * Float64(M * M))))) / d) / d)
	t_1 = Float64(Float64(Float64(c0 * d) / D) * Float64(Float64(d / Float64(w * h)) * Float64(Float64(c0 / D) / w)))
	t_2 = Float64(Float64(M * Float64(0.25 * Float64(D * Float64(D * M)))) / d)
	tmp = 0.0
	if (w <= -3.4e+109)
		tmp = Float64(h * Float64(t_2 / d));
	elseif (w <= -2.5e+44)
		tmp = t_1;
	elseif (w <= -5e-131)
		tmp = t_0;
	elseif (w <= -8.2e-231)
		tmp = t_1;
	elseif (w <= 3.1e-189)
		tmp = t_0;
	elseif (w <= 2.45e+126)
		tmp = t_1;
	else
		tmp = Float64(Float64(h / d) * t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = ((D * (0.25 * (D * (h * (M * M))))) / d) / d;
	t_1 = ((c0 * d) / D) * ((d / (w * h)) * ((c0 / D) / w));
	t_2 = (M * (0.25 * (D * (D * M)))) / d;
	tmp = 0.0;
	if (w <= -3.4e+109)
		tmp = h * (t_2 / d);
	elseif (w <= -2.5e+44)
		tmp = t_1;
	elseif (w <= -5e-131)
		tmp = t_0;
	elseif (w <= -8.2e-231)
		tmp = t_1;
	elseif (w <= 3.1e-189)
		tmp = t_0;
	elseif (w <= 2.45e+126)
		tmp = t_1;
	else
		tmp = (h / d) * t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(N[(D * N[(0.25 * N[(D * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(c0 * d), $MachinePrecision] / D), $MachinePrecision] * N[(N[(d / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[(N[(c0 / D), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(M * N[(0.25 * N[(D * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[w, -3.4e+109], N[(h * N[(t$95$2 / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, -2.5e+44], t$95$1, If[LessEqual[w, -5e-131], t$95$0, If[LessEqual[w, -8.2e-231], t$95$1, If[LessEqual[w, 3.1e-189], t$95$0, If[LessEqual[w, 2.45e+126], t$95$1, N[(N[(h / d), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if w < -3.40000000000000006e109

   1. Initial program 6.3%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   4. 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. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}}\right)\right) \]

      5. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0}{w}\right)\right) \]

      8. div0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot 0\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + 0\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \left(-1 + \color{blue}{1}\right)\right)\right) \]

   5. Simplified 12.5%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \left(0.25 \cdot \frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}}{c0 \cdot c0} + 0\right)} \]

   6. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \color{blue}{\left({d}^{2}\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)\right), \left({d}^{2}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right) \cdot h\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(h \cdot \left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot \left(M \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left(\left({D}^{2} \cdot M\right) \cdot M\right)\right)\right), \left({d}^{2}\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right) \cdot M\right)\right), \left({d}^{2}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(M \cdot \left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(\left(D \cdot D\right) \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \left(D \cdot M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \left(M \cdot D\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(M \cdot D\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(D \cdot M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

      20. *-lowering-*.f64 55.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   8. Simplified 55.1%

      \[\leadsto \color{blue}{\frac{h \cdot \left(M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right)}{d \cdot d}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto h \cdot \color{blue}{\frac{M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d \cdot d}} \]

      2. *-commutative N/A

         \[\leadsto \frac{M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d \cdot d} \cdot \color{blue}{h} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d \cdot d}\right), \color{blue}{h}\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}}{d}\right), h\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}\right), d\right), h\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right), d\right), d\right), h\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right), d\right), d\right), h\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \left(D \cdot M\right)\right)\right)\right), d\right), d\right), h\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(D \cdot M\right)\right)\right)\right), d\right), d\right), h\right) \]

      10. *-lowering-*.f64 64.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right), d\right), d\right), h\right) \]

   10. Applied egg-rr 64.5%

       \[\leadsto \color{blue}{\frac{\frac{M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}}{d} \cdot h} \]

   if -3.40000000000000006e109 < w < -2.4999999999999998e44 or -5.0000000000000004e-131 < w < -8.2000000000000003e-231 or 3.1e-189 < w < 2.45e126

   1. Initial program 32.7%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({c0}^{2} \cdot {d}^{2}\right), \color{blue}{\left({D}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left({d}^{2} \cdot {c0}^{2}\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({d}^{2}\right), \left({c0}^{2}\right)\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(d \cdot d\right), \left({c0}^{2}\right)\right), \left({\color{blue}{D}}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \left({c0}^{2}\right)\right), \left({\color{blue}{D}}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \left(c0 \cdot c0\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(h \cdot {w}^{2}\right)}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{h} \cdot {w}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{h} \cdot {w}^{2}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \color{blue}{\left({w}^{2}\right)}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(w \cdot \color{blue}{w}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 35.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \mathsf{*.f64}\left(c0, c0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w, \color{blue}{w}\right)\right)\right)\right) \]

   5. Simplified 35.6%

      \[\leadsto \color{blue}{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}} \]
   6. Step-by-step derivation

      1. unswap-sqr N/A

         \[\leadsto \frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{\color{blue}{\left(D \cdot D\right)} \cdot \left(h \cdot \left(w \cdot w\right)\right)} \]

      2. associate-*l* N/A

         \[\leadsto \frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{D \cdot \color{blue}{\left(D \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)}} \]

