Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.0% → 62.2%

Time: 26.4s

Alternatives: 19

Speedup: 151.0×

• could not determine a ground truth

  1. c0 = 6.741885255771015e+305
  2. w = -4.0601698872088055e-252
  3. h = 1.700369561690123e-246
  4. D = 1.1495535754806119e-290
  5. d = -2.3981478984091047e+270
  6. M = 2.5932518674077528e-138

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 24.0% 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: 62.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\left(\frac{0.5}{\frac{w}{c0}} \cdot \frac{\frac{2 \cdot \frac{d}{\frac{w}{c0}}}{D}}{\frac{h}{d}}\right) \cdot \frac{1}{D}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(h \cdot \left(M \cdot M\right)\right) \cdot \frac{D}{\frac{d}{D}}}{\frac{d}{0.25}}\\ \end{array} \end{array} \]

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)
     (*
      (* (/ 0.5 (/ w c0)) (/ (/ (* 2.0 (/ d (/ w c0))) D) (/ h d)))
      (/ 1.0 D))
     (/ (* (* h (* M M)) (/ D (/ d D))) (/ d 0.25)))))

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 = ((0.5 / (w / c0)) * (((2.0 * (d / (w / c0))) / D) / (h / d))) * (1.0 / D);
	} else {
		tmp = ((h * (M * M)) * (D / (d / D))) / (d / 0.25);
	}
	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 = ((0.5 / (w / c0)) * (((2.0 * (d / (w / c0))) / D) / (h / d))) * (1.0 / D);
	} else {
		tmp = ((h * (M * M)) * (D / (d / D))) / (d / 0.25);
	}
	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 = ((0.5 / (w / c0)) * (((2.0 * (d / (w / c0))) / D) / (h / d))) * (1.0 / D)
	else:
		tmp = ((h * (M * M)) * (D / (d / D))) / (d / 0.25)
	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(0.5 / Float64(w / c0)) * Float64(Float64(Float64(2.0 * Float64(d / Float64(w / c0))) / D) / Float64(h / d))) * Float64(1.0 / D));
	else
		tmp = Float64(Float64(Float64(h * Float64(M * M)) * Float64(D / Float64(d / D))) / Float64(d / 0.25));
	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 = ((0.5 / (w / c0)) * (((2.0 * (d / (w / c0))) / D) / (h / d))) * (1.0 / D);
	else
		tmp = ((h * (M * M)) * (D / (d / D))) / (d / 0.25);
	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[(0.5 / N[(w / c0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(2.0 * N[(d / N[(w / c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision] / N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / D), $MachinePrecision]), $MachinePrecision], N[(N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] * N[(D / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d / 0.25), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 68.3%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. unpow2 N/A

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 69.3%

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

   5. Simplified 69.3%

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. frac-times N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. associate-/r* N/A

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

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

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

      10. associate-/l* N/A

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

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

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

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

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

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

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

      14. *-lowering-*.f64 73.0%

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

   7. Applied egg-rr 73.0%

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

      1. associate-*r/ N/A

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

      2. associate-/r* N/A

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

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

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

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

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

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

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

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

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

      9. associate-*r/ N/A

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

      10. clear-num N/A

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

      11. un-div-inv N/A

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

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

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

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

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

      14. *-lowering-*.f64 75.1%

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

   9. Applied egg-rr 75.1%

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

       1. associate-*r/ N/A

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

       2. div-inv N/A

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

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

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

   11. Applied egg-rr 82.1%

       \[\leadsto \color{blue}{\left(\frac{0.5}{\frac{w}{c0}} \cdot \frac{\frac{2 \cdot \frac{d}{\frac{w}{c0}}}{D}}{\frac{h}{d}}\right) \cdot \frac{1}{D}} \]

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

   1. Initial program 0.0%

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

      1. associate-*l/ N/A

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

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

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

   3. Simplified 0.1%

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

   5. Taylor expanded in c0 around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

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

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

   7. Simplified 20.3%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\left(\frac{-0.5 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot w\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)} + 0\right) \cdot \left(-c0\right)\right)}}{2 \cdot w} \]

   8. Taylor expanded in c0 around 0

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 38.8%

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

   10. Simplified 38.8%

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

       1. *-commutative N/A

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

       2. times-frac N/A

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

       3. clear-num N/A

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

       4. un-div-inv N/A

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

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

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

       6. *-commutative N/A

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

       7. associate-/l* N/A

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

       8. associate-*r/ N/A

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

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

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

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

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

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

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

       12. clear-num N/A

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

       13. un-div-inv N/A

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

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

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

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

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

       16. /-lowering-/.f64 48.9%

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

   12. Applied egg-rr 48.9%

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

Alternative 2: 38.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(D \cdot 0.25\right) \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot D\right)\right)}{d \cdot d}\\ \mathbf{if}\;w \leq -2.1 \cdot 10^{+229}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;w \leq -1.75 \cdot 10^{-85}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot \frac{d}{D \cdot D}\right)\\ \mathbf{elif}\;w \leq -1.05 \cdot 10^{-111}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;w \leq -4.2 \cdot 10^{-208}:\\ \;\;\;\;\frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{D \cdot \left(D \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)}\\ \mathbf{elif}\;w \leq 7.8 \cdot 10^{-119}:\\ \;\;\;\;\frac{D \cdot \left(D \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot 0.25\right)\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d \cdot \left(c0 \cdot d\right)}{w}}{D} \cdot \frac{c0}{w \cdot \left(h \cdot D\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* (* D 0.25) (* (* M M) (* h D))) (* d d))))
   (if (<= w -2.1e+229)
     t_0
     (if (<= w -1.75e-85)
       (* (/ c0 (* 2.0 w)) (* (* 2.0 (/ (* c0 (/ d w)) h)) (/ d (* D D))))
       (if (<= w -1.05e-111)
         t_0
         (if (<= w -4.2e-208)
           (/ (* c0 (* c0 (* d d))) (* D (* D (* h (* w w)))))
           (if (<= w 7.8e-119)
             (/ (* D (* D (* (* h (* M M)) 0.25))) (* d d))
             (* (/ (/ (* d (* c0 d)) w) D) (/ c0 (* w (* h D)))))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((D * 0.25) * ((M * M) * (h * D))) / (d * d);
	double tmp;
	if (w <= -2.1e+229) {
		tmp = t_0;
	} else if (w <= -1.75e-85) {
		tmp = (c0 / (2.0 * w)) * ((2.0 * ((c0 * (d / w)) / h)) * (d / (D * D)));
	} else if (w <= -1.05e-111) {
		tmp = t_0;
	} else if (w <= -4.2e-208) {
		tmp = (c0 * (c0 * (d * d))) / (D * (D * (h * (w * w))));
	} else if (w <= 7.8e-119) {
		tmp = (D * (D * ((h * (M * M)) * 0.25))) / (d * d);
	} else {
		tmp = (((d * (c0 * d)) / w) / D) * (c0 / (w * (h * D)));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((d * 0.25d0) * ((m * m) * (h * d))) / (d_1 * d_1)
    if (w <= (-2.1d+229)) then
        tmp = t_0
    else if (w <= (-1.75d-85)) then
        tmp = (c0 / (2.0d0 * w)) * ((2.0d0 * ((c0 * (d_1 / w)) / h)) * (d_1 / (d * d)))
    else if (w <= (-1.05d-111)) then
        tmp = t_0
    else if (w <= (-4.2d-208)) then
        tmp = (c0 * (c0 * (d_1 * d_1))) / (d * (d * (h * (w * w))))
    else if (w <= 7.8d-119) then
        tmp = (d * (d * ((h * (m * m)) * 0.25d0))) / (d_1 * d_1)
    else
        tmp = (((d_1 * (c0 * d_1)) / w) / d) * (c0 / (w * (h * d)))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((D * 0.25) * ((M * M) * (h * D))) / (d * d);
	double tmp;
	if (w <= -2.1e+229) {
		tmp = t_0;
	} else if (w <= -1.75e-85) {
		tmp = (c0 / (2.0 * w)) * ((2.0 * ((c0 * (d / w)) / h)) * (d / (D * D)));
	} else if (w <= -1.05e-111) {
		tmp = t_0;
	} else if (w <= -4.2e-208) {
		tmp = (c0 * (c0 * (d * d))) / (D * (D * (h * (w * w))));
	} else if (w <= 7.8e-119) {
		tmp = (D * (D * ((h * (M * M)) * 0.25))) / (d * d);
	} else {
		tmp = (((d * (c0 * d)) / w) / D) * (c0 / (w * (h * D)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = ((D * 0.25) * ((M * M) * (h * D))) / (d * d)
	tmp = 0
	if w <= -2.1e+229:
		tmp = t_0
	elif w <= -1.75e-85:
		tmp = (c0 / (2.0 * w)) * ((2.0 * ((c0 * (d / w)) / h)) * (d / (D * D)))
	elif w <= -1.05e-111:
		tmp = t_0
	elif w <= -4.2e-208:
		tmp = (c0 * (c0 * (d * d))) / (D * (D * (h * (w * w))))
	elif w <= 7.8e-119:
		tmp = (D * (D * ((h * (M * M)) * 0.25))) / (d * d)
	else:
		tmp = (((d * (c0 * d)) / w) / D) * (c0 / (w * (h * D)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(Float64(D * 0.25) * Float64(Float64(M * M) * Float64(h * D))) / Float64(d * d))
	tmp = 0.0
	if (w <= -2.1e+229)
		tmp = t_0;
	elseif (w <= -1.75e-85)
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(Float64(2.0 * Float64(Float64(c0 * Float64(d / w)) / h)) * Float64(d / Float64(D * D))));
	elseif (w <= -1.05e-111)
		tmp = t_0;
	elseif (w <= -4.2e-208)
		tmp = Float64(Float64(c0 * Float64(c0 * Float64(d * d))) / Float64(D * Float64(D * Float64(h * Float64(w * w)))));
	elseif (w <= 7.8e-119)
		tmp = Float64(Float64(D * Float64(D * Float64(Float64(h * Float64(M * M)) * 0.25))) / Float64(d * d));
	else
		tmp = Float64(Float64(Float64(Float64(d * Float64(c0 * d)) / w) / D) * Float64(c0 / Float64(w * Float64(h * D))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = ((D * 0.25) * ((M * M) * (h * D))) / (d * d);
	tmp = 0.0;
	if (w <= -2.1e+229)
		tmp = t_0;
	elseif (w <= -1.75e-85)
		tmp = (c0 / (2.0 * w)) * ((2.0 * ((c0 * (d / w)) / h)) * (d / (D * D)));
	elseif (w <= -1.05e-111)
		tmp = t_0;
	elseif (w <= -4.2e-208)
		tmp = (c0 * (c0 * (d * d))) / (D * (D * (h * (w * w))));
	elseif (w <= 7.8e-119)
		tmp = (D * (D * ((h * (M * M)) * 0.25))) / (d * d);
	else
		tmp = (((d * (c0 * d)) / w) / D) * (c0 / (w * (h * D)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(N[(D * 0.25), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, -2.1e+229], t$95$0, If[LessEqual[w, -1.75e-85], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[(N[(c0 * N[(d / w), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision] * N[(d / N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, -1.05e-111], t$95$0, If[LessEqual[w, -4.2e-208], N[(N[(c0 * N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(D * N[(D * N[(h * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 7.8e-119], N[(N[(D * N[(D * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(d * N[(c0 * d), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision] / D), $MachinePrecision] * N[(c0 / N[(w * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if w < -2.09999999999999988e229 or -1.74999999999999989e-85 < w < -1.0499999999999999e-111

   1. Initial program 0.8%

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

      1. associate-*l/ N/A

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

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

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

   3. Simplified 0.8%

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

   5. Taylor expanded in c0 around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

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

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

   7. Simplified 20.8%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\left(\frac{-0.5 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot w\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)} + 0\right) \cdot \left(-c0\right)\right)}}{2 \cdot w} \]

   8. Taylor expanded in c0 around 0

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 50.5%

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

   10. Simplified 50.5%

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

       1. associate-*l* N/A

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

       2. associate-*r* N/A

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

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

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

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

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

       5. associate-*r* N/A

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

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

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

       7. *-commutative N/A

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

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

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

       9. *-lowering-*.f64 68.9%

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

   12. Applied egg-rr 68.9%

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

   if -2.09999999999999988e229 < w < -1.74999999999999989e-85

   1. Initial program 33.1%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. unpow2 N/A

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 41.5%

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

   5. Simplified 41.5%

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. frac-times N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. associate-/r* N/A

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

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

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

      10. associate-/l* N/A

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

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

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

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

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

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

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

      14. *-lowering-*.f64 50.6%

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

   7. Applied egg-rr 50.6%

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

   if -1.0499999999999999e-111 < w < -4.20000000000000024e-208

   1. Initial program 44.7%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. unpow2 N/A

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 57.3%

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

   5. Simplified 57.3%

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. frac-times N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. associate-/r* N/A

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

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

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

      10. associate-/l* N/A

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

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

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

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

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

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

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

      14. *-lowering-*.f64 57.0%

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

   7. Applied egg-rr 57.0%

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

   8. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]
   9. 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. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

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

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

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

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 69.4%

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

   10. Simplified 69.4%

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

   if -4.20000000000000024e-208 < w < 7.7999999999999998e-119

   1. Initial program 14.8%

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

      1. associate-*l/ N/A

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

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

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

   3. Simplified 15.0%

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

   5. Taylor expanded in c0 around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

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

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

   7. Simplified 19.5%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\left(\frac{-0.5 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot w\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)} + 0\right) \cdot \left(-c0\right)\right)}}{2 \cdot w} \]

   8. Taylor expanded in c0 around 0

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 47.6%

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

   10. Simplified 47.6%

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

   11. Taylor expanded in D around 0

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

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

       4. associate-*r* N/A

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

       5. associate-*l* N/A

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

       8. associate-*l* N/A

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

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

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

       10. associate-*r* N/A

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

       11. associate-*l* N/A

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

       12. *-commutative N/A

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

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

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

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

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

       15. *-commutative N/A

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

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

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

       17. unpow2 N/A

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

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

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

       19. unpow2 N/A

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

       20. *-lowering-*.f64 55.8%

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

   13. Simplified 55.8%

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

   if 7.7999999999999998e-119 < w

   1. Initial program 26.4%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. unpow2 N/A

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 30.7%

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

   5. Simplified 30.7%

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. frac-times N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. associate-/r* N/A

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

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

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

      10. associate-/l* N/A

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

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

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

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

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

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

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

      14. *-lowering-*.f64 39.8%

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

   7. Applied egg-rr 39.8%

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

      1. associate-*l* N/A

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

      2. associate-*r/ N/A

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

      3. associate-/l/ N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. clear-num N/A