      3. times-frac N/A

         \[\leadsto \frac{d \cdot c0}{D} \cdot \color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{d \cdot c0}{D}\right), \color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{c0 \cdot d}{D}\right), \left(\frac{\color{blue}{d} \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(c0 \cdot d\right), D\right), \left(\frac{\color{blue}{d \cdot c0}}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(d \cdot c0\right), D\right), \left(\frac{\color{blue}{d} \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \left(\frac{\color{blue}{d} \cdot c0}{D \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \left(\frac{c0 \cdot d}{\color{blue}{D} \cdot \left(h \cdot \left(w \cdot w\right)\right)}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\left(c0 \cdot d\right), \color{blue}{\left(D \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\left(d \cdot c0\right), \left(\color{blue}{D} \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \left(\color{blue}{D} \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \color{blue}{\left(h \cdot \left(w \cdot w\right)\right)}\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \left(\left(w \cdot w\right) \cdot \color{blue}{h}\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \left(w \cdot \color{blue}{\left(w \cdot h\right)}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, \color{blue}{\left(w \cdot h\right)}\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, \left(h \cdot \color{blue}{w}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 59.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(h, \color{blue}{w}\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 59.5%

      \[\leadsto \color{blue}{\frac{d \cdot c0}{D} \cdot \frac{d \cdot c0}{D \cdot \left(w \cdot \left(h \cdot w\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \left(\frac{\frac{d \cdot c0}{D}}{\color{blue}{w \cdot \left(h \cdot w\right)}}\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \left(\frac{d \cdot \frac{c0}{D}}{\color{blue}{w} \cdot \left(h \cdot w\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \left(\frac{d \cdot \frac{c0}{D}}{\left(h \cdot w\right) \cdot \color{blue}{w}}\right)\right) \]

      4. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \left(\frac{d}{h \cdot w} \cdot \color{blue}{\frac{\frac{c0}{D}}{w}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{*.f64}\left(\left(\frac{d}{h \cdot w}\right), \color{blue}{\left(\frac{\frac{c0}{D}}{w}\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(h \cdot w\right)\right), \left(\frac{\color{blue}{\frac{c0}{D}}}{w}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(w \cdot h\right)\right), \left(\frac{\frac{c0}{\color{blue}{D}}}{w}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(w, h\right)\right), \left(\frac{\frac{c0}{\color{blue}{D}}}{w}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(\left(\frac{c0}{D}\right), \color{blue}{w}\right)\right)\right) \]

      10. /-lowering-/.f64 60.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(d, c0\right), D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, D\right), w\right)\right)\right) \]

   9. Applied egg-rr 60.2%

      \[\leadsto \frac{d \cdot c0}{D} \cdot \color{blue}{\left(\frac{d}{w \cdot h} \cdot \frac{\frac{c0}{D}}{w}\right)} \]

   if -2.4999999999999998e44 < w < -5.0000000000000004e-131 or -8.2000000000000003e-231 < w < 3.1e-189

   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. Add Preprocessing

   3. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   4. 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. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}}\right)\right) \]

      5. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0}{w}\right)\right) \]

      8. div0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot 0\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + 0\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \left(-1 + \color{blue}{1}\right)\right)\right) \]

   5. Simplified 17.6%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \left(0.25 \cdot \frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}}{c0 \cdot c0} + 0\right)} \]

   6. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \color{blue}{\left({d}^{2}\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)\right), \left({d}^{2}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right) \cdot h\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(h \cdot \left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot \left(M \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left(\left({D}^{2} \cdot M\right) \cdot M\right)\right)\right), \left({d}^{2}\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right) \cdot M\right)\right), \left({d}^{2}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(M \cdot \left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(\left(D \cdot D\right) \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \left(D \cdot M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \left(M \cdot D\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(M \cdot D\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(D \cdot M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

      20. *-lowering-*.f64 44.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   8. Simplified 44.0%

      \[\leadsto \color{blue}{\frac{h \cdot \left(M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right)}{d \cdot d}} \]

   9. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2}}} \]

       2. unpow2 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{d \cdot \color{blue}{d}} \]

       3. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{d}}{\color{blue}{d}} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{d}\right), \color{blue}{d}\right) \]

   11. Simplified 62.0%

       \[\leadsto \color{blue}{\frac{\frac{D \cdot \left(0.25 \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d}}{d}} \]

   if 2.45e126 < w

   1. Initial program 13.1%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   4. 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. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}}\right)\right) \]

      5. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0}{w}\right)\right) \]

      8. div0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot 0\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + 0\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \left(-1 + \color{blue}{1}\right)\right)\right) \]

   5. Simplified 29.7%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \left(0.25 \cdot \frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}}{c0 \cdot c0} + 0\right)} \]

   6. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \color{blue}{\left({d}^{2}\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)\right), \left({d}^{2}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right) \cdot h\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(h \cdot \left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot \left(M \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left(\left({D}^{2} \cdot M\right) \cdot M\right)\right)\right), \left({d}^{2}\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right) \cdot M\right)\right), \left({d}^{2}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(M \cdot \left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(\left(D \cdot D\right) \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \left(D \cdot M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \left(M \cdot D\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(M \cdot D\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(D \cdot M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

      20. *-lowering-*.f64 67.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   8. Simplified 67.2%

      \[\leadsto \color{blue}{\frac{h \cdot \left(M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right)}{d \cdot d}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right) \cdot h}{\color{blue}{d} \cdot d} \]

      2. times-frac N/A

         \[\leadsto \frac{M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d} \cdot \color{blue}{\frac{h}{d}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}\right), \color{blue}{\left(\frac{h}{d}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right), d\right), \left(\frac{\color{blue}{h}}{d}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right), d\right), \left(\frac{h}{d}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \left(D \cdot M\right)\right)\right)\right), d\right), \left(\frac{h}{d}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(D \cdot M\right)\right)\right)\right), d\right), \left(\frac{h}{d}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right), d\right), \left(\frac{h}{d}\right)\right) \]

      9. /-lowering-/.f64 68.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right), d\right), \mathsf{/.f64}\left(h, \color{blue}{d}\right)\right) \]

   10. Applied egg-rr 68.2%

       \[\leadsto \color{blue}{\frac{M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d} \cdot \frac{h}{d}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -3.4 \cdot 10^{+109}:\\ \;\;\;\;h \cdot \frac{\frac{M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}}{d}\\ \mathbf{elif}\;w \leq -2.5 \cdot 10^{+44}:\\ \;\;\;\;\frac{c0 \cdot d}{D} \cdot \left(\frac{d}{w \cdot h} \cdot \frac{\frac{c0}{D}}{w}\right)\\ \mathbf{elif}\;w \leq -5 \cdot 10^{-131}:\\ \;\;\;\;\frac{\frac{D \cdot \left(0.25 \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d}}{d}\\ \mathbf{elif}\;w \leq -8.2 \cdot 10^{-231}:\\ \;\;\;\;\frac{c0 \cdot d}{D} \cdot \left(\frac{d}{w \cdot h} \cdot \frac{\frac{c0}{D}}{w}\right)\\ \mathbf{elif}\;w \leq 3.1 \cdot 10^{-189}:\\ \;\;\;\;\frac{\frac{D \cdot \left(0.25 \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d}}{d}\\ \mathbf{elif}\;w \leq 2.45 \cdot 10^{+126}:\\ \;\;\;\;\frac{c0 \cdot d}{D} \cdot \left(\frac{d}{w \cdot h} \cdot \frac{\frac{c0}{D}}{w}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{h}{d} \cdot \frac{M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 47.0% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq -1.85 \cdot 10^{-125}:\\ \;\;\;\;\frac{h}{d} \cdot \frac{M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}\\ \mathbf{elif}\;h \leq 5.2 \cdot 10^{-109}:\\ \;\;\;\;\frac{\frac{D \cdot \left(0.25 \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \left(M \cdot \frac{\frac{h \cdot M}{d}}{d}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= h -1.85e-125)
   (* (/ h d) (/ (* M (* 0.25 (* D (* D M)))) d))
   (if (<= h 5.2e-109)
     (/ (/ (* D (* 0.25 (* D (* h (* M M))))) d) d)
     (* 0.25 (* D (* D (* M (/ (/ (* h M) d) d))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (h <= -1.85e-125) {
		tmp = (h / d) * ((M * (0.25 * (D * (D * M)))) / d);
	} else if (h <= 5.2e-109) {
		tmp = ((D * (0.25 * (D * (h * (M * M))))) / d) / d;
	} else {
		tmp = 0.25 * (D * (D * (M * (((h * 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 (h <= (-1.85d-125)) then
        tmp = (h / d_1) * ((m * (0.25d0 * (d * (d * m)))) / d_1)
    else if (h <= 5.2d-109) then
        tmp = ((d * (0.25d0 * (d * (h * (m * m))))) / d_1) / d_1
    else
        tmp = 0.25d0 * (d * (d * (m * (((h * 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 (h <= -1.85e-125) {
		tmp = (h / d) * ((M * (0.25 * (D * (D * M)))) / d);
	} else if (h <= 5.2e-109) {
		tmp = ((D * (0.25 * (D * (h * (M * M))))) / d) / d;
	} else {
		tmp = 0.25 * (D * (D * (M * (((h * M) / d) / d))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if h <= -1.85e-125:
		tmp = (h / d) * ((M * (0.25 * (D * (D * M)))) / d)
	elif h <= 5.2e-109:
		tmp = ((D * (0.25 * (D * (h * (M * M))))) / d) / d
	else:
		tmp = 0.25 * (D * (D * (M * (((h * M) / d) / d))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (h <= -1.85e-125)
		tmp = Float64(Float64(h / d) * Float64(Float64(M * Float64(0.25 * Float64(D * Float64(D * M)))) / d));
	elseif (h <= 5.2e-109)
		tmp = Float64(Float64(Float64(D * Float64(0.25 * Float64(D * Float64(h * Float64(M * M))))) / d) / d);
	else
		tmp = Float64(0.25 * Float64(D * Float64(D * Float64(M * Float64(Float64(Float64(h * M) / d) / d)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (h <= -1.85e-125)
		tmp = (h / d) * ((M * (0.25 * (D * (D * M)))) / d);
	elseif (h <= 5.2e-109)
		tmp = ((D * (0.25 * (D * (h * (M * M))))) / d) / d;
	else
		tmp = 0.25 * (D * (D * (M * (((h * M) / d) / d))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[h, -1.85e-125], N[(N[(h / d), $MachinePrecision] * N[(N[(M * N[(0.25 * N[(D * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 5.2e-109], N[(N[(N[(D * N[(0.25 * N[(D * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision], N[(0.25 * N[(D * N[(D * N[(M * N[(N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.85e-125

   1. Initial program 19.3%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   4. 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. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}}\right)\right) \]

      5. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0}{w}\right)\right) \]

      8. div0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot 0\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + 0\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \left(-1 + \color{blue}{1}\right)\right)\right) \]