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

      10. un-div-inv N/A

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

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

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

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

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

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

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

   9. Applied egg-rr 47.5%

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

   11. Applied egg-rr 46.4%

       \[\leadsto \color{blue}{\frac{-\frac{d \cdot \left(c0 \cdot d\right)}{w}}{D} \cdot \frac{c0}{w \cdot \left(h \cdot \left(0 - D\right)\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -2.1 \cdot 10^{+229}:\\ \;\;\;\;\frac{\left(D \cdot 0.25\right) \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot D\right)\right)}{d \cdot d}\\ \mathbf{elif}\;w \leq -1.75 \cdot 10^{-85}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot \frac{d}{D \cdot D}\right)\\ \mathbf{elif}\;w \leq -1.05 \cdot 10^{-111}:\\ \;\;\;\;\frac{\left(D \cdot 0.25\right) \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot D\right)\right)}{d \cdot d}\\ \mathbf{elif}\;w \leq -4.2 \cdot 10^{-208}:\\ \;\;\;\;\frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{D \cdot \left(D \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)}\\ \mathbf{elif}\;w \leq 7.8 \cdot 10^{-119}:\\ \;\;\;\;\frac{D \cdot \left(D \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot 0.25\right)\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d \cdot \left(c0 \cdot d\right)}{w}}{D} \cdot \frac{c0}{w \cdot \left(h \cdot D\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 41.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d}{w \cdot h}\\ t_1 := \frac{c0}{2 \cdot w}\\ \mathbf{if}\;w \leq -1.25 \cdot 10^{-214}:\\ \;\;\;\;t\_1 \cdot \frac{\frac{2 \cdot \left(c0 \cdot t\_0\right)}{\frac{D}{d}}}{D}\\ \mathbf{elif}\;w \leq 3.8 \cdot 10^{-119}:\\ \;\;\;\;\frac{D \cdot \left(D \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot 0.25\right)\right)}{d \cdot d}\\ \mathbf{elif}\;w \leq 1.75 \cdot 10^{+30}:\\ \;\;\;\;t\_1 \cdot \left(\frac{c0 \cdot 2}{D} \cdot \frac{t\_0}{\frac{D}{d}}\right)\\ \mathbf{elif}\;w \leq 1.5 \cdot 10^{+41}:\\ \;\;\;\;\frac{0.25 \cdot \left(\left(h \cdot M\right) \cdot \left(\left(D \cdot D\right) \cdot M\right)\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{\frac{d}{D} \cdot \frac{2}{\frac{w}{c0 \cdot d}}}{h \cdot D}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ d (* w h))) (t_1 (/ c0 (* 2.0 w))))
   (if (<= w -1.25e-214)
     (* t_1 (/ (/ (* 2.0 (* c0 t_0)) (/ D d)) D))
     (if (<= w 3.8e-119)
       (/ (* D (* D (* (* h (* M M)) 0.25))) (* d d))
       (if (<= w 1.75e+30)
         (* t_1 (* (/ (* c0 2.0) D) (/ t_0 (/ D d))))
         (if (<= w 1.5e+41)
           (/ (* 0.25 (* (* h M) (* (* D D) M))) (* d d))
           (* t_1 (/ (* (/ d D) (/ 2.0 (/ w (* c0 d)))) (* h D)))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = d / (w * h);
	double t_1 = c0 / (2.0 * w);
	double tmp;
	if (w <= -1.25e-214) {
		tmp = t_1 * (((2.0 * (c0 * t_0)) / (D / d)) / D);
	} else if (w <= 3.8e-119) {
		tmp = (D * (D * ((h * (M * M)) * 0.25))) / (d * d);
	} else if (w <= 1.75e+30) {
		tmp = t_1 * (((c0 * 2.0) / D) * (t_0 / (D / d)));
	} else if (w <= 1.5e+41) {
		tmp = (0.25 * ((h * M) * ((D * D) * M))) / (d * d);
	} else {
		tmp = t_1 * (((d / D) * (2.0 / (w / (c0 * d)))) / (h * 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_1 / (w * h)
    t_1 = c0 / (2.0d0 * w)
    if (w <= (-1.25d-214)) then
        tmp = t_1 * (((2.0d0 * (c0 * t_0)) / (d / d_1)) / d)
    else if (w <= 3.8d-119) then
        tmp = (d * (d * ((h * (m * m)) * 0.25d0))) / (d_1 * d_1)
    else if (w <= 1.75d+30) then
        tmp = t_1 * (((c0 * 2.0d0) / d) * (t_0 / (d / d_1)))
    else if (w <= 1.5d+41) then
        tmp = (0.25d0 * ((h * m) * ((d * d) * m))) / (d_1 * d_1)
    else
        tmp = t_1 * (((d_1 / d) * (2.0d0 / (w / (c0 * d_1)))) / (h * d))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = d / (w * h);
	double t_1 = c0 / (2.0 * w);
	double tmp;
	if (w <= -1.25e-214) {
		tmp = t_1 * (((2.0 * (c0 * t_0)) / (D / d)) / D);
	} else if (w <= 3.8e-119) {
		tmp = (D * (D * ((h * (M * M)) * 0.25))) / (d * d);
	} else if (w <= 1.75e+30) {
		tmp = t_1 * (((c0 * 2.0) / D) * (t_0 / (D / d)));
	} else if (w <= 1.5e+41) {
		tmp = (0.25 * ((h * M) * ((D * D) * M))) / (d * d);
	} else {
		tmp = t_1 * (((d / D) * (2.0 / (w / (c0 * d)))) / (h * D));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = d / (w * h)
	t_1 = c0 / (2.0 * w)
	tmp = 0
	if w <= -1.25e-214:
		tmp = t_1 * (((2.0 * (c0 * t_0)) / (D / d)) / D)
	elif w <= 3.8e-119:
		tmp = (D * (D * ((h * (M * M)) * 0.25))) / (d * d)
	elif w <= 1.75e+30:
		tmp = t_1 * (((c0 * 2.0) / D) * (t_0 / (D / d)))
	elif w <= 1.5e+41:
		tmp = (0.25 * ((h * M) * ((D * D) * M))) / (d * d)
	else:
		tmp = t_1 * (((d / D) * (2.0 / (w / (c0 * d)))) / (h * D))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(d / Float64(w * h))
	t_1 = Float64(c0 / Float64(2.0 * w))
	tmp = 0.0
	if (w <= -1.25e-214)
		tmp = Float64(t_1 * Float64(Float64(Float64(2.0 * Float64(c0 * t_0)) / Float64(D / d)) / D));
	elseif (w <= 3.8e-119)
		tmp = Float64(Float64(D * Float64(D * Float64(Float64(h * Float64(M * M)) * 0.25))) / Float64(d * d));
	elseif (w <= 1.75e+30)
		tmp = Float64(t_1 * Float64(Float64(Float64(c0 * 2.0) / D) * Float64(t_0 / Float64(D / d))));
	elseif (w <= 1.5e+41)
		tmp = Float64(Float64(0.25 * Float64(Float64(h * M) * Float64(Float64(D * D) * M))) / Float64(d * d));
	else
		tmp = Float64(t_1 * Float64(Float64(Float64(d / D) * Float64(2.0 / Float64(w / Float64(c0 * d)))) / Float64(h * D)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = d / (w * h);
	t_1 = c0 / (2.0 * w);
	tmp = 0.0;
	if (w <= -1.25e-214)
		tmp = t_1 * (((2.0 * (c0 * t_0)) / (D / d)) / D);
	elseif (w <= 3.8e-119)
		tmp = (D * (D * ((h * (M * M)) * 0.25))) / (d * d);
	elseif (w <= 1.75e+30)
		tmp = t_1 * (((c0 * 2.0) / D) * (t_0 / (D / d)));
	elseif (w <= 1.5e+41)
		tmp = (0.25 * ((h * M) * ((D * D) * M))) / (d * d);
	else
		tmp = t_1 * (((d / D) * (2.0 / (w / (c0 * d)))) / (h * D));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(d / N[(w * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, -1.25e-214], N[(t$95$1 * N[(N[(N[(2.0 * N[(c0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(D / d), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 3.8e-119], N[(N[(D * N[(D * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 1.75e+30], N[(t$95$1 * N[(N[(N[(c0 * 2.0), $MachinePrecision] / D), $MachinePrecision] * N[(t$95$0 / N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 1.5e+41], N[(N[(0.25 * N[(N[(h * M), $MachinePrecision] * N[(N[(D * D), $MachinePrecision] * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(N[(d / D), $MachinePrecision] * N[(2.0 / N[(w / N[(c0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if w < -1.2499999999999999e-214

   1. Initial program 30.9%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. unpow2 N/A

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 40.1%

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

   5. Simplified 40.1%

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. frac-times N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. associate-/r* N/A

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

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

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

      10. associate-/l* N/A

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

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

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

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

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

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

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

      14. *-lowering-*.f64 44.7%

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

   7. Applied egg-rr 44.7%

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

      1. associate-/r* N/A

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

      2. associate-*r/ N/A

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

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

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

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

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

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

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

      8. associate-/l* N/A

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

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

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

      10. associate-/l/ N/A

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

      11. *-commutative N/A

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

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

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

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

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

      14. /-lowering-/.f64 54.8%

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

   9. Applied egg-rr 54.8%

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

   if -1.2499999999999999e-214 < w < 3.79999999999999975e-119

   1. Initial program 14.8%

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

      1. associate-*l/ N/A

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

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

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

   3. Simplified 15.0%

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

   5. Taylor expanded in c0 around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

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

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

   7. Simplified 19.5%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\left(\frac{-0.5 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot w\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)} + 0\right) \cdot \left(-c0\right)\right)}}{2 \cdot w} \]

   8. Taylor expanded in c0 around 0

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 47.6%

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

   10. Simplified 47.6%

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

   11. Taylor expanded in D around 0

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

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

       4. associate-*r* N/A

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

       5. associate-*l* N/A

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

       8. associate-*l* N/A

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

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

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

       10. associate-*r* N/A

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

       11. associate-*l* N/A

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

       12. *-commutative N/A

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

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

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

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

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

       15. *-commutative N/A

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

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

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

       17. unpow2 N/A

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

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

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

       19. unpow2 N/A

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

       20. *-lowering-*.f64 55.8%

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

   13. Simplified 55.8%

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

   if 3.79999999999999975e-119 < w < 1.75000000000000011e30

   1. Initial program 23.6%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. unpow2 N/A

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 35.1%

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

   5. Simplified 35.1%

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. frac-times N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. associate-/r* N/A

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

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

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

      10. associate-/l* N/A

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

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

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

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

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

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

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

      14. *-lowering-*.f64 43.9%

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

   7. Applied egg-rr 43.9%

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

      1. associate-/r* N/A

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

      2. associate-*r/ N/A

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

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

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

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

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

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

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

      8. associate-/l* N/A

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

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

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

      10. associate-/l/ N/A

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

      11. *-commutative N/A

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

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

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

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

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

      14. /-lowering-/.f64 61.4%

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

   9. Applied egg-rr 61.4%

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

       1. associate-/l/ N/A

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

       2. associate-*r* N/A

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

       3. times-frac N/A

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

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

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

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

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

       6. *-commutative N/A

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

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

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

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

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

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

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

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

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

       11. /-lowering-/.f64 61.5%

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

   11. Applied egg-rr 61.5%

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

   if 1.75000000000000011e30 < w < 1.4999999999999999e41

   1. Initial program 33.3%

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

      1. associate-*l/ N/A

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

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

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

   3. Simplified 16.9%

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

   5. Taylor expanded in c0 around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

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

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

   7. Simplified 83.8%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\left(\frac{-0.5 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot w\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)} + 0\right) \cdot \left(-c0\right)\right)}}{2 \cdot w} \]

   8. Taylor expanded in c0 around 0

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 83.8%

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

   10. Simplified 83.8%

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

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

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

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

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

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

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

       7. *-lowering-*.f64 99.7%

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

   12. Applied egg-rr 99.7%

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

   if 1.4999999999999999e41 < w

   1. Initial program 28.0%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. unpow2 N/A

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 31.2%

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

   5. Simplified 31.2%

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. frac-times N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. associate-/r* N/A