   5. Simplified 9.0%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \left(0.25 \cdot \frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}}{c0 \cdot c0} + 0\right)} \]

   6. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \color{blue}{\left({d}^{2}\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)\right), \left({d}^{2}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right) \cdot h\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(h \cdot \left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot \left(M \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left(\left({D}^{2} \cdot M\right) \cdot M\right)\right)\right), \left({d}^{2}\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right) \cdot M\right)\right), \left({d}^{2}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(M \cdot \left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(\left(D \cdot D\right) \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \left(D \cdot M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \left(M \cdot D\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(M \cdot D\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(D \cdot M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

      20. *-lowering-*.f64 43.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   8. Simplified 43.8%

      \[\leadsto \color{blue}{\frac{h \cdot \left(M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right)}{d \cdot d}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right) \cdot h}{\color{blue}{d} \cdot d} \]

      2. times-frac N/A

         \[\leadsto \frac{M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d} \cdot \color{blue}{\frac{h}{d}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}\right), \color{blue}{\left(\frac{h}{d}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right), d\right), \left(\frac{\color{blue}{h}}{d}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right), d\right), \left(\frac{h}{d}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \left(D \cdot M\right)\right)\right)\right), d\right), \left(\frac{h}{d}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(D \cdot M\right)\right)\right)\right), d\right), \left(\frac{h}{d}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right), d\right), \left(\frac{h}{d}\right)\right) \]

      9. /-lowering-/.f64 53.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right), d\right), \mathsf{/.f64}\left(h, \color{blue}{d}\right)\right) \]

   10. Applied egg-rr 53.0%

       \[\leadsto \color{blue}{\frac{M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d} \cdot \frac{h}{d}} \]

   if -1.85e-125 < h < 5.1999999999999997e-109

   1. Initial program 16.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   4. 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. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}}\right)\right) \]

      5. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0}{w}\right)\right) \]

      8. div0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot 0\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + 0\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \left(-1 + \color{blue}{1}\right)\right)\right) \]

   5. Simplified 15.6%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \left(0.25 \cdot \frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}}{c0 \cdot c0} + 0\right)} \]

   6. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \color{blue}{\left({d}^{2}\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)\right), \left({d}^{2}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right) \cdot h\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(h \cdot \left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot \left(M \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left(\left({D}^{2} \cdot M\right) \cdot M\right)\right)\right), \left({d}^{2}\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right) \cdot M\right)\right), \left({d}^{2}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(M \cdot \left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(\left(D \cdot D\right) \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \left(D \cdot M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \left(M \cdot D\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(M \cdot D\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(D \cdot M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

      20. *-lowering-*.f64 39.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   8. Simplified 39.7%

      \[\leadsto \color{blue}{\frac{h \cdot \left(M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right)}{d \cdot d}} \]

   9. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2}}} \]

       2. unpow2 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{d \cdot \color{blue}{d}} \]

       3. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{d}}{\color{blue}{d}} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{d}\right), \color{blue}{d}\right) \]

   11. Simplified 53.5%

       \[\leadsto \color{blue}{\frac{\frac{D \cdot \left(0.25 \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d}}{d}} \]

   if 5.1999999999999997e-109 < h

   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. Add Preprocessing

   3. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   4. 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. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}}\right)\right) \]

      5. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0}{w}\right)\right) \]

      8. div0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot 0\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + 0\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \left(-1 + \color{blue}{1}\right)\right)\right) \]

   5. Simplified 17.0%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \left(0.25 \cdot \frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}}{c0 \cdot c0} + 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \left(\frac{1}{4} \cdot \color{blue}{\frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}}{c0 \cdot c0}}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \left(\frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}}{c0 \cdot c0} \cdot \color{blue}{\frac{1}{4}}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(c0 \cdot c0\right) \cdot \frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}}{c0 \cdot c0}\right) \cdot \color{blue}{\frac{1}{4}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(c0 \cdot c0\right) \cdot \frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}}{c0 \cdot c0}\right), \color{blue}{\frac{1}{4}}\right) \]

   7. Applied egg-rr 28.2%

      \[\leadsto \color{blue}{\left(\left(c0 \cdot c0\right) \cdot \left(\left(D \cdot \left(D \cdot M\right)\right) \cdot \left(M \cdot \frac{h}{d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)}\right)\right)\right) \cdot 0.25} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right)}, \frac{1}{4}\right) \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right), \frac{1}{4}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(D \cdot D\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right), \frac{1}{4}\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(D \cdot \left(D \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right), \frac{1}{4}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \left(D \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right), \frac{1}{4}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(\frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right)\right), \frac{1}{4}\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(\frac{\left(M \cdot M\right) \cdot h}{{d}^{2}}\right)\right)\right), \frac{1}{4}\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(\frac{M \cdot \left(M \cdot h\right)}{{d}^{2}}\right)\right)\right), \frac{1}{4}\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(M \cdot \frac{M \cdot h}{{d}^{2}}\right)\right)\right), \frac{1}{4}\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, \left(\frac{M \cdot h}{{d}^{2}}\right)\right)\right)\right), \frac{1}{4}\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, \left(\frac{h \cdot M}{{d}^{2}}\right)\right)\right)\right), \frac{1}{4}\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, \left(\frac{h \cdot M}{d \cdot d}\right)\right)\right)\right), \frac{1}{4}\right) \]

      12. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, \left(\frac{\frac{h \cdot M}{d}}{d}\right)\right)\right)\right), \frac{1}{4}\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, \left(\frac{\frac{M \cdot h}{d}}{d}\right)\right)\right)\right), \frac{1}{4}\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, \mathsf{/.f64}\left(\left(\frac{M \cdot h}{d}\right), d\right)\right)\right)\right), \frac{1}{4}\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, \mathsf{/.f64}\left(\left(\frac{h \cdot M}{d}\right), d\right)\right)\right)\right), \frac{1}{4}\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(h \cdot M\right), d\right), d\right)\right)\right)\right), \frac{1}{4}\right) \]

      17. *-lowering-*.f64 57.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, M\right), d\right), d\right)\right)\right)\right), \frac{1}{4}\right) \]

   10. Simplified 57.3%

       \[\leadsto \color{blue}{\left(D \cdot \left(D \cdot \left(M \cdot \frac{\frac{h \cdot M}{d}}{d}\right)\right)\right)} \cdot 0.25 \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1.85 \cdot 10^{-125}:\\ \;\;\;\;\frac{h}{d} \cdot \frac{M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}\\ \mathbf{elif}\;h \leq 5.2 \cdot 10^{-109}:\\ \;\;\;\;\frac{\frac{D \cdot \left(0.25 \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \left(M \cdot \frac{\frac{h \cdot M}{d}}{d}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 47.2% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq -1.42 \cdot 10^{-282}:\\ \;\;\;\;h \cdot \frac{\frac{M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \left(M \cdot \frac{\frac{h \cdot M}{d}}{d}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= h -1.42e-282)
   (* h (/ (/ (* M (* 0.25 (* D (* D M)))) d) d))
   (* 0.25 (* D (* D (* M (/ (/ (* h M) d) d)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (h <= -1.42e-282) {
		tmp = h * (((M * (0.25 * (D * (D * M)))) / d) / d);
	} else {
		tmp = 0.25 * (D * (D * (M * (((h * 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 (h <= (-1.42d-282)) then
        tmp = h * (((m * (0.25d0 * (d * (d * m)))) / d_1) / d_1)
    else
        tmp = 0.25d0 * (d * (d * (m * (((h * 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 (h <= -1.42e-282) {
		tmp = h * (((M * (0.25 * (D * (D * M)))) / d) / d);
	} else {
		tmp = 0.25 * (D * (D * (M * (((h * M) / d) / d))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if h <= -1.42e-282:
		tmp = h * (((M * (0.25 * (D * (D * M)))) / d) / d)
	else:
		tmp = 0.25 * (D * (D * (M * (((h * M) / d) / d))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (h <= -1.42e-282)
		tmp = Float64(h * Float64(Float64(Float64(M * Float64(0.25 * Float64(D * Float64(D * M)))) / d) / d));
	else
		tmp = Float64(0.25 * Float64(D * Float64(D * Float64(M * Float64(Float64(Float64(h * M) / d) / d)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (h <= -1.42e-282)
		tmp = h * (((M * (0.25 * (D * (D * M)))) / d) / d);
	else
		tmp = 0.25 * (D * (D * (M * (((h * M) / d) / d))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[h, -1.42e-282], N[(h * N[(N[(N[(M * N[(0.25 * N[(D * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(D * N[(D * N[(M * N[(N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -1.42e-282

   1. Initial program 18.0%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   4. 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. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}}\right)\right) \]

      5. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0}{w}\right)\right) \]

      8. div0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot 0\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + 0\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \left(-1 + \color{blue}{1}\right)\right)\right) \]

   5. Simplified 11.3%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \left(0.25 \cdot \frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}}{c0 \cdot c0} + 0\right)} \]

   6. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \color{blue}{\left({d}^{2}\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)\right), \left({d}^{2}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right) \cdot h\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(h \cdot \left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot \left(M \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left(\left({D}^{2} \cdot M\right) \cdot M\right)\right)\right), \left({d}^{2}\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right) \cdot M\right)\right), \left({d}^{2}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(M \cdot \left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(\left(D \cdot D\right) \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \left(D \cdot M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \left(M \cdot D\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(M \cdot D\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(D \cdot M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

      20. *-lowering-*.f64 42.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   8. Simplified 42.7%

      \[\leadsto \color{blue}{\frac{h \cdot \left(M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right)}{d \cdot d}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto h \cdot \color{blue}{\frac{M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d \cdot d}} \]

      2. *-commutative N/A

         \[\leadsto \frac{M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d \cdot d} \cdot \color{blue}{h} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d \cdot d}\right), \color{blue}{h}\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}}{d}\right), h\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}\right), d\right), h\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right), d\right), d\right), h\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right), d\right), d\right), h\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \left(D \cdot M\right)\right)\right)\right), d\right), d\right), h\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(D \cdot M\right)\right)\right)\right), d\right), d\right), h\right) \]

      10. *-lowering-*.f64 50.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right), d\right), d\right), h\right) \]

   10. Applied egg-rr 50.6%

       \[\leadsto \color{blue}{\frac{\frac{M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}}{d} \cdot h} \]

   if -1.42e-282 < h

   1. Initial program 22.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   4. 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. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}}\right)\right) \]

      5. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0}{w}\right)\right) \]

      8. div0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot 0\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + 0\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \left(-1 + \color{blue}{1}\right)\right)\right) \]