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

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

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

      10. associate-/l* N/A

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

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

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

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

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

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

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

      14. *-lowering-*.f64 41.9%

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

   7. Applied egg-rr 41.9%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. associate-*r/ N/A

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

      4. frac-times N/A

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

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

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

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

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

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

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

      8. associate-*r/ N/A

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

      9. clear-num N/A

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

      10. un-div-inv N/A

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

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

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

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

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

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

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 55.1%

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

   9. Applied egg-rr 55.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{\frac{d}{D} \cdot \frac{2}{\frac{w}{c0 \cdot d}}}{h \cdot D}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -1.25 \cdot 10^{-214}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \frac{\frac{2 \cdot \left(c0 \cdot \frac{d}{w \cdot h}\right)}{\frac{D}{d}}}{D}\\ \mathbf{elif}\;w \leq 3.8 \cdot 10^{-119}:\\ \;\;\;\;\frac{D \cdot \left(D \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot 0.25\right)\right)}{d \cdot d}\\ \mathbf{elif}\;w \leq 1.75 \cdot 10^{+30}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot 2}{D} \cdot \frac{\frac{d}{w \cdot h}}{\frac{D}{d}}\right)\\ \mathbf{elif}\;w \leq 1.5 \cdot 10^{+41}:\\ \;\;\;\;\frac{0.25 \cdot \left(\left(h \cdot M\right) \cdot \left(\left(D \cdot D\right) \cdot M\right)\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \frac{\frac{d}{D} \cdot \frac{2}{\frac{w}{c0 \cdot d}}}{h \cdot D}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 41.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d}{w \cdot h}\\ t_1 := \frac{c0}{2 \cdot w}\\ \mathbf{if}\;w \leq -1.25 \cdot 10^{-214}:\\ \;\;\;\;t\_1 \cdot \left(\frac{d}{D} \cdot \frac{2 \cdot \left(c0 \cdot t\_0\right)}{D}\right)\\ \mathbf{elif}\;w \leq 3.9 \cdot 10^{-119}:\\ \;\;\;\;\frac{D \cdot \left(D \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot 0.25\right)\right)}{d \cdot d}\\ \mathbf{elif}\;w \leq 10^{+30}:\\ \;\;\;\;t\_1 \cdot \left(\frac{c0 \cdot 2}{D} \cdot \frac{t\_0}{\frac{D}{d}}\right)\\ \mathbf{elif}\;w \leq 3.4 \cdot 10^{+43}:\\ \;\;\;\;\frac{0.25 \cdot \left(\left(h \cdot M\right) \cdot \left(\left(D \cdot D\right) \cdot M\right)\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{\frac{d}{D} \cdot \frac{2}{\frac{w}{c0 \cdot d}}}{h \cdot D}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ d (* w h))) (t_1 (/ c0 (* 2.0 w))))
   (if (<= w -1.25e-214)
     (* t_1 (* (/ d D) (/ (* 2.0 (* c0 t_0)) D)))
     (if (<= w 3.9e-119)
       (/ (* D (* D (* (* h (* M M)) 0.25))) (* d d))
       (if (<= w 1e+30)
         (* t_1 (* (/ (* c0 2.0) D) (/ t_0 (/ D d))))
         (if (<= w 3.4e+43)
           (/ (* 0.25 (* (* h M) (* (* D D) M))) (* d d))
           (* t_1 (/ (* (/ d D) (/ 2.0 (/ w (* c0 d)))) (* h D)))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = d / (w * h);
	double t_1 = c0 / (2.0 * w);
	double tmp;
	if (w <= -1.25e-214) {
		tmp = t_1 * ((d / D) * ((2.0 * (c0 * t_0)) / D));
	} else if (w <= 3.9e-119) {
		tmp = (D * (D * ((h * (M * M)) * 0.25))) / (d * d);
	} else if (w <= 1e+30) {
		tmp = t_1 * (((c0 * 2.0) / D) * (t_0 / (D / d)));
	} else if (w <= 3.4e+43) {
		tmp = (0.25 * ((h * M) * ((D * D) * M))) / (d * d);
	} else {
		tmp = t_1 * (((d / D) * (2.0 / (w / (c0 * d)))) / (h * 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_1 / (w * h)
    t_1 = c0 / (2.0d0 * w)
    if (w <= (-1.25d-214)) then
        tmp = t_1 * ((d_1 / d) * ((2.0d0 * (c0 * t_0)) / d))
    else if (w <= 3.9d-119) then
        tmp = (d * (d * ((h * (m * m)) * 0.25d0))) / (d_1 * d_1)
    else if (w <= 1d+30) then
        tmp = t_1 * (((c0 * 2.0d0) / d) * (t_0 / (d / d_1)))
    else if (w <= 3.4d+43) then
        tmp = (0.25d0 * ((h * m) * ((d * d) * m))) / (d_1 * d_1)
    else
        tmp = t_1 * (((d_1 / d) * (2.0d0 / (w / (c0 * d_1)))) / (h * d))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = d / (w * h);
	double t_1 = c0 / (2.0 * w);
	double tmp;
	if (w <= -1.25e-214) {
		tmp = t_1 * ((d / D) * ((2.0 * (c0 * t_0)) / D));
	} else if (w <= 3.9e-119) {
		tmp = (D * (D * ((h * (M * M)) * 0.25))) / (d * d);
	} else if (w <= 1e+30) {
		tmp = t_1 * (((c0 * 2.0) / D) * (t_0 / (D / d)));
	} else if (w <= 3.4e+43) {
		tmp = (0.25 * ((h * M) * ((D * D) * M))) / (d * d);
	} else {
		tmp = t_1 * (((d / D) * (2.0 / (w / (c0 * d)))) / (h * D));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = d / (w * h)
	t_1 = c0 / (2.0 * w)
	tmp = 0
	if w <= -1.25e-214:
		tmp = t_1 * ((d / D) * ((2.0 * (c0 * t_0)) / D))
	elif w <= 3.9e-119:
		tmp = (D * (D * ((h * (M * M)) * 0.25))) / (d * d)
	elif w <= 1e+30:
		tmp = t_1 * (((c0 * 2.0) / D) * (t_0 / (D / d)))
	elif w <= 3.4e+43:
		tmp = (0.25 * ((h * M) * ((D * D) * M))) / (d * d)
	else:
		tmp = t_1 * (((d / D) * (2.0 / (w / (c0 * d)))) / (h * D))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(d / Float64(w * h))
	t_1 = Float64(c0 / Float64(2.0 * w))
	tmp = 0.0
	if (w <= -1.25e-214)
		tmp = Float64(t_1 * Float64(Float64(d / D) * Float64(Float64(2.0 * Float64(c0 * t_0)) / D)));
	elseif (w <= 3.9e-119)
		tmp = Float64(Float64(D * Float64(D * Float64(Float64(h * Float64(M * M)) * 0.25))) / Float64(d * d));
	elseif (w <= 1e+30)
		tmp = Float64(t_1 * Float64(Float64(Float64(c0 * 2.0) / D) * Float64(t_0 / Float64(D / d))));
	elseif (w <= 3.4e+43)
		tmp = Float64(Float64(0.25 * Float64(Float64(h * M) * Float64(Float64(D * D) * M))) / Float64(d * d));
	else
		tmp = Float64(t_1 * Float64(Float64(Float64(d / D) * Float64(2.0 / Float64(w / Float64(c0 * d)))) / Float64(h * D)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = d / (w * h);
	t_1 = c0 / (2.0 * w);
	tmp = 0.0;
	if (w <= -1.25e-214)
		tmp = t_1 * ((d / D) * ((2.0 * (c0 * t_0)) / D));
	elseif (w <= 3.9e-119)
		tmp = (D * (D * ((h * (M * M)) * 0.25))) / (d * d);
	elseif (w <= 1e+30)
		tmp = t_1 * (((c0 * 2.0) / D) * (t_0 / (D / d)));
	elseif (w <= 3.4e+43)
		tmp = (0.25 * ((h * M) * ((D * D) * M))) / (d * d);
	else
		tmp = t_1 * (((d / D) * (2.0 / (w / (c0 * d)))) / (h * D));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(d / N[(w * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, -1.25e-214], N[(t$95$1 * N[(N[(d / D), $MachinePrecision] * N[(N[(2.0 * N[(c0 * t$95$0), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 3.9e-119], N[(N[(D * N[(D * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 1e+30], N[(t$95$1 * N[(N[(N[(c0 * 2.0), $MachinePrecision] / D), $MachinePrecision] * N[(t$95$0 / N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 3.4e+43], N[(N[(0.25 * N[(N[(h * M), $MachinePrecision] * N[(N[(D * D), $MachinePrecision] * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(N[(d / D), $MachinePrecision] * N[(2.0 / N[(w / N[(c0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if w < -1.2499999999999999e-214

   1. Initial program 30.9%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. unpow2 N/A

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 40.1%

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

   5. Simplified 40.1%

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. frac-times N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. associate-/r* N/A

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

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

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

      10. associate-/l* N/A

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

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

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

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

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

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

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

      14. *-lowering-*.f64 44.7%

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

   7. Applied egg-rr 44.7%

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

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

      6. associate-/l* N/A

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

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

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

      8. associate-/l/ N/A

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

      9. *-commutative N/A

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

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

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

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

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

      12. /-lowering-/.f64 53.8%

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

   9. Applied egg-rr 53.8%

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

   if -1.2499999999999999e-214 < w < 3.8999999999999999e-119

   1. Initial program 14.8%

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

      1. associate-*l/ N/A

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

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

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

   3. Simplified 15.0%

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

   5. Taylor expanded in c0 around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

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

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

   7. Simplified 19.5%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\left(\frac{-0.5 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot w\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)} + 0\right) \cdot \left(-c0\right)\right)}}{2 \cdot w} \]

   8. Taylor expanded in c0 around 0

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 47.6%

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

   10. Simplified 47.6%

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

   11. Taylor expanded in D around 0

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

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

       4. associate-*r* N/A

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

       5. associate-*l* N/A

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

       8. associate-*l* N/A

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

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

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

       10. associate-*r* N/A

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

       11. associate-*l* N/A

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

       12. *-commutative N/A

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

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

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

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

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

       15. *-commutative N/A

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

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

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

       17. unpow2 N/A

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

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       19. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

       20. *-lowering-*.f64 55.8%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   13. Simplified 55.8%

       \[\leadsto \color{blue}{\frac{D \cdot \left(D \cdot \left(0.25 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot d}} \]

   if 3.8999999999999999e-119 < w < 1e30

   1. Initial program 23.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\color{blue}{{D}^{2} \cdot \left(h \cdot w\right)}}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(2 \cdot \left(c0 \cdot {d}^{2}\right)\right), \color{blue}{\left({D}^{2} \cdot \left(h \cdot w\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot {d}^{2}\right)\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left({d}^{2}\right)\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(d \cdot d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(h \cdot w\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{h} \cdot w\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{h} \cdot w\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(w \cdot \color{blue}{h}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 35.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(w, \color{blue}{h}\right)\right)\right)\right) \]

   5. Simplified 35.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(D \cdot D\right)} \cdot \left(w \cdot h\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}}\right)\right) \]

      4. frac-times N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \left(\frac{c0 \cdot d}{w \cdot h} \cdot \color{blue}{\frac{d}{D \cdot D}}\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\left(2 \cdot \frac{c0 \cdot d}{w \cdot h}\right) \cdot \color{blue}{\frac{d}{D \cdot D}}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \frac{c0 \cdot d}{w \cdot h}\right), \color{blue}{\left(\frac{d}{D \cdot D}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{c0 \cdot d}{w \cdot h}\right)\right), \left(\frac{\color{blue}{d}}{D \cdot D}\right)\right)\right) \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{\frac{c0 \cdot d}{w}}{h}\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{c0 \cdot d}{w}\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(c0 \cdot \frac{d}{w}\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{d}{w}\right)\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \mathsf{/.f64}\left(d, \color{blue}{\left(D \cdot D\right)}\right)\right)\right) \]

      14. *-lowering-*.f64 43.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(D, \color{blue}{D}\right)\right)\right)\right) \]

   7. Applied egg-rr 43.9%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot \frac{d}{D \cdot D}\right)} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot \frac{\frac{d}{D}}{\color{blue}{D}}\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot \frac{d}{D}}{\color{blue}{D}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot \frac{d}{D}\right), \color{blue}{D}\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot \frac{1}{\frac{D}{d}}\right), D\right)\right) \]

      5. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}}{\frac{D}{d}}\right), D\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right), \left(\frac{D}{d}\right)\right), D\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{c0 \cdot \frac{d}{w}}{h}\right)\right), \left(\frac{D}{d}\right)\right), D\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot \frac{\frac{d}{w}}{h}\right)\right), \left(\frac{D}{d}\right)\right), D\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(\frac{\frac{d}{w}}{h}\right)\right)\right), \left(\frac{D}{d}\right)\right), D\right)\right) \]

      10. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(\frac{d}{h \cdot w}\right)\right)\right), \left(\frac{D}{d}\right)\right), D\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(\frac{d}{w \cdot h}\right)\right)\right), \left(\frac{D}{d}\right)\right), D\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, \left(w \cdot h\right)\right)\right)\right), \left(\frac{D}{d}\right)\right), D\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(w, h\right)\right)\right)\right), \left(\frac{D}{d}\right)\right), D\right)\right) \]

      14. /-lowering-/.f64 61.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(w, h\right)\right)\right)\right), \mathsf{/.f64}\left(D, d\right)\right), D\right)\right) \]

   9. Applied egg-rr 61.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{\frac{2 \cdot \left(c0 \cdot \frac{d}{w \cdot h}\right)}{\frac{D}{d}}}{D}} \]
   10. Step-by-step derivation

       1. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{2 \cdot \left(c0 \cdot \frac{d}{w \cdot h}\right)}{\color{blue}{D \cdot \frac{D}{d}}}\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{\left(2 \cdot c0\right) \cdot \frac{d}{w \cdot h}}{\color{blue}{D} \cdot \frac{D}{d}}\right)\right) \]

       3. times-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{2 \cdot c0}{D} \cdot \color{blue}{\frac{\frac{d}{w \cdot h}}{\frac{D}{d}}}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\left(\frac{2 \cdot c0}{D}\right), \color{blue}{\left(\frac{\frac{d}{w \cdot h}}{\frac{D}{d}}\right)}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot c0\right), D\right), \left(\frac{\color{blue}{\frac{d}{w \cdot h}}}{\frac{D}{d}}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(c0 \cdot 2\right), D\right), \left(\frac{\frac{\color{blue}{d}}{w \cdot h}}{\frac{D}{d}}\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, 2\right), D\right), \left(\frac{\frac{\color{blue}{d}}{w \cdot h}}{\frac{D}{d}}\right)\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, 2\right), D\right), \mathsf{/.f64}\left(\left(\frac{d}{w \cdot h}\right), \color{blue}{\left(\frac{D}{d}\right)}\right)\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, 2\right), D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \left(w \cdot h\right)\right), \left(\frac{\color{blue}{D}}{d}\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, 2\right), D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(w, h\right)\right), \left(\frac{D}{d}\right)\right)\right)\right) \]

       11. /-lowering-/.f64 61.5%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, 2\right), D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(D, \color{blue}{d}\right)\right)\right)\right) \]

   11. Applied egg-rr 61.5%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\frac{c0 \cdot 2}{D} \cdot \frac{\frac{d}{w \cdot h}}{\frac{D}{d}}\right)} \]

   if 1e30 < w < 3.40000000000000012e43

   1. Initial program 33.3%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}{\color{blue}{2 \cdot w}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right), \color{blue}{\left(2 \cdot w\right)}\right) \]

   3. Simplified 16.9%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot \frac{\frac{\frac{d \cdot d}{w} \cdot c0}{h} \cdot c0}{D \cdot D}}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(\mathsf{neg}\left(c0\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 83.8%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\left(\frac{-0.5 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot w\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)} + 0\right) \cdot \left(-c0\right)\right)}}{2 \cdot w} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. 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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left({D}^{2}\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(D \cdot D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot {M}^{2}\right)\right)\right), \left({d}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

      12. *-lowering-*.f64 83.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   10. Simplified 83.8%

       \[\leadsto \color{blue}{\frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot d}} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \left(D \cdot D\right)\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\left(\left(h \cdot M\right) \cdot M\right) \cdot \left(D \cdot D\right)\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\left(h \cdot M\right) \cdot \left(M \cdot \left(D \cdot D\right)\right)\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(h \cdot M\right), \left(M \cdot \left(D \cdot D\right)\right)\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \left(M \cdot \left(D \cdot D\right)\right)\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \mathsf{*.f64}\left(M, \left(D \cdot D\right)\right)\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

       7. *-lowering-*.f64 99.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, D\right)\right)\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

   12. Applied egg-rr 99.7%

       \[\leadsto \frac{0.25 \cdot \color{blue}{\left(\left(h \cdot M\right) \cdot \left(M \cdot \left(D \cdot D\right)\right)\right)}}{d \cdot d} \]

   if 3.40000000000000012e43 < w

   1. Initial program 28.0%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\color{blue}{{D}^{2} \cdot \left(h \cdot w\right)}}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(2 \cdot \left(c0 \cdot {d}^{2}\right)\right), \color{blue}{\left({D}^{2} \cdot \left(h \cdot w\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot {d}^{2}\right)\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left({d}^{2}\right)\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(d \cdot d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(h \cdot w\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{h} \cdot w\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{h} \cdot w\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(w \cdot \color{blue}{h}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 31.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(w, \color{blue}{h}\right)\right)\right)\right) \]