   5. Simplified 16.9%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \left(0.25 \cdot \frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}}{c0 \cdot c0} + 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \left(\frac{1}{4} \cdot \color{blue}{\frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}}{c0 \cdot c0}}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \left(\frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}}{c0 \cdot c0} \cdot \color{blue}{\frac{1}{4}}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(c0 \cdot c0\right) \cdot \frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}}{c0 \cdot c0}\right) \cdot \color{blue}{\frac{1}{4}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(c0 \cdot c0\right) \cdot \frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}}{c0 \cdot c0}\right), \color{blue}{\frac{1}{4}}\right) \]

   7. Applied egg-rr 24.2%

      \[\leadsto \color{blue}{\left(\left(c0 \cdot c0\right) \cdot \left(\left(D \cdot \left(D \cdot M\right)\right) \cdot \left(M \cdot \frac{h}{d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)}\right)\right)\right) \cdot 0.25} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right)}, \frac{1}{4}\right) \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right), \frac{1}{4}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(D \cdot D\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right), \frac{1}{4}\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(D \cdot \left(D \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right), \frac{1}{4}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \left(D \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right), \frac{1}{4}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(\frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right)\right), \frac{1}{4}\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(\frac{\left(M \cdot M\right) \cdot h}{{d}^{2}}\right)\right)\right), \frac{1}{4}\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(\frac{M \cdot \left(M \cdot h\right)}{{d}^{2}}\right)\right)\right), \frac{1}{4}\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(M \cdot \frac{M \cdot h}{{d}^{2}}\right)\right)\right), \frac{1}{4}\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, \left(\frac{M \cdot h}{{d}^{2}}\right)\right)\right)\right), \frac{1}{4}\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, \left(\frac{h \cdot M}{{d}^{2}}\right)\right)\right)\right), \frac{1}{4}\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, \left(\frac{h \cdot M}{d \cdot d}\right)\right)\right)\right), \frac{1}{4}\right) \]

      12. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, \left(\frac{\frac{h \cdot M}{d}}{d}\right)\right)\right)\right), \frac{1}{4}\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, \left(\frac{\frac{M \cdot h}{d}}{d}\right)\right)\right)\right), \frac{1}{4}\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, \mathsf{/.f64}\left(\left(\frac{M \cdot h}{d}\right), d\right)\right)\right)\right), \frac{1}{4}\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, \mathsf{/.f64}\left(\left(\frac{h \cdot M}{d}\right), d\right)\right)\right)\right), \frac{1}{4}\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(h \cdot M\right), d\right), d\right)\right)\right)\right), \frac{1}{4}\right) \]

      17. *-lowering-*.f64 51.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, M\right), d\right), d\right)\right)\right)\right), \frac{1}{4}\right) \]

   10. Simplified 51.5%

       \[\leadsto \color{blue}{\left(D \cdot \left(D \cdot \left(M \cdot \frac{\frac{h \cdot M}{d}}{d}\right)\right)\right)} \cdot 0.25 \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1.42 \cdot 10^{-282}:\\ \;\;\;\;h \cdot \frac{\frac{M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \left(M \cdot \frac{\frac{h \cdot M}{d}}{d}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 47.1% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq -8 \cdot 10^{-179}:\\ \;\;\;\;\frac{h}{d} \cdot \frac{M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \left(M \cdot \frac{\frac{h \cdot M}{d}}{d}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= h -8e-179)
   (* (/ h d) (/ (* M (* 0.25 (* D (* D M)))) d))
   (* 0.25 (* D (* D (* M (/ (/ (* h M) d) d)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (h <= -8e-179) {
		tmp = (h / d) * ((M * (0.25 * (D * (D * M)))) / d);
	} else {
		tmp = 0.25 * (D * (D * (M * (((h * 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 (h <= (-8d-179)) then
        tmp = (h / d_1) * ((m * (0.25d0 * (d * (d * m)))) / d_1)
    else
        tmp = 0.25d0 * (d * (d * (m * (((h * 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 (h <= -8e-179) {
		tmp = (h / d) * ((M * (0.25 * (D * (D * M)))) / d);
	} else {
		tmp = 0.25 * (D * (D * (M * (((h * M) / d) / d))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if h <= -8e-179:
		tmp = (h / d) * ((M * (0.25 * (D * (D * M)))) / d)
	else:
		tmp = 0.25 * (D * (D * (M * (((h * M) / d) / d))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (h <= -8e-179)
		tmp = Float64(Float64(h / d) * Float64(Float64(M * Float64(0.25 * Float64(D * Float64(D * M)))) / d));
	else
		tmp = Float64(0.25 * Float64(D * Float64(D * Float64(M * Float64(Float64(Float64(h * M) / d) / d)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (h <= -8e-179)
		tmp = (h / d) * ((M * (0.25 * (D * (D * M)))) / d);
	else
		tmp = 0.25 * (D * (D * (M * (((h * M) / d) / d))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[h, -8e-179], N[(N[(h / d), $MachinePrecision] * N[(N[(M * N[(0.25 * N[(D * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(D * N[(D * N[(M * N[(N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -8.0000000000000002e-179

   1. Initial program 19.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   4. 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. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}}\right)\right) \]

      5. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0}{w}\right)\right) \]

      8. div0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot 0\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + 0\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \left(-1 + \color{blue}{1}\right)\right)\right) \]

   5. Simplified 9.9%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \left(0.25 \cdot \frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}}{c0 \cdot c0} + 0\right)} \]

   6. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \color{blue}{\left({d}^{2}\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)\right), \left({d}^{2}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right) \cdot h\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(h \cdot \left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot \left(M \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{1}{4} \cdot \left(\left({D}^{2} \cdot M\right) \cdot M\right)\right)\right), \left({d}^{2}\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right) \cdot M\right)\right), \left({d}^{2}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(M \cdot \left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \left(\frac{1}{4} \cdot \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(\left(D \cdot D\right) \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \left(D \cdot M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \left(M \cdot D\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(M \cdot D\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(D \cdot M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

      20. *-lowering-*.f64 42.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   8. Simplified 42.6%

      \[\leadsto \color{blue}{\frac{h \cdot \left(M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right)}{d \cdot d}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right) \cdot h}{\color{blue}{d} \cdot d} \]

      2. times-frac N/A

         \[\leadsto \frac{M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d} \cdot \color{blue}{\frac{h}{d}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}\right), \color{blue}{\left(\frac{h}{d}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right), d\right), \left(\frac{\color{blue}{h}}{d}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)\right), d\right), \left(\frac{h}{d}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \left(D \cdot M\right)\right)\right)\right), d\right), \left(\frac{h}{d}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(D \cdot M\right)\right)\right)\right), d\right), \left(\frac{h}{d}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right), d\right), \left(\frac{h}{d}\right)\right) \]

      9. /-lowering-/.f64 51.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, M\right)\right)\right)\right), d\right), \mathsf{/.f64}\left(h, \color{blue}{d}\right)\right) \]

   10. Applied egg-rr 51.4%

       \[\leadsto \color{blue}{\frac{M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d} \cdot \frac{h}{d}} \]

   if -8.0000000000000002e-179 < h

   1. Initial program 20.8%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
   4. 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. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}}\right)\right) \]

      5. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0}{w}\right)\right) \]

      8. div0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot 0\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + 0\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \left(-1 + \color{blue}{1}\right)\right)\right) \]

   5. Simplified 16.3%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \left(0.25 \cdot \frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}}{c0 \cdot c0} + 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \left(\frac{1}{4} \cdot \color{blue}{\frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}}{c0 \cdot c0}}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \left(\frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}}{c0 \cdot c0} \cdot \color{blue}{\frac{1}{4}}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(c0 \cdot c0\right) \cdot \frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}}{c0 \cdot c0}\right) \cdot \color{blue}{\frac{1}{4}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(c0 \cdot c0\right) \cdot \frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}}{c0 \cdot c0}\right), \color{blue}{\frac{1}{4}}\right) \]

   7. Applied egg-rr 22.8%

      \[\leadsto \color{blue}{\left(\left(c0 \cdot c0\right) \cdot \left(\left(D \cdot \left(D \cdot M\right)\right) \cdot \left(M \cdot \frac{h}{d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)}\right)\right)\right) \cdot 0.25} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right)}, \frac{1}{4}\right) \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right), \frac{1}{4}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(D \cdot D\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right), \frac{1}{4}\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(D \cdot \left(D \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right), \frac{1}{4}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \left(D \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right), \frac{1}{4}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(\frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right)\right), \frac{1}{4}\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(\frac{\left(M \cdot M\right) \cdot h}{{d}^{2}}\right)\right)\right), \frac{1}{4}\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(\frac{M \cdot \left(M \cdot h\right)}{{d}^{2}}\right)\right)\right), \frac{1}{4}\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(M \cdot \frac{M \cdot h}{{d}^{2}}\right)\right)\right), \frac{1}{4}\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, \left(\frac{M \cdot h}{{d}^{2}}\right)\right)\right)\right), \frac{1}{4}\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, \left(\frac{h \cdot M}{{d}^{2}}\right)\right)\right)\right), \frac{1}{4}\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, \left(\frac{h \cdot M}{d \cdot d}\right)\right)\right)\right), \frac{1}{4}\right) \]

      12. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, \left(\frac{\frac{h \cdot M}{d}}{d}\right)\right)\right)\right), \frac{1}{4}\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, \left(\frac{\frac{M \cdot h}{d}}{d}\right)\right)\right)\right), \frac{1}{4}\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, \mathsf{/.f64}\left(\left(\frac{M \cdot h}{d}\right), d\right)\right)\right)\right), \frac{1}{4}\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, \mathsf{/.f64}\left(\left(\frac{h \cdot M}{d}\right), d\right)\right)\right)\right), \frac{1}{4}\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(h \cdot M\right), d\right), d\right)\right)\right)\right), \frac{1}{4}\right) \]

      17. *-lowering-*.f64 50.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, M\right), d\right), d\right)\right)\right)\right), \frac{1}{4}\right) \]

   10. Simplified 50.4%

       \[\leadsto \color{blue}{\left(D \cdot \left(D \cdot \left(M \cdot \frac{\frac{h \cdot M}{d}}{d}\right)\right)\right)} \cdot 0.25 \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -8 \cdot 10^{-179}:\\ \;\;\;\;\frac{h}{d} \cdot \frac{M \cdot \left(0.25 \cdot \left(D \cdot \left(D \cdot M\right)\right)\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \left(M \cdot \frac{\frac{h \cdot M}{d}}{d}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 48.2% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ 0.25 \cdot \left(D \cdot \left(D \cdot \left(M \cdot \frac{\frac{h \cdot M}{d}}{d}\right)\right)\right) \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (* 0.25 (* D (* D (* M (/ (/ (* h M) d) d))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return 0.25 * (D * (D * (M * (((h * 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 * (m * (((h * 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 * (M * (((h * M) / d) / d))));
}

Python

def code(c0, w, h, D, d, M):
	return 0.25 * (D * (D * (M * (((h * M) / d) / d))))