   5. Simplified 31.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(D \cdot D\right)} \cdot \left(w \cdot h\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}}\right)\right) \]

      4. frac-times N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \left(\frac{c0 \cdot d}{w \cdot h} \cdot \color{blue}{\frac{d}{D \cdot D}}\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\left(2 \cdot \frac{c0 \cdot d}{w \cdot h}\right) \cdot \color{blue}{\frac{d}{D \cdot D}}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \frac{c0 \cdot d}{w \cdot h}\right), \color{blue}{\left(\frac{d}{D \cdot D}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{c0 \cdot d}{w \cdot h}\right)\right), \left(\frac{\color{blue}{d}}{D \cdot D}\right)\right)\right) \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{\frac{c0 \cdot d}{w}}{h}\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{c0 \cdot d}{w}\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(c0 \cdot \frac{d}{w}\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{d}{w}\right)\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \mathsf{/.f64}\left(d, \color{blue}{\left(D \cdot D\right)}\right)\right)\right) \]

      14. *-lowering-*.f64 41.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(D, \color{blue}{D}\right)\right)\right)\right) \]

   7. Applied egg-rr 41.9%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot \frac{d}{D \cdot D}\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{d}{D \cdot D} \cdot \color{blue}{\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right)}\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{\frac{d}{D}}{D} \cdot \left(\color{blue}{2} \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{\frac{d}{D}}{D} \cdot \frac{2 \cdot \left(c0 \cdot \frac{d}{w}\right)}{\color{blue}{h}}\right)\right) \]

      4. frac-times N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{\frac{d}{D} \cdot \left(2 \cdot \left(c0 \cdot \frac{d}{w}\right)\right)}{\color{blue}{D \cdot h}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(\frac{d}{D} \cdot \left(2 \cdot \left(c0 \cdot \frac{d}{w}\right)\right)\right), \color{blue}{\left(D \cdot h\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{d}{D}\right), \left(2 \cdot \left(c0 \cdot \frac{d}{w}\right)\right)\right), \left(\color{blue}{D} \cdot h\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(d, D\right), \left(2 \cdot \left(c0 \cdot \frac{d}{w}\right)\right)\right), \left(D \cdot h\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(d, D\right), \left(2 \cdot \frac{c0 \cdot d}{w}\right)\right), \left(D \cdot h\right)\right)\right) \]

      9. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(d, D\right), \left(2 \cdot \frac{1}{\frac{w}{c0 \cdot d}}\right)\right), \left(D \cdot h\right)\right)\right) \]

      10. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(d, D\right), \left(\frac{2}{\frac{w}{c0 \cdot d}}\right)\right), \left(D \cdot h\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{/.f64}\left(2, \left(\frac{w}{c0 \cdot d}\right)\right)\right), \left(D \cdot h\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(w, \left(c0 \cdot d\right)\right)\right)\right), \left(D \cdot h\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(w, \mathsf{*.f64}\left(c0, d\right)\right)\right)\right), \left(D \cdot h\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(w, \mathsf{*.f64}\left(c0, d\right)\right)\right)\right), \left(h \cdot \color{blue}{D}\right)\right)\right) \]

      15. *-lowering-*.f64 55.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(d, D\right), \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(w, \mathsf{*.f64}\left(c0, d\right)\right)\right)\right), \mathsf{*.f64}\left(h, \color{blue}{D}\right)\right)\right) \]

   9. Applied egg-rr 55.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{\frac{d}{D} \cdot \frac{2}{\frac{w}{c0 \cdot d}}}{h \cdot D}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -1.25 \cdot 10^{-214}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{d}{D} \cdot \frac{2 \cdot \left(c0 \cdot \frac{d}{w \cdot h}\right)}{D}\right)\\ \mathbf{elif}\;w \leq 3.9 \cdot 10^{-119}:\\ \;\;\;\;\frac{D \cdot \left(D \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot 0.25\right)\right)}{d \cdot d}\\ \mathbf{elif}\;w \leq 10^{+30}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot 2}{D} \cdot \frac{\frac{d}{w \cdot h}}{\frac{D}{d}}\right)\\ \mathbf{elif}\;w \leq 3.4 \cdot 10^{+43}:\\ \;\;\;\;\frac{0.25 \cdot \left(\left(h \cdot M\right) \cdot \left(\left(D \cdot D\right) \cdot M\right)\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \frac{\frac{d}{D} \cdot \frac{2}{\frac{w}{c0 \cdot d}}}{h \cdot D}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 44.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w} \cdot \left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot \frac{\frac{d}{D}}{D}\right)\\ \mathbf{if}\;d \leq 7.5 \cdot 10^{-109}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.75 \cdot 10^{-67}:\\ \;\;\;\;\frac{D \cdot \left(D \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot 0.25\right)\right)}{d \cdot d}\\ \mathbf{elif}\;d \leq 1.55 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{d \cdot \left(c0 \cdot d\right)}{w}}{w \cdot \left(h \cdot D\right)} \cdot \frac{c0}{D}\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{+161}:\\ \;\;\;\;\frac{\left(D \cdot 0.25\right) \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot D\right)\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0
         (* (/ c0 (* 2.0 w)) (* (* 2.0 (/ (* c0 (/ d w)) h)) (/ (/ d D) D)))))
   (if (<= d 7.5e-109)
     t_0
     (if (<= d 1.75e-67)
       (/ (* D (* D (* (* h (* M M)) 0.25))) (* d d))
       (if (<= d 1.55e-11)
         (* (/ (/ (* d (* c0 d)) w) (* w (* h D))) (/ c0 D))
         (if (<= d 4.5e+161)
           (/ (* (* D 0.25) (* (* M M) (* h D))) (* d d))
           t_0))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 / (2.0 * w)) * ((2.0 * ((c0 * (d / w)) / h)) * ((d / D) / D));
	double tmp;
	if (d <= 7.5e-109) {
		tmp = t_0;
	} else if (d <= 1.75e-67) {
		tmp = (D * (D * ((h * (M * M)) * 0.25))) / (d * d);
	} else if (d <= 1.55e-11) {
		tmp = (((d * (c0 * d)) / w) / (w * (h * D))) * (c0 / D);
	} else if (d <= 4.5e+161) {
		tmp = ((D * 0.25) * ((M * M) * (h * D))) / (d * d);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (c0 / (2.0d0 * w)) * ((2.0d0 * ((c0 * (d_1 / w)) / h)) * ((d_1 / d) / d))
    if (d_1 <= 7.5d-109) then
        tmp = t_0
    else if (d_1 <= 1.75d-67) then
        tmp = (d * (d * ((h * (m * m)) * 0.25d0))) / (d_1 * d_1)
    else if (d_1 <= 1.55d-11) then
        tmp = (((d_1 * (c0 * d_1)) / w) / (w * (h * d))) * (c0 / d)
    else if (d_1 <= 4.5d+161) then
        tmp = ((d * 0.25d0) * ((m * m) * (h * d))) / (d_1 * d_1)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 / (2.0 * w)) * ((2.0 * ((c0 * (d / w)) / h)) * ((d / D) / D));
	double tmp;
	if (d <= 7.5e-109) {
		tmp = t_0;
	} else if (d <= 1.75e-67) {
		tmp = (D * (D * ((h * (M * M)) * 0.25))) / (d * d);
	} else if (d <= 1.55e-11) {
		tmp = (((d * (c0 * d)) / w) / (w * (h * D))) * (c0 / D);
	} else if (d <= 4.5e+161) {
		tmp = ((D * 0.25) * ((M * M) * (h * D))) / (d * d);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 / (2.0 * w)) * ((2.0 * ((c0 * (d / w)) / h)) * ((d / D) / D))
	tmp = 0
	if d <= 7.5e-109:
		tmp = t_0
	elif d <= 1.75e-67:
		tmp = (D * (D * ((h * (M * M)) * 0.25))) / (d * d)
	elif d <= 1.55e-11:
		tmp = (((d * (c0 * d)) / w) / (w * (h * D))) * (c0 / D)
	elif d <= 4.5e+161:
		tmp = ((D * 0.25) * ((M * M) * (h * D))) / (d * d)
	else:
		tmp = t_0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(Float64(2.0 * Float64(Float64(c0 * Float64(d / w)) / h)) * Float64(Float64(d / D) / D)))
	tmp = 0.0
	if (d <= 7.5e-109)
		tmp = t_0;
	elseif (d <= 1.75e-67)
		tmp = Float64(Float64(D * Float64(D * Float64(Float64(h * Float64(M * M)) * 0.25))) / Float64(d * d));
	elseif (d <= 1.55e-11)
		tmp = Float64(Float64(Float64(Float64(d * Float64(c0 * d)) / w) / Float64(w * Float64(h * D))) * Float64(c0 / D));
	elseif (d <= 4.5e+161)
		tmp = Float64(Float64(Float64(D * 0.25) * Float64(Float64(M * M) * Float64(h * D))) / Float64(d * d));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 / (2.0 * w)) * ((2.0 * ((c0 * (d / w)) / h)) * ((d / D) / D));
	tmp = 0.0;
	if (d <= 7.5e-109)
		tmp = t_0;
	elseif (d <= 1.75e-67)
		tmp = (D * (D * ((h * (M * M)) * 0.25))) / (d * d);
	elseif (d <= 1.55e-11)
		tmp = (((d * (c0 * d)) / w) / (w * (h * D))) * (c0 / D);
	elseif (d <= 4.5e+161)
		tmp = ((D * 0.25) * ((M * M) * (h * D))) / (d * d);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[(N[(c0 * N[(d / w), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision] * N[(N[(d / D), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, 7.5e-109], t$95$0, If[LessEqual[d, 1.75e-67], N[(N[(D * N[(D * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.55e-11], N[(N[(N[(N[(d * N[(c0 * d), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision] / N[(w * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c0 / D), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.5e+161], N[(N[(N[(D * 0.25), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if d < 7.49999999999999982e-109 or 4.49999999999999992e161 < d

   1. Initial program 25.2%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\color{blue}{{D}^{2} \cdot \left(h \cdot w\right)}}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(2 \cdot \left(c0 \cdot {d}^{2}\right)\right), \color{blue}{\left({D}^{2} \cdot \left(h \cdot w\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot {d}^{2}\right)\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left({d}^{2}\right)\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(d \cdot d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(h \cdot w\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{h} \cdot w\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{h} \cdot w\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(w \cdot \color{blue}{h}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 30.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(w, \color{blue}{h}\right)\right)\right)\right) \]

   5. Simplified 30.9%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(D \cdot D\right)} \cdot \left(w \cdot h\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}}\right)\right) \]

      4. frac-times N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \left(\frac{c0 \cdot d}{w \cdot h} \cdot \color{blue}{\frac{d}{D \cdot D}}\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\left(2 \cdot \frac{c0 \cdot d}{w \cdot h}\right) \cdot \color{blue}{\frac{d}{D \cdot D}}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \frac{c0 \cdot d}{w \cdot h}\right), \color{blue}{\left(\frac{d}{D \cdot D}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{c0 \cdot d}{w \cdot h}\right)\right), \left(\frac{\color{blue}{d}}{D \cdot D}\right)\right)\right) \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{\frac{c0 \cdot d}{w}}{h}\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{c0 \cdot d}{w}\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(c0 \cdot \frac{d}{w}\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{d}{w}\right)\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \mathsf{/.f64}\left(d, \color{blue}{\left(D \cdot D\right)}\right)\right)\right) \]

      14. *-lowering-*.f64 40.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(D, \color{blue}{D}\right)\right)\right)\right) \]

   7. Applied egg-rr 40.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot \frac{d}{D \cdot D}\right)} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \left(\frac{\frac{d}{D}}{\color{blue}{D}}\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \mathsf{/.f64}\left(\left(\frac{d}{D}\right), \color{blue}{D}\right)\right)\right) \]

      3. /-lowering-/.f64 48.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), D\right)\right)\right) \]

   9. Applied egg-rr 48.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot \color{blue}{\frac{\frac{d}{D}}{D}}\right) \]

   if 7.49999999999999982e-109 < d < 1.75e-67

   1. Initial program 1.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}{\color{blue}{2 \cdot w}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right), \color{blue}{\left(2 \cdot w\right)}\right) \]

   3. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot \frac{\frac{\frac{d \cdot d}{w} \cdot c0}{h} \cdot c0}{D \cdot D}}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(\mathsf{neg}\left(c0\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 1.9%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\left(\frac{-0.5 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot w\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)} + 0\right) \cdot \left(-c0\right)\right)}}{2 \cdot w} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. 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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left({D}^{2}\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(D \cdot D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot {M}^{2}\right)\right)\right), \left({d}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

      12. *-lowering-*.f64 3.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   10. Simplified 3.0%

       \[\leadsto \color{blue}{\frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot d}} \]

   11. Taylor expanded in D around 0

       \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   12. 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. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right) \cdot \frac{1}{4}\right), \left({\color{blue}{d}}^{2}\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right) \cdot \frac{1}{4}\right), \left({d}^{2}\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \left(h \cdot \frac{1}{4}\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(D \cdot D\right) \cdot {M}^{2}\right) \cdot \left(h \cdot \frac{1}{4}\right)\right), \left({d}^{2}\right)\right) \]

       7. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(D \cdot \left(D \cdot {M}^{2}\right)\right) \cdot \left(h \cdot \frac{1}{4}\right)\right), \left({d}^{2}\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(D \cdot \left(\left(D \cdot {M}^{2}\right) \cdot \left(h \cdot \frac{1}{4}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(\left(D \cdot {M}^{2}\right) \cdot \left(h \cdot \frac{1}{4}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

       10. associate-*r* N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(D \cdot \left({M}^{2} \cdot \left(h \cdot \frac{1}{4}\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(D \cdot \left(\left({M}^{2} \cdot h\right) \cdot \frac{1}{4}\right)\right)\right), \left({d}^{2}\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(D \cdot \left(\frac{1}{4} \cdot \left({M}^{2} \cdot h\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(\frac{1}{4} \cdot \left({M}^{2} \cdot h\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \left({M}^{2} \cdot h\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \left(h \cdot {M}^{2}\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       18. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       19. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

       20. *-lowering-*.f64 36.5%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   13. Simplified 36.5%

       \[\leadsto \color{blue}{\frac{D \cdot \left(D \cdot \left(0.25 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot d}} \]

   if 1.75e-67 < d < 1.55000000000000014e-11

   1. Initial program 55.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\color{blue}{{D}^{2} \cdot \left(h \cdot w\right)}}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(2 \cdot \left(c0 \cdot {d}^{2}\right)\right), \color{blue}{\left({D}^{2} \cdot \left(h \cdot w\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot {d}^{2}\right)\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left({d}^{2}\right)\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(d \cdot d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(h \cdot w\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{h} \cdot w\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{h} \cdot w\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(w \cdot \color{blue}{h}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 48.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(w, \color{blue}{h}\right)\right)\right)\right) \]