Julia

function code(c0, w, h, D, d, M)
	return Float64(0.25 * Float64(D * Float64(D * Float64(M * Float64(Float64(Float64(h * M) / d) / d)))))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = 0.25 * (D * (D * (M * (((h * M) / d) / d))));
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(0.25 * N[(D * N[(D * N[(M * N[(N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 20.4%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
2. Add Preprocessing

3. Taylor expanded in c0 around -inf

   \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
4. 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. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}}\right)\right) \]

   5. distribute-lft1-in N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right)\right) \]

   7. mul0-lft N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot \frac{0}{w}\right)\right) \]

   8. div0 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{-1}{2} \cdot 0\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + 0\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \left(-1 + \color{blue}{1}\right)\right)\right) \]

5. Simplified 14.2%

   \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \left(0.25 \cdot \frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}}{c0 \cdot c0} + 0\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto \left(c0 \cdot c0\right) \cdot \left(\frac{1}{4} \cdot \color{blue}{\frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}}{c0 \cdot c0}}\right) \]

   2. *-commutative N/A

      \[\leadsto \left(c0 \cdot c0\right) \cdot \left(\frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}}{c0 \cdot c0} \cdot \color{blue}{\frac{1}{4}}\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(\left(c0 \cdot c0\right) \cdot \frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}}{c0 \cdot c0}\right) \cdot \color{blue}{\frac{1}{4}} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(c0 \cdot c0\right) \cdot \frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}}{c0 \cdot c0}\right), \color{blue}{\frac{1}{4}}\right) \]

7. Applied egg-rr 20.1%

   \[\leadsto \color{blue}{\left(\left(c0 \cdot c0\right) \cdot \left(\left(D \cdot \left(D \cdot M\right)\right) \cdot \left(M \cdot \frac{h}{d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)}\right)\right)\right) \cdot 0.25} \]

8. Taylor expanded in c0 around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right)}, \frac{1}{4}\right) \]
9. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \mathsf{*.f64}\left(\left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right), \frac{1}{4}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(D \cdot D\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right), \frac{1}{4}\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(D \cdot \left(D \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right), \frac{1}{4}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \left(D \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right), \frac{1}{4}\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(\frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right)\right), \frac{1}{4}\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(\frac{\left(M \cdot M\right) \cdot h}{{d}^{2}}\right)\right)\right), \frac{1}{4}\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(\frac{M \cdot \left(M \cdot h\right)}{{d}^{2}}\right)\right)\right), \frac{1}{4}\right) \]

   8. associate-/l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(M \cdot \frac{M \cdot h}{{d}^{2}}\right)\right)\right), \frac{1}{4}\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, \left(\frac{M \cdot h}{{d}^{2}}\right)\right)\right)\right), \frac{1}{4}\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, \left(\frac{h \cdot M}{{d}^{2}}\right)\right)\right)\right), \frac{1}{4}\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, \left(\frac{h \cdot M}{d \cdot d}\right)\right)\right)\right), \frac{1}{4}\right) \]

   12. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, \left(\frac{\frac{h \cdot M}{d}}{d}\right)\right)\right)\right), \frac{1}{4}\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, \left(\frac{\frac{M \cdot h}{d}}{d}\right)\right)\right)\right), \frac{1}{4}\right) \]

   14. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, \mathsf{/.f64}\left(\left(\frac{M \cdot h}{d}\right), d\right)\right)\right)\right), \frac{1}{4}\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, \mathsf{/.f64}\left(\left(\frac{h \cdot M}{d}\right), d\right)\right)\right)\right), \frac{1}{4}\right) \]

   16. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(h \cdot M\right), d\right), d\right)\right)\right)\right), \frac{1}{4}\right) \]

   17. *-lowering-*.f64 47.7%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, M\right), d\right), d\right)\right)\right)\right), \frac{1}{4}\right) \]

10. Simplified 47.7%

    \[\leadsto \color{blue}{\left(D \cdot \left(D \cdot \left(M \cdot \frac{\frac{h \cdot M}{d}}{d}\right)\right)\right)} \cdot 0.25 \]

11. Final simplification 47.7%

    \[\leadsto 0.25 \cdot \left(D \cdot \left(D \cdot \left(M \cdot \frac{\frac{h \cdot M}{d}}{d}\right)\right)\right) \]
12. Add Preprocessing

Alternative 12: 33.5% 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 20.4%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
2. Add Preprocessing

3. Taylor expanded in c0 around -inf

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{{c0}^{2} \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}} \]
4. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \frac{-1}{2} \cdot \left({c0}^{2} \cdot \color{blue}{\frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}}\right) \]

   2. distribute-lft1-in N/A

      \[\leadsto \frac{-1}{2} \cdot \left({c0}^{2} \cdot \frac{\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right) \]

   3. metadata-eval N/A

      \[\leadsto \frac{-1}{2} \cdot \left({c0}^{2} \cdot \frac{0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right) \]

   4. mul0-lft N/A

      \[\leadsto \frac{-1}{2} \cdot \left({c0}^{2} \cdot \frac{0}{w}\right) \]

   5. div0 N/A

      \[\leadsto \frac{-1}{2} \cdot \left({c0}^{2} \cdot 0\right) \]

   6. mul0-rgt N/A

      \[\leadsto \frac{-1}{2} \cdot 0 \]

   7. metadata-eval 36.1%

      \[\leadsto 0 \]

5. Simplified 36.1%

   \[\leadsto \color{blue}{0} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024191 
(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))))))