   5. Simplified 48.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(D \cdot D\right)} \cdot \left(w \cdot h\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}}\right)\right) \]

      4. frac-times N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \left(\frac{c0 \cdot d}{w \cdot h} \cdot \color{blue}{\frac{d}{D \cdot D}}\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\left(2 \cdot \frac{c0 \cdot d}{w \cdot h}\right) \cdot \color{blue}{\frac{d}{D \cdot D}}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \frac{c0 \cdot d}{w \cdot h}\right), \color{blue}{\left(\frac{d}{D \cdot D}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{c0 \cdot d}{w \cdot h}\right)\right), \left(\frac{\color{blue}{d}}{D \cdot D}\right)\right)\right) \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{\frac{c0 \cdot d}{w}}{h}\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{c0 \cdot d}{w}\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(c0 \cdot \frac{d}{w}\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{d}{w}\right)\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \mathsf{/.f64}\left(d, \color{blue}{\left(D \cdot D\right)}\right)\right)\right) \]

      14. *-lowering-*.f64 48.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(D, \color{blue}{D}\right)\right)\right)\right) \]

   7. Applied egg-rr 48.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot \frac{d}{D \cdot D}\right)} \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{c0 \cdot \frac{d}{w}}{h} \cdot \frac{d}{D \cdot D}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\frac{c0 \cdot d}{w}}{h} \cdot \frac{d}{D \cdot D}\right)\right) \]

      3. associate-/l/ N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0 \cdot d}{h \cdot w} \cdot \frac{\color{blue}{d}}{D \cdot D}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0 \cdot d}{w \cdot h} \cdot \frac{d}{D \cdot D}\right)\right) \]

      5. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}}\right) \]

      6. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(w \cdot h\right)} \cdot \left(D \cdot D\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \color{blue}{\left(w \cdot h\right)}}\right) \]

      8. associate-/l* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}} \]

      9. clear-num N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{1}{\color{blue}{\frac{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}}} \]

      10. un-div-inv N/A

          \[\leadsto \frac{\frac{c0}{2 \cdot w}}{\color{blue}{\frac{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}}} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{c0}{2 \cdot w}\right), \color{blue}{\left(\frac{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}\right)}\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \left(2 \cdot w\right)\right), \left(\frac{\color{blue}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{\left(D \cdot D\right) \cdot \color{blue}{\left(w \cdot h\right)}}{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}\right)\right) \]

   9. Applied egg-rr 56.2%

      \[\leadsto \color{blue}{\frac{\frac{c0}{2 \cdot w}}{\frac{D}{2 \cdot \left(c0 \cdot d\right)} \cdot \frac{D \cdot \left(w \cdot h\right)}{d}}} \]

   11. Applied egg-rr 63.0%

       \[\leadsto \color{blue}{\frac{-\frac{d \cdot \left(c0 \cdot d\right)}{w}}{w \cdot \left(h \cdot \left(0 - D\right)\right)} \cdot \frac{c0}{D}} \]

   if 1.55000000000000014e-11 < d < 4.49999999999999992e161

   1. Initial program 16.0%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}{\color{blue}{2 \cdot w}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right), \color{blue}{\left(2 \cdot w\right)}\right) \]

   3. Simplified 11.7%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot \frac{\frac{\frac{d \cdot d}{w} \cdot c0}{h} \cdot c0}{D \cdot D}}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(\mathsf{neg}\left(c0\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 18.8%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\left(\frac{-0.5 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot w\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)} + 0\right) \cdot \left(-c0\right)\right)}}{2 \cdot w} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. 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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left({D}^{2}\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(D \cdot D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot {M}^{2}\right)\right)\right), \left({d}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

      12. *-lowering-*.f64 42.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   10. Simplified 42.9%

       \[\leadsto \color{blue}{\frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot d}} \]
   11. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{4} \cdot D\right) \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{d}, d\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{4} \cdot D\right), \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{d}, d\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, D\right), \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, D\right), \left(\left(D \cdot h\right) \cdot \left(M \cdot M\right)\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, D\right), \mathsf{*.f64}\left(\left(D \cdot h\right), \left(M \cdot M\right)\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, D\right), \mathsf{*.f64}\left(\left(h \cdot D\right), \left(M \cdot M\right)\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, D\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, D\right), \left(M \cdot M\right)\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

       9. *-lowering-*.f64 56.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, D\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, D\right), \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

   12. Applied egg-rr 56.0%

       \[\leadsto \frac{\color{blue}{\left(0.25 \cdot D\right) \cdot \left(\left(h \cdot D\right) \cdot \left(M \cdot M\right)\right)}}{d \cdot d} \]
3. Recombined 4 regimes into one program.

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 7.5 \cdot 10^{-109}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot \frac{\frac{d}{D}}{D}\right)\\ \mathbf{elif}\;d \leq 1.75 \cdot 10^{-67}:\\ \;\;\;\;\frac{D \cdot \left(D \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot 0.25\right)\right)}{d \cdot d}\\ \mathbf{elif}\;d \leq 1.55 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{d \cdot \left(c0 \cdot d\right)}{w}}{w \cdot \left(h \cdot D\right)} \cdot \frac{c0}{D}\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{+161}:\\ \;\;\;\;\frac{\left(D \cdot 0.25\right) \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot D\right)\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot \frac{\frac{d}{D}}{D}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 41.3% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := h \cdot \left(M \cdot M\right)\\ t_1 := \frac{d}{w \cdot h}\\ t_2 := \frac{c0}{2 \cdot w}\\ \mathbf{if}\;w \leq -3.8 \cdot 10^{-215}:\\ \;\;\;\;t\_2 \cdot \left(\frac{d}{D} \cdot \frac{2 \cdot \left(c0 \cdot t\_1\right)}{D}\right)\\ \mathbf{elif}\;w \leq 4.3 \cdot 10^{-118}:\\ \;\;\;\;\frac{D \cdot \left(D \cdot \left(t\_0 \cdot 0.25\right)\right)}{d \cdot d}\\ \mathbf{elif}\;w \leq 1.45 \cdot 10^{+30}:\\ \;\;\;\;t\_2 \cdot \left(\frac{c0 \cdot 2}{D} \cdot \frac{t\_1}{\frac{D}{d}}\right)\\ \mathbf{elif}\;w \leq 1.1 \cdot 10^{+99}:\\ \;\;\;\;\frac{\frac{\left(D \cdot D\right) \cdot t\_0}{\frac{d}{0.25}}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d \cdot \left(c0 \cdot d\right)}{w}}{D} \cdot \frac{c0}{w \cdot \left(h \cdot D\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* h (* M M))) (t_1 (/ d (* w h))) (t_2 (/ c0 (* 2.0 w))))
   (if (<= w -3.8e-215)
     (* t_2 (* (/ d D) (/ (* 2.0 (* c0 t_1)) D)))
     (if (<= w 4.3e-118)
       (/ (* D (* D (* t_0 0.25))) (* d d))
       (if (<= w 1.45e+30)
         (* t_2 (* (/ (* c0 2.0) D) (/ t_1 (/ D d))))
         (if (<= w 1.1e+99)
           (/ (/ (* (* D D) t_0) (/ d 0.25)) d)
           (* (/ (/ (* d (* c0 d)) w) D) (/ c0 (* w (* h D))))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = h * (M * M);
	double t_1 = d / (w * h);
	double t_2 = c0 / (2.0 * w);
	double tmp;
	if (w <= -3.8e-215) {
		tmp = t_2 * ((d / D) * ((2.0 * (c0 * t_1)) / D));
	} else if (w <= 4.3e-118) {
		tmp = (D * (D * (t_0 * 0.25))) / (d * d);
	} else if (w <= 1.45e+30) {
		tmp = t_2 * (((c0 * 2.0) / D) * (t_1 / (D / d)));
	} else if (w <= 1.1e+99) {
		tmp = (((D * D) * t_0) / (d / 0.25)) / d;
	} else {
		tmp = (((d * (c0 * d)) / w) / D) * (c0 / (w * (h * 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) :: t_2
    real(8) :: tmp
    t_0 = h * (m * m)
    t_1 = d_1 / (w * h)
    t_2 = c0 / (2.0d0 * w)
    if (w <= (-3.8d-215)) then
        tmp = t_2 * ((d_1 / d) * ((2.0d0 * (c0 * t_1)) / d))
    else if (w <= 4.3d-118) then
        tmp = (d * (d * (t_0 * 0.25d0))) / (d_1 * d_1)
    else if (w <= 1.45d+30) then
        tmp = t_2 * (((c0 * 2.0d0) / d) * (t_1 / (d / d_1)))
    else if (w <= 1.1d+99) then
        tmp = (((d * d) * t_0) / (d_1 / 0.25d0)) / d_1
    else
        tmp = (((d_1 * (c0 * d_1)) / w) / d) * (c0 / (w * (h * d)))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = h * (M * M);
	double t_1 = d / (w * h);
	double t_2 = c0 / (2.0 * w);
	double tmp;
	if (w <= -3.8e-215) {
		tmp = t_2 * ((d / D) * ((2.0 * (c0 * t_1)) / D));
	} else if (w <= 4.3e-118) {
		tmp = (D * (D * (t_0 * 0.25))) / (d * d);
	} else if (w <= 1.45e+30) {
		tmp = t_2 * (((c0 * 2.0) / D) * (t_1 / (D / d)));
	} else if (w <= 1.1e+99) {
		tmp = (((D * D) * t_0) / (d / 0.25)) / d;
	} else {
		tmp = (((d * (c0 * d)) / w) / D) * (c0 / (w * (h * D)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = h * (M * M)
	t_1 = d / (w * h)
	t_2 = c0 / (2.0 * w)
	tmp = 0
	if w <= -3.8e-215:
		tmp = t_2 * ((d / D) * ((2.0 * (c0 * t_1)) / D))
	elif w <= 4.3e-118:
		tmp = (D * (D * (t_0 * 0.25))) / (d * d)
	elif w <= 1.45e+30:
		tmp = t_2 * (((c0 * 2.0) / D) * (t_1 / (D / d)))
	elif w <= 1.1e+99:
		tmp = (((D * D) * t_0) / (d / 0.25)) / d
	else:
		tmp = (((d * (c0 * d)) / w) / D) * (c0 / (w * (h * D)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(h * Float64(M * M))
	t_1 = Float64(d / Float64(w * h))
	t_2 = Float64(c0 / Float64(2.0 * w))
	tmp = 0.0
	if (w <= -3.8e-215)
		tmp = Float64(t_2 * Float64(Float64(d / D) * Float64(Float64(2.0 * Float64(c0 * t_1)) / D)));
	elseif (w <= 4.3e-118)
		tmp = Float64(Float64(D * Float64(D * Float64(t_0 * 0.25))) / Float64(d * d));
	elseif (w <= 1.45e+30)
		tmp = Float64(t_2 * Float64(Float64(Float64(c0 * 2.0) / D) * Float64(t_1 / Float64(D / d))));
	elseif (w <= 1.1e+99)
		tmp = Float64(Float64(Float64(Float64(D * D) * t_0) / Float64(d / 0.25)) / d);
	else
		tmp = Float64(Float64(Float64(Float64(d * Float64(c0 * d)) / w) / D) * Float64(c0 / Float64(w * Float64(h * D))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = h * (M * M);
	t_1 = d / (w * h);
	t_2 = c0 / (2.0 * w);
	tmp = 0.0;
	if (w <= -3.8e-215)
		tmp = t_2 * ((d / D) * ((2.0 * (c0 * t_1)) / D));
	elseif (w <= 4.3e-118)
		tmp = (D * (D * (t_0 * 0.25))) / (d * d);
	elseif (w <= 1.45e+30)
		tmp = t_2 * (((c0 * 2.0) / D) * (t_1 / (D / d)));
	elseif (w <= 1.1e+99)
		tmp = (((D * D) * t_0) / (d / 0.25)) / d;
	else
		tmp = (((d * (c0 * d)) / w) / D) * (c0 / (w * (h * D)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(d / N[(w * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, -3.8e-215], N[(t$95$2 * N[(N[(d / D), $MachinePrecision] * N[(N[(2.0 * N[(c0 * t$95$1), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 4.3e-118], N[(N[(D * N[(D * N[(t$95$0 * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 1.45e+30], N[(t$95$2 * N[(N[(N[(c0 * 2.0), $MachinePrecision] / D), $MachinePrecision] * N[(t$95$1 / N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 1.1e+99], N[(N[(N[(N[(D * D), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(d / 0.25), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], N[(N[(N[(N[(d * N[(c0 * d), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision] / D), $MachinePrecision] * N[(c0 / N[(w * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if w < -3.79999999999999977e-215

   1. Initial program 30.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\color{blue}{{D}^{2} \cdot \left(h \cdot w\right)}}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(2 \cdot \left(c0 \cdot {d}^{2}\right)\right), \color{blue}{\left({D}^{2} \cdot \left(h \cdot w\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot {d}^{2}\right)\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left({d}^{2}\right)\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(d \cdot d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(h \cdot w\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{h} \cdot w\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{h} \cdot w\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(w \cdot \color{blue}{h}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 40.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(w, \color{blue}{h}\right)\right)\right)\right) \]

   5. Simplified 40.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(D \cdot D\right)} \cdot \left(w \cdot h\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}}\right)\right) \]

      4. frac-times N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \left(\frac{c0 \cdot d}{w \cdot h} \cdot \color{blue}{\frac{d}{D \cdot D}}\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\left(2 \cdot \frac{c0 \cdot d}{w \cdot h}\right) \cdot \color{blue}{\frac{d}{D \cdot D}}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \frac{c0 \cdot d}{w \cdot h}\right), \color{blue}{\left(\frac{d}{D \cdot D}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{c0 \cdot d}{w \cdot h}\right)\right), \left(\frac{\color{blue}{d}}{D \cdot D}\right)\right)\right) \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{\frac{c0 \cdot d}{w}}{h}\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{c0 \cdot d}{w}\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(c0 \cdot \frac{d}{w}\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{d}{w}\right)\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \mathsf{/.f64}\left(d, \color{blue}{\left(D \cdot D\right)}\right)\right)\right) \]

      14. *-lowering-*.f64 44.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(D, \color{blue}{D}\right)\right)\right)\right) \]

   7. Applied egg-rr 44.7%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot \frac{d}{D \cdot D}\right)} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot d}{\color{blue}{D \cdot D}}\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}}{D} \cdot \color{blue}{\frac{d}{D}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\left(\frac{2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}}{D}\right), \color{blue}{\left(\frac{d}{D}\right)}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right), D\right), \left(\frac{\color{blue}{d}}{D}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{c0 \cdot \frac{d}{w}}{h}\right)\right), D\right), \left(\frac{d}{D}\right)\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot \frac{\frac{d}{w}}{h}\right)\right), D\right), \left(\frac{d}{D}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(\frac{\frac{d}{w}}{h}\right)\right)\right), D\right), \left(\frac{d}{D}\right)\right)\right) \]

      8. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(\frac{d}{h \cdot w}\right)\right)\right), D\right), \left(\frac{d}{D}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(\frac{d}{w \cdot h}\right)\right)\right), D\right), \left(\frac{d}{D}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, \left(w \cdot h\right)\right)\right)\right), D\right), \left(\frac{d}{D}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(w, h\right)\right)\right)\right), D\right), \left(\frac{d}{D}\right)\right)\right) \]

      12. /-lowering-/.f64 53.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(w, h\right)\right)\right)\right), D\right), \mathsf{/.f64}\left(d, \color{blue}{D}\right)\right)\right) \]

   9. Applied egg-rr 53.8%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\frac{2 \cdot \left(c0 \cdot \frac{d}{w \cdot h}\right)}{D} \cdot \frac{d}{D}\right)} \]

   if -3.79999999999999977e-215 < w < 4.30000000000000018e-118

   1. Initial program 14.8%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}{\color{blue}{2 \cdot w}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right), \color{blue}{\left(2 \cdot w\right)}\right) \]

   3. Simplified 15.0%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot \frac{\frac{\frac{d \cdot d}{w} \cdot c0}{h} \cdot c0}{D \cdot D}}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(\mathsf{neg}\left(c0\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 19.5%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\left(\frac{-0.5 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot w\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)} + 0\right) \cdot \left(-c0\right)\right)}}{2 \cdot w} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. 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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left({D}^{2}\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(D \cdot D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot {M}^{2}\right)\right)\right), \left({d}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

      12. *-lowering-*.f64 47.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   10. Simplified 47.6%

       \[\leadsto \color{blue}{\frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot d}} \]

   11. Taylor expanded in D around 0

       \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   12. 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. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right) \cdot \frac{1}{4}\right), \left({\color{blue}{d}}^{2}\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right) \cdot \frac{1}{4}\right), \left({d}^{2}\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \left(h \cdot \frac{1}{4}\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(D \cdot D\right) \cdot {M}^{2}\right) \cdot \left(h \cdot \frac{1}{4}\right)\right), \left({d}^{2}\right)\right) \]

       7. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(D \cdot \left(D \cdot {M}^{2}\right)\right) \cdot \left(h \cdot \frac{1}{4}\right)\right), \left({d}^{2}\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(D \cdot \left(\left(D \cdot {M}^{2}\right) \cdot \left(h \cdot \frac{1}{4}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(\left(D \cdot {M}^{2}\right) \cdot \left(h \cdot \frac{1}{4}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

       10. associate-*r* N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(D \cdot \left({M}^{2} \cdot \left(h \cdot \frac{1}{4}\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(D \cdot \left(\left({M}^{2} \cdot h\right) \cdot \frac{1}{4}\right)\right)\right), \left({d}^{2}\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(D \cdot \left(\frac{1}{4} \cdot \left({M}^{2} \cdot h\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(\frac{1}{4} \cdot \left({M}^{2} \cdot h\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \left({M}^{2} \cdot h\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \left(h \cdot {M}^{2}\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       18. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       19. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

       20. *-lowering-*.f64 55.8%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   13. Simplified 55.8%

       \[\leadsto \color{blue}{\frac{D \cdot \left(D \cdot \left(0.25 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot d}} \]

   if 4.30000000000000018e-118 < w < 1.4499999999999999e30

   1. Initial program 23.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\color{blue}{{D}^{2} \cdot \left(h \cdot w\right)}}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(2 \cdot \left(c0 \cdot {d}^{2}\right)\right), \color{blue}{\left({D}^{2} \cdot \left(h \cdot w\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot {d}^{2}\right)\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left({d}^{2}\right)\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(d \cdot d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(h \cdot w\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{h} \cdot w\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{h} \cdot w\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(w \cdot \color{blue}{h}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 35.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(w, \color{blue}{h}\right)\right)\right)\right) \]

   5. Simplified 35.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(D \cdot D\right)} \cdot \left(w \cdot h\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}}\right)\right) \]

      4. frac-times N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \left(\frac{c0 \cdot d}{w \cdot h} \cdot \color{blue}{\frac{d}{D \cdot D}}\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\left(2 \cdot \frac{c0 \cdot d}{w \cdot h}\right) \cdot \color{blue}{\frac{d}{D \cdot D}}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \frac{c0 \cdot d}{w \cdot h}\right), \color{blue}{\left(\frac{d}{D \cdot D}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{c0 \cdot d}{w \cdot h}\right)\right), \left(\frac{\color{blue}{d}}{D \cdot D}\right)\right)\right) \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{\frac{c0 \cdot d}{w}}{h}\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{c0 \cdot d}{w}\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(c0 \cdot \frac{d}{w}\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{d}{w}\right)\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \mathsf{/.f64}\left(d, \color{blue}{\left(D \cdot D\right)}\right)\right)\right) \]

      14. *-lowering-*.f64 43.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(D, \color{blue}{D}\right)\right)\right)\right) \]

   7. Applied egg-rr 43.9%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot \frac{d}{D \cdot D}\right)} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot \frac{\frac{d}{D}}{\color{blue}{D}}\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot \frac{d}{D}}{\color{blue}{D}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot \frac{d}{D}\right), \color{blue}{D}\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot \frac{1}{\frac{D}{d}}\right), D\right)\right) \]

      5. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}}{\frac{D}{d}}\right), D\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right), \left(\frac{D}{d}\right)\right), D\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{c0 \cdot \frac{d}{w}}{h}\right)\right), \left(\frac{D}{d}\right)\right), D\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot \frac{\frac{d}{w}}{h}\right)\right), \left(\frac{D}{d}\right)\right), D\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(\frac{\frac{d}{w}}{h}\right)\right)\right), \left(\frac{D}{d}\right)\right), D\right)\right) \]

      10. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(\frac{d}{h \cdot w}\right)\right)\right), \left(\frac{D}{d}\right)\right), D\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(\frac{d}{w \cdot h}\right)\right)\right), \left(\frac{D}{d}\right)\right), D\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, \left(w \cdot h\right)\right)\right)\right), \left(\frac{D}{d}\right)\right), D\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(w, h\right)\right)\right)\right), \left(\frac{D}{d}\right)\right), D\right)\right) \]

      14. /-lowering-/.f64 61.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(w, h\right)\right)\right)\right), \mathsf{/.f64}\left(D, d\right)\right), D\right)\right) \]

   9. Applied egg-rr 61.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{\frac{2 \cdot \left(c0 \cdot \frac{d}{w \cdot h}\right)}{\frac{D}{d}}}{D}} \]
   10. Step-by-step derivation

       1. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{2 \cdot \left(c0 \cdot \frac{d}{w \cdot h}\right)}{\color{blue}{D \cdot \frac{D}{d}}}\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{\left(2 \cdot c0\right) \cdot \frac{d}{w \cdot h}}{\color{blue}{D} \cdot \frac{D}{d}}\right)\right) \]

       3. times-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{2 \cdot c0}{D} \cdot \color{blue}{\frac{\frac{d}{w \cdot h}}{\frac{D}{d}}}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\left(\frac{2 \cdot c0}{D}\right), \color{blue}{\left(\frac{\frac{d}{w \cdot h}}{\frac{D}{d}}\right)}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot c0\right), D\right), \left(\frac{\color{blue}{\frac{d}{w \cdot h}}}{\frac{D}{d}}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(c0 \cdot 2\right), D\right), \left(\frac{\frac{\color{blue}{d}}{w \cdot h}}{\frac{D}{d}}\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, 2\right), D\right), \left(\frac{\frac{\color{blue}{d}}{w \cdot h}}{\frac{D}{d}}\right)\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, 2\right), D\right), \mathsf{/.f64}\left(\left(\frac{d}{w \cdot h}\right), \color{blue}{\left(\frac{D}{d}\right)}\right)\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, 2\right), D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \left(w \cdot h\right)\right), \left(\frac{\color{blue}{D}}{d}\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, 2\right), D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(w, h\right)\right), \left(\frac{D}{d}\right)\right)\right)\right) \]

       11. /-lowering-/.f64 61.5%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, 2\right), D\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(w, h\right)\right), \mathsf{/.f64}\left(D, \color{blue}{d}\right)\right)\right)\right) \]

   11. Applied egg-rr 61.5%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\frac{c0 \cdot 2}{D} \cdot \frac{\frac{d}{w \cdot h}}{\frac{D}{d}}\right)} \]

   if 1.4499999999999999e30 < w < 1.09999999999999989e99

   1. Initial program 34.1%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}{\color{blue}{2 \cdot w}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right), \color{blue}{\left(2 \cdot w\right)}\right) \]

   3. Simplified 21.1%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot \frac{\frac{\frac{d \cdot d}{w} \cdot c0}{h} \cdot c0}{D \cdot D}}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(\mathsf{neg}\left(c0\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 53.8%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\left(\frac{-0.5 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot w\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)} + 0\right) \cdot \left(-c0\right)\right)}}{2 \cdot w} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. 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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left({D}^{2}\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(D \cdot D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot {M}^{2}\right)\right)\right), \left({d}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

      12. *-lowering-*.f64 60.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   10. Simplified 60.5%

       \[\leadsto \color{blue}{\frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot d}} \]
   11. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d}}{\color{blue}{d}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d}\right), \color{blue}{d}\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{1}{4}}{d}\right), d\right) \]

       4. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right), d\right) \]

       5. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{1}{\frac{d}{\frac{1}{4}}}\right), d\right) \]

       6. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d}{\frac{1}{4}}}\right), d\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right), \left(\frac{d}{\frac{1}{4}}\right)\right), d\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(D \cdot D\right), \left(h \cdot \left(M \cdot M\right)\right)\right), \left(\frac{d}{\frac{1}{4}}\right)\right), d\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot \left(M \cdot M\right)\right)\right), \left(\frac{d}{\frac{1}{4}}\right)\right), d\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right), \left(\frac{d}{\frac{1}{4}}\right)\right), d\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \left(\frac{d}{\frac{1}{4}}\right)\right), d\right) \]

       12. /-lowering-/.f64 73.8%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{/.f64}\left(d, \frac{1}{4}\right)\right), d\right) \]

   12. Applied egg-rr 73.8%

       \[\leadsto \color{blue}{\frac{\frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d}{0.25}}}{d}} \]

   if 1.09999999999999989e99 < w

   1. Initial program 25.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\color{blue}{{D}^{2} \cdot \left(h \cdot w\right)}}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(2 \cdot \left(c0 \cdot {d}^{2}\right)\right), \color{blue}{\left({D}^{2} \cdot \left(h \cdot w\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot {d}^{2}\right)\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left({d}^{2}\right)\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(d \cdot d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(h \cdot w\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{h} \cdot w\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{h} \cdot w\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(w \cdot \color{blue}{h}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 33.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(w, \color{blue}{h}\right)\right)\right)\right) \]

   5. Simplified 33.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(D \cdot D\right)} \cdot \left(w \cdot h\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}}\right)\right) \]

      4. frac-times N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \left(\frac{c0 \cdot d}{w \cdot h} \cdot \color{blue}{\frac{d}{D \cdot D}}\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\left(2 \cdot \frac{c0 \cdot d}{w \cdot h}\right) \cdot \color{blue}{\frac{d}{D \cdot D}}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \frac{c0 \cdot d}{w \cdot h}\right), \color{blue}{\left(\frac{d}{D \cdot D}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{c0 \cdot d}{w \cdot h}\right)\right), \left(\frac{\color{blue}{d}}{D \cdot D}\right)\right)\right) \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{\frac{c0 \cdot d}{w}}{h}\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{c0 \cdot d}{w}\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(c0 \cdot \frac{d}{w}\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{d}{w}\right)\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \mathsf{/.f64}\left(d, \color{blue}{\left(D \cdot D\right)}\right)\right)\right) \]

      14. *-lowering-*.f64 40.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(D, \color{blue}{D}\right)\right)\right)\right) \]

   7. Applied egg-rr 40.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot \frac{d}{D \cdot D}\right)} \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{c0 \cdot \frac{d}{w}}{h} \cdot \frac{d}{D \cdot D}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\frac{c0 \cdot d}{w}}{h} \cdot \frac{d}{D \cdot D}\right)\right) \]

      3. associate-/l/ N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0 \cdot d}{h \cdot w} \cdot \frac{\color{blue}{d}}{D \cdot D}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0 \cdot d}{w \cdot h} \cdot \frac{d}{D \cdot D}\right)\right) \]

      5. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}}\right) \]

      6. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(w \cdot h\right)} \cdot \left(D \cdot D\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \color{blue}{\left(w \cdot h\right)}}\right) \]

      8. associate-/l* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}} \]

      9. clear-num N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{1}{\color{blue}{\frac{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}}} \]

      10. un-div-inv N/A

          \[\leadsto \frac{\frac{c0}{2 \cdot w}}{\color{blue}{\frac{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}}} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{c0}{2 \cdot w}\right), \color{blue}{\left(\frac{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}\right)}\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \left(2 \cdot w\right)\right), \left(\frac{\color{blue}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{\left(D \cdot D\right) \cdot \color{blue}{\left(w \cdot h\right)}}{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}\right)\right) \]

   9. Applied egg-rr 44.0%

      \[\leadsto \color{blue}{\frac{\frac{c0}{2 \cdot w}}{\frac{D}{2 \cdot \left(c0 \cdot d\right)} \cdot \frac{D \cdot \left(w \cdot h\right)}{d}}} \]

   11. Applied egg-rr 51.5%

       \[\leadsto \color{blue}{\frac{-\frac{d \cdot \left(c0 \cdot d\right)}{w}}{D} \cdot \frac{c0}{w \cdot \left(h \cdot \left(0 - D\right)\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -3.8 \cdot 10^{-215}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{d}{D} \cdot \frac{2 \cdot \left(c0 \cdot \frac{d}{w \cdot h}\right)}{D}\right)\\ \mathbf{elif}\;w \leq 4.3 \cdot 10^{-118}:\\ \;\;\;\;\frac{D \cdot \left(D \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot 0.25\right)\right)}{d \cdot d}\\ \mathbf{elif}\;w \leq 1.45 \cdot 10^{+30}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot 2}{D} \cdot \frac{\frac{d}{w \cdot h}}{\frac{D}{d}}\right)\\ \mathbf{elif}\;w \leq 1.1 \cdot 10^{+99}:\\ \;\;\;\;\frac{\frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d}{0.25}}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d \cdot \left(c0 \cdot d\right)}{w}}{D} \cdot \frac{c0}{w \cdot \left(h \cdot D\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 41.6% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := h \cdot \left(M \cdot M\right)\\ t_1 := \frac{c0}{2 \cdot w} \cdot \left(\frac{d}{D} \cdot \frac{2 \cdot \left(c0 \cdot \frac{d}{w \cdot h}\right)}{D}\right)\\ \mathbf{if}\;w \leq -4.6 \cdot 10^{-214}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;w \leq 7.8 \cdot 10^{-119}:\\ \;\;\;\;\frac{D \cdot \left(D \cdot \left(t\_0 \cdot 0.25\right)\right)}{d \cdot d}\\ \mathbf{elif}\;w \leq 4 \cdot 10^{+29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;w \leq 3.1 \cdot 10^{+98}:\\ \;\;\;\;\frac{\frac{\left(D \cdot D\right) \cdot t\_0}{\frac{d}{0.25}}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d \cdot \left(c0 \cdot d\right)}{w}}{D} \cdot \frac{c0}{w \cdot \left(h \cdot D\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* h (* M M)))
        (t_1
         (* (/ c0 (* 2.0 w)) (* (/ d D) (/ (* 2.0 (* c0 (/ d (* w h)))) D)))))
   (if (<= w -4.6e-214)
     t_1
     (if (<= w 7.8e-119)
       (/ (* D (* D (* t_0 0.25))) (* d d))
       (if (<= w 4e+29)
         t_1
         (if (<= w 3.1e+98)
           (/ (/ (* (* D D) t_0) (/ d 0.25)) d)
           (* (/ (/ (* d (* c0 d)) w) D) (/ c0 (* w (* h D))))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = h * (M * M);
	double t_1 = (c0 / (2.0 * w)) * ((d / D) * ((2.0 * (c0 * (d / (w * h)))) / D));
	double tmp;
	if (w <= -4.6e-214) {
		tmp = t_1;
	} else if (w <= 7.8e-119) {
		tmp = (D * (D * (t_0 * 0.25))) / (d * d);
	} else if (w <= 4e+29) {
		tmp = t_1;
	} else if (w <= 3.1e+98) {
		tmp = (((D * D) * t_0) / (d / 0.25)) / d;
	} else {
		tmp = (((d * (c0 * d)) / w) / D) * (c0 / (w * (h * D)));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = h * (m * m)
    t_1 = (c0 / (2.0d0 * w)) * ((d_1 / d) * ((2.0d0 * (c0 * (d_1 / (w * h)))) / d))
    if (w <= (-4.6d-214)) then
        tmp = t_1
    else if (w <= 7.8d-119) then
        tmp = (d * (d * (t_0 * 0.25d0))) / (d_1 * d_1)
    else if (w <= 4d+29) then
        tmp = t_1
    else if (w <= 3.1d+98) then
        tmp = (((d * d) * t_0) / (d_1 / 0.25d0)) / d_1
    else
        tmp = (((d_1 * (c0 * d_1)) / w) / d) * (c0 / (w * (h * d)))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = h * (M * M);
	double t_1 = (c0 / (2.0 * w)) * ((d / D) * ((2.0 * (c0 * (d / (w * h)))) / D));
	double tmp;
	if (w <= -4.6e-214) {
		tmp = t_1;
	} else if (w <= 7.8e-119) {
		tmp = (D * (D * (t_0 * 0.25))) / (d * d);
	} else if (w <= 4e+29) {
		tmp = t_1;
	} else if (w <= 3.1e+98) {
		tmp = (((D * D) * t_0) / (d / 0.25)) / d;
	} else {
		tmp = (((d * (c0 * d)) / w) / D) * (c0 / (w * (h * D)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = h * (M * M)
	t_1 = (c0 / (2.0 * w)) * ((d / D) * ((2.0 * (c0 * (d / (w * h)))) / D))
	tmp = 0
	if w <= -4.6e-214:
		tmp = t_1
	elif w <= 7.8e-119:
		tmp = (D * (D * (t_0 * 0.25))) / (d * d)
	elif w <= 4e+29:
		tmp = t_1
	elif w <= 3.1e+98:
		tmp = (((D * D) * t_0) / (d / 0.25)) / d
	else:
		tmp = (((d * (c0 * d)) / w) / D) * (c0 / (w * (h * D)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(h * Float64(M * M))
	t_1 = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(Float64(d / D) * Float64(Float64(2.0 * Float64(c0 * Float64(d / Float64(w * h)))) / D)))
	tmp = 0.0
	if (w <= -4.6e-214)
		tmp = t_1;
	elseif (w <= 7.8e-119)
		tmp = Float64(Float64(D * Float64(D * Float64(t_0 * 0.25))) / Float64(d * d));
	elseif (w <= 4e+29)
		tmp = t_1;
	elseif (w <= 3.1e+98)
		tmp = Float64(Float64(Float64(Float64(D * D) * t_0) / Float64(d / 0.25)) / d);
	else
		tmp = Float64(Float64(Float64(Float64(d * Float64(c0 * d)) / w) / D) * Float64(c0 / Float64(w * Float64(h * D))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = h * (M * M);
	t_1 = (c0 / (2.0 * w)) * ((d / D) * ((2.0 * (c0 * (d / (w * h)))) / D));
	tmp = 0.0;
	if (w <= -4.6e-214)
		tmp = t_1;
	elseif (w <= 7.8e-119)
		tmp = (D * (D * (t_0 * 0.25))) / (d * d);
	elseif (w <= 4e+29)
		tmp = t_1;
	elseif (w <= 3.1e+98)
		tmp = (((D * D) * t_0) / (d / 0.25)) / d;
	else
		tmp = (((d * (c0 * d)) / w) / D) * (c0 / (w * (h * D)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[(d / D), $MachinePrecision] * N[(N[(2.0 * N[(c0 * N[(d / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, -4.6e-214], t$95$1, If[LessEqual[w, 7.8e-119], N[(N[(D * N[(D * N[(t$95$0 * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 4e+29], t$95$1, If[LessEqual[w, 3.1e+98], N[(N[(N[(N[(D * D), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(d / 0.25), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], N[(N[(N[(N[(d * N[(c0 * d), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision] / D), $MachinePrecision] * N[(c0 / N[(w * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if w < -4.60000000000000022e-214 or 7.7999999999999998e-119 < w < 3.99999999999999966e29

   1. Initial program 29.0%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\color{blue}{{D}^{2} \cdot \left(h \cdot w\right)}}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(2 \cdot \left(c0 \cdot {d}^{2}\right)\right), \color{blue}{\left({D}^{2} \cdot \left(h \cdot w\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot {d}^{2}\right)\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left({d}^{2}\right)\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(d \cdot d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(h \cdot w\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{h} \cdot w\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{h} \cdot w\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(w \cdot \color{blue}{h}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 38.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(w, \color{blue}{h}\right)\right)\right)\right) \]

   5. Simplified 38.8%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(D \cdot D\right)} \cdot \left(w \cdot h\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}}\right)\right) \]

      4. frac-times N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \left(\frac{c0 \cdot d}{w \cdot h} \cdot \color{blue}{\frac{d}{D \cdot D}}\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\left(2 \cdot \frac{c0 \cdot d}{w \cdot h}\right) \cdot \color{blue}{\frac{d}{D \cdot D}}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \frac{c0 \cdot d}{w \cdot h}\right), \color{blue}{\left(\frac{d}{D \cdot D}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{c0 \cdot d}{w \cdot h}\right)\right), \left(\frac{\color{blue}{d}}{D \cdot D}\right)\right)\right) \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{\frac{c0 \cdot d}{w}}{h}\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{c0 \cdot d}{w}\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(c0 \cdot \frac{d}{w}\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{d}{w}\right)\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \mathsf{/.f64}\left(d, \color{blue}{\left(D \cdot D\right)}\right)\right)\right) \]

      14. *-lowering-*.f64 44.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(D, \color{blue}{D}\right)\right)\right)\right) \]

   7. Applied egg-rr 44.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot \frac{d}{D \cdot D}\right)} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot d}{\color{blue}{D \cdot D}}\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}}{D} \cdot \color{blue}{\frac{d}{D}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\left(\frac{2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}}{D}\right), \color{blue}{\left(\frac{d}{D}\right)}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right), D\right), \left(\frac{\color{blue}{d}}{D}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{c0 \cdot \frac{d}{w}}{h}\right)\right), D\right), \left(\frac{d}{D}\right)\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot \frac{\frac{d}{w}}{h}\right)\right), D\right), \left(\frac{d}{D}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(\frac{\frac{d}{w}}{h}\right)\right)\right), D\right), \left(\frac{d}{D}\right)\right)\right) \]

      8. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(\frac{d}{h \cdot w}\right)\right)\right), D\right), \left(\frac{d}{D}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(\frac{d}{w \cdot h}\right)\right)\right), D\right), \left(\frac{d}{D}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, \left(w \cdot h\right)\right)\right)\right), D\right), \left(\frac{d}{D}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(w, h\right)\right)\right)\right), D\right), \left(\frac{d}{D}\right)\right)\right) \]

      12. /-lowering-/.f64 55.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(w, h\right)\right)\right)\right), D\right), \mathsf{/.f64}\left(d, \color{blue}{D}\right)\right)\right) \]

   9. Applied egg-rr 55.7%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\frac{2 \cdot \left(c0 \cdot \frac{d}{w \cdot h}\right)}{D} \cdot \frac{d}{D}\right)} \]

   if -4.60000000000000022e-214 < w < 7.7999999999999998e-119

   1. Initial program 14.8%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}{\color{blue}{2 \cdot w}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right), \color{blue}{\left(2 \cdot w\right)}\right) \]

   3. Simplified 15.0%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot \frac{\frac{\frac{d \cdot d}{w} \cdot c0}{h} \cdot c0}{D \cdot D}}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(\mathsf{neg}\left(c0\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 19.5%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\left(\frac{-0.5 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot w\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)} + 0\right) \cdot \left(-c0\right)\right)}}{2 \cdot w} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. 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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left({D}^{2}\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(D \cdot D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot {M}^{2}\right)\right)\right), \left({d}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

      12. *-lowering-*.f64 47.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   10. Simplified 47.6%

       \[\leadsto \color{blue}{\frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot d}} \]

   11. Taylor expanded in D around 0

       \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   12. 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. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right) \cdot \frac{1}{4}\right), \left({\color{blue}{d}}^{2}\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right) \cdot \frac{1}{4}\right), \left({d}^{2}\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \left(h \cdot \frac{1}{4}\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(D \cdot D\right) \cdot {M}^{2}\right) \cdot \left(h \cdot \frac{1}{4}\right)\right), \left({d}^{2}\right)\right) \]

       7. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(D \cdot \left(D \cdot {M}^{2}\right)\right) \cdot \left(h \cdot \frac{1}{4}\right)\right), \left({d}^{2}\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(D \cdot \left(\left(D \cdot {M}^{2}\right) \cdot \left(h \cdot \frac{1}{4}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(\left(D \cdot {M}^{2}\right) \cdot \left(h \cdot \frac{1}{4}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

       10. associate-*r* N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(D \cdot \left({M}^{2} \cdot \left(h \cdot \frac{1}{4}\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(D \cdot \left(\left({M}^{2} \cdot h\right) \cdot \frac{1}{4}\right)\right)\right), \left({d}^{2}\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(D \cdot \left(\frac{1}{4} \cdot \left({M}^{2} \cdot h\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(\frac{1}{4} \cdot \left({M}^{2} \cdot h\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \left({M}^{2} \cdot h\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \left(h \cdot {M}^{2}\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       18. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       19. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

       20. *-lowering-*.f64 55.8%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   13. Simplified 55.8%

       \[\leadsto \color{blue}{\frac{D \cdot \left(D \cdot \left(0.25 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot d}} \]

   if 3.99999999999999966e29 < w < 3.10000000000000019e98

   1. Initial program 34.1%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}{\color{blue}{2 \cdot w}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right), \color{blue}{\left(2 \cdot w\right)}\right) \]

   3. Simplified 21.1%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot \frac{\frac{\frac{d \cdot d}{w} \cdot c0}{h} \cdot c0}{D \cdot D}}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(\mathsf{neg}\left(c0\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 53.8%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\left(\frac{-0.5 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot w\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)} + 0\right) \cdot \left(-c0\right)\right)}}{2 \cdot w} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. 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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left({D}^{2}\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(D \cdot D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot {M}^{2}\right)\right)\right), \left({d}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

      12. *-lowering-*.f64 60.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   10. Simplified 60.5%

       \[\leadsto \color{blue}{\frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot d}} \]
   11. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d}}{\color{blue}{d}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d}\right), \color{blue}{d}\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{1}{4}}{d}\right), d\right) \]

       4. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right), d\right) \]

       5. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{1}{\frac{d}{\frac{1}{4}}}\right), d\right) \]

       6. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d}{\frac{1}{4}}}\right), d\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right), \left(\frac{d}{\frac{1}{4}}\right)\right), d\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(D \cdot D\right), \left(h \cdot \left(M \cdot M\right)\right)\right), \left(\frac{d}{\frac{1}{4}}\right)\right), d\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot \left(M \cdot M\right)\right)\right), \left(\frac{d}{\frac{1}{4}}\right)\right), d\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right), \left(\frac{d}{\frac{1}{4}}\right)\right), d\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \left(\frac{d}{\frac{1}{4}}\right)\right), d\right) \]

       12. /-lowering-/.f64 73.8%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{/.f64}\left(d, \frac{1}{4}\right)\right), d\right) \]

   12. Applied egg-rr 73.8%

       \[\leadsto \color{blue}{\frac{\frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d}{0.25}}}{d}} \]

   if 3.10000000000000019e98 < w

   1. Initial program 25.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\color{blue}{{D}^{2} \cdot \left(h \cdot w\right)}}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(2 \cdot \left(c0 \cdot {d}^{2}\right)\right), \color{blue}{\left({D}^{2} \cdot \left(h \cdot w\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot {d}^{2}\right)\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left({d}^{2}\right)\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(d \cdot d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(h \cdot w\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{h} \cdot w\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{h} \cdot w\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(w \cdot \color{blue}{h}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 33.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(w, \color{blue}{h}\right)\right)\right)\right) \]

   5. Simplified 33.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(D \cdot D\right)} \cdot \left(w \cdot h\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}}\right)\right) \]

      4. frac-times N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \left(\frac{c0 \cdot d}{w \cdot h} \cdot \color{blue}{\frac{d}{D \cdot D}}\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\left(2 \cdot \frac{c0 \cdot d}{w \cdot h}\right) \cdot \color{blue}{\frac{d}{D \cdot D}}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \frac{c0 \cdot d}{w \cdot h}\right), \color{blue}{\left(\frac{d}{D \cdot D}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{c0 \cdot d}{w \cdot h}\right)\right), \left(\frac{\color{blue}{d}}{D \cdot D}\right)\right)\right) \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{\frac{c0 \cdot d}{w}}{h}\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{c0 \cdot d}{w}\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(c0 \cdot \frac{d}{w}\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{d}{w}\right)\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \mathsf{/.f64}\left(d, \color{blue}{\left(D \cdot D\right)}\right)\right)\right) \]

      14. *-lowering-*.f64 40.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(D, \color{blue}{D}\right)\right)\right)\right) \]

   7. Applied egg-rr 40.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot \frac{d}{D \cdot D}\right)} \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{c0 \cdot \frac{d}{w}}{h} \cdot \frac{d}{D \cdot D}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\frac{c0 \cdot d}{w}}{h} \cdot \frac{d}{D \cdot D}\right)\right) \]

      3. associate-/l/ N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0 \cdot d}{h \cdot w} \cdot \frac{\color{blue}{d}}{D \cdot D}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0 \cdot d}{w \cdot h} \cdot \frac{d}{D \cdot D}\right)\right) \]

      5. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}}\right) \]

      6. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(w \cdot h\right)} \cdot \left(D \cdot D\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \color{blue}{\left(w \cdot h\right)}}\right) \]

      8. associate-/l* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}} \]

      9. clear-num N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{1}{\color{blue}{\frac{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}}} \]

      10. un-div-inv N/A

          \[\leadsto \frac{\frac{c0}{2 \cdot w}}{\color{blue}{\frac{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}}} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{c0}{2 \cdot w}\right), \color{blue}{\left(\frac{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}\right)}\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \left(2 \cdot w\right)\right), \left(\frac{\color{blue}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{\left(D \cdot D\right) \cdot \color{blue}{\left(w \cdot h\right)}}{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}\right)\right) \]

   9. Applied egg-rr 44.0%

      \[\leadsto \color{blue}{\frac{\frac{c0}{2 \cdot w}}{\frac{D}{2 \cdot \left(c0 \cdot d\right)} \cdot \frac{D \cdot \left(w \cdot h\right)}{d}}} \]

   11. Applied egg-rr 51.5%

       \[\leadsto \color{blue}{\frac{-\frac{d \cdot \left(c0 \cdot d\right)}{w}}{D} \cdot \frac{c0}{w \cdot \left(h \cdot \left(0 - D\right)\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -4.6 \cdot 10^{-214}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{d}{D} \cdot \frac{2 \cdot \left(c0 \cdot \frac{d}{w \cdot h}\right)}{D}\right)\\ \mathbf{elif}\;w \leq 7.8 \cdot 10^{-119}:\\ \;\;\;\;\frac{D \cdot \left(D \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot 0.25\right)\right)}{d \cdot d}\\ \mathbf{elif}\;w \leq 4 \cdot 10^{+29}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{d}{D} \cdot \frac{2 \cdot \left(c0 \cdot \frac{d}{w \cdot h}\right)}{D}\right)\\ \mathbf{elif}\;w \leq 3.1 \cdot 10^{+98}:\\ \;\;\;\;\frac{\frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d}{0.25}}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d \cdot \left(c0 \cdot d\right)}{w}}{D} \cdot \frac{c0}{w \cdot \left(h \cdot D\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 42.7% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 5.5 \cdot 10^{-54}:\\ \;\;\;\;\frac{\left(h \cdot \left(M \cdot M\right)\right) \cdot \frac{D}{\frac{d}{D}}}{\frac{d}{0.25}}\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{+49}:\\ \;\;\;\;\frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{D \cdot \left(D \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)}\\ \mathbf{elif}\;d \leq 4.8 \cdot 10^{+160}:\\ \;\;\;\;\frac{\left(D \cdot 0.25\right) \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot D\right)\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left(d \cdot \frac{\frac{\frac{c0 \cdot \frac{c0}{D \cdot D}}{w}}{w}}{h}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= d 5.5e-54)
   (/ (* (* h (* M M)) (/ D (/ d D))) (/ d 0.25))
   (if (<= d 3.5e+49)
     (/ (* c0 (* c0 (* d d))) (* D (* D (* h (* w w)))))
     (if (<= d 4.8e+160)
       (/ (* (* D 0.25) (* (* M M) (* h D))) (* d d))
       (* d (* d (/ (/ (/ (* c0 (/ c0 (* D D))) w) w) h)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (d <= 5.5e-54) {
		tmp = ((h * (M * M)) * (D / (d / D))) / (d / 0.25);
	} else if (d <= 3.5e+49) {
		tmp = (c0 * (c0 * (d * d))) / (D * (D * (h * (w * w))));
	} else if (d <= 4.8e+160) {
		tmp = ((D * 0.25) * ((M * M) * (h * D))) / (d * d);
	} else {
		tmp = d * (d * ((((c0 * (c0 / (D * D))) / w) / w) / h));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (d_1 <= 5.5d-54) then
        tmp = ((h * (m * m)) * (d / (d_1 / d))) / (d_1 / 0.25d0)
    else if (d_1 <= 3.5d+49) then
        tmp = (c0 * (c0 * (d_1 * d_1))) / (d * (d * (h * (w * w))))
    else if (d_1 <= 4.8d+160) then
        tmp = ((d * 0.25d0) * ((m * m) * (h * d))) / (d_1 * d_1)
    else
        tmp = d_1 * (d_1 * ((((c0 * (c0 / (d * d))) / w) / w) / h))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (d <= 5.5e-54) {
		tmp = ((h * (M * M)) * (D / (d / D))) / (d / 0.25);
	} else if (d <= 3.5e+49) {
		tmp = (c0 * (c0 * (d * d))) / (D * (D * (h * (w * w))));
	} else if (d <= 4.8e+160) {
		tmp = ((D * 0.25) * ((M * M) * (h * D))) / (d * d);
	} else {
		tmp = d * (d * ((((c0 * (c0 / (D * D))) / w) / w) / h));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if d <= 5.5e-54:
		tmp = ((h * (M * M)) * (D / (d / D))) / (d / 0.25)
	elif d <= 3.5e+49:
		tmp = (c0 * (c0 * (d * d))) / (D * (D * (h * (w * w))))
	elif d <= 4.8e+160:
		tmp = ((D * 0.25) * ((M * M) * (h * D))) / (d * d)
	else:
		tmp = d * (d * ((((c0 * (c0 / (D * D))) / w) / w) / h))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (d <= 5.5e-54)
		tmp = Float64(Float64(Float64(h * Float64(M * M)) * Float64(D / Float64(d / D))) / Float64(d / 0.25));
	elseif (d <= 3.5e+49)
		tmp = Float64(Float64(c0 * Float64(c0 * Float64(d * d))) / Float64(D * Float64(D * Float64(h * Float64(w * w)))));
	elseif (d <= 4.8e+160)
		tmp = Float64(Float64(Float64(D * 0.25) * Float64(Float64(M * M) * Float64(h * D))) / Float64(d * d));
	else
		tmp = Float64(d * Float64(d * Float64(Float64(Float64(Float64(c0 * Float64(c0 / Float64(D * D))) / w) / w) / h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (d <= 5.5e-54)
		tmp = ((h * (M * M)) * (D / (d / D))) / (d / 0.25);
	elseif (d <= 3.5e+49)
		tmp = (c0 * (c0 * (d * d))) / (D * (D * (h * (w * w))));
	elseif (d <= 4.8e+160)
		tmp = ((D * 0.25) * ((M * M) * (h * D))) / (d * d);
	else
		tmp = d * (d * ((((c0 * (c0 / (D * D))) / w) / w) / h));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[d, 5.5e-54], N[(N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] * N[(D / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d / 0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.5e+49], N[(N[(c0 * N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(D * N[(D * N[(h * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.8e+160], N[(N[(N[(D * 0.25), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], N[(d * N[(d * N[(N[(N[(N[(c0 * N[(c0 / N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision] / w), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < 5.50000000000000046e-54

   1. Initial program 24.2%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}{\color{blue}{2 \cdot w}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right), \color{blue}{\left(2 \cdot w\right)}\right) \]

   3. Simplified 22.6%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot \frac{\frac{\frac{d \cdot d}{w} \cdot c0}{h} \cdot c0}{D \cdot D}}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(\mathsf{neg}\left(c0\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 12.8%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\left(\frac{-0.5 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot w\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)} + 0\right) \cdot \left(-c0\right)\right)}}{2 \cdot w} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. 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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left({D}^{2}\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(D \cdot D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot {M}^{2}\right)\right)\right), \left({d}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

      12. *-lowering-*.f64 29.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   10. Simplified 29.8%

       \[\leadsto \color{blue}{\frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot d}} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{1}{4}}{\color{blue}{d} \cdot d} \]

       2. times-frac N/A

          \[\leadsto \frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d} \cdot \color{blue}{\frac{\frac{1}{4}}{d}} \]

       3. clear-num N/A

          \[\leadsto \frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d} \cdot \frac{1}{\color{blue}{\frac{d}{\frac{1}{4}}}} \]

       4. un-div-inv N/A

          \[\leadsto \frac{\frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d}}{\color{blue}{\frac{d}{\frac{1}{4}}}} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d}\right), \color{blue}{\left(\frac{d}{\frac{1}{4}}\right)}\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(h \cdot \left(M \cdot M\right)\right) \cdot \left(D \cdot D\right)}{d}\right), \left(\frac{d}{\frac{1}{4}}\right)\right) \]

       7. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \frac{D \cdot D}{d}\right), \left(\frac{\color{blue}{d}}{\frac{1}{4}}\right)\right) \]

       8. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \left(D \cdot \frac{D}{d}\right)\right), \left(\frac{d}{\frac{1}{4}}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(h \cdot \left(M \cdot M\right)\right), \left(D \cdot \frac{D}{d}\right)\right), \left(\frac{\color{blue}{d}}{\frac{1}{4}}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \left(M \cdot M\right)\right), \left(D \cdot \frac{D}{d}\right)\right), \left(\frac{d}{\frac{1}{4}}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), \left(D \cdot \frac{D}{d}\right)\right), \left(\frac{d}{\frac{1}{4}}\right)\right) \]

       12. clear-num N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), \left(D \cdot \frac{1}{\frac{d}{D}}\right)\right), \left(\frac{d}{\frac{1}{4}}\right)\right) \]

       13. un-div-inv N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), \left(\frac{D}{\frac{d}{D}}\right)\right), \left(\frac{d}{\frac{1}{4}}\right)\right) \]

       14. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), \mathsf{/.f64}\left(D, \left(\frac{d}{D}\right)\right)\right), \left(\frac{d}{\frac{1}{4}}\right)\right) \]

       15. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), \mathsf{/.f64}\left(D, \mathsf{/.f64}\left(d, D\right)\right)\right), \left(\frac{d}{\frac{1}{4}}\right)\right) \]

       16. /-lowering-/.f64 37.7%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), \mathsf{/.f64}\left(D, \mathsf{/.f64}\left(d, D\right)\right)\right), \mathsf{/.f64}\left(d, \color{blue}{\frac{1}{4}}\right)\right) \]

   12. Applied egg-rr 37.7%

       \[\leadsto \color{blue}{\frac{\left(h \cdot \left(M \cdot M\right)\right) \cdot \frac{D}{\frac{d}{D}}}{\frac{d}{0.25}}} \]

   if 5.50000000000000046e-54 < d < 3.49999999999999975e49

   1. Initial program 39.0%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\color{blue}{{D}^{2} \cdot \left(h \cdot w\right)}}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(2 \cdot \left(c0 \cdot {d}^{2}\right)\right), \color{blue}{\left({D}^{2} \cdot \left(h \cdot w\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot {d}^{2}\right)\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left({d}^{2}\right)\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(d \cdot d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(h \cdot w\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{h} \cdot w\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{h} \cdot w\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(w \cdot \color{blue}{h}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 44.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(w, \color{blue}{h}\right)\right)\right)\right) \]

   5. Simplified 44.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(D \cdot D\right)} \cdot \left(w \cdot h\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}}\right)\right) \]

      4. frac-times N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \left(\frac{c0 \cdot d}{w \cdot h} \cdot \color{blue}{\frac{d}{D \cdot D}}\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\left(2 \cdot \frac{c0 \cdot d}{w \cdot h}\right) \cdot \color{blue}{\frac{d}{D \cdot D}}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \frac{c0 \cdot d}{w \cdot h}\right), \color{blue}{\left(\frac{d}{D \cdot D}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{c0 \cdot d}{w \cdot h}\right)\right), \left(\frac{\color{blue}{d}}{D \cdot D}\right)\right)\right) \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{\frac{c0 \cdot d}{w}}{h}\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{c0 \cdot d}{w}\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(c0 \cdot \frac{d}{w}\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{d}{w}\right)\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \mathsf{/.f64}\left(d, \color{blue}{\left(D \cdot D\right)}\right)\right)\right) \]

      14. *-lowering-*.f64 48.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(D, \color{blue}{D}\right)\right)\right)\right) \]

   7. Applied egg-rr 48.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot \frac{d}{D \cdot D}\right)} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]
   9. 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. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(c0 \cdot c0\right) \cdot {d}^{2}\right), \left({\color{blue}{D}}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(c0 \cdot {d}^{2}\right)\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot {d}^{2}\right)\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, \left({d}^{2}\right)\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, \left(d \cdot d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \left(\left(D \cdot D\right) \cdot \left(\color{blue}{h} \cdot {w}^{2}\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \left(D \cdot \color{blue}{\left(D \cdot \left(h \cdot {w}^{2}\right)\right)}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(D, \color{blue}{\left(D \cdot \left(h \cdot {w}^{2}\right)\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \color{blue}{\left(h \cdot {w}^{2}\right)}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(h, \color{blue}{\left({w}^{2}\right)}\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(h, \left(w \cdot \color{blue}{w}\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 53.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w, \color{blue}{w}\right)\right)\right)\right)\right) \]

   10. Simplified 53.4%

       \[\leadsto \color{blue}{\frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{D \cdot \left(D \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)}} \]

   if 3.49999999999999975e49 < d < 4.8000000000000003e160

   1. Initial program 14.8%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}{\color{blue}{2 \cdot w}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right), \color{blue}{\left(2 \cdot w\right)}\right) \]

   3. Simplified 9.3%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot \frac{\frac{\frac{d \cdot d}{w} \cdot c0}{h} \cdot c0}{D \cdot D}}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(\mathsf{neg}\left(c0\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 21.5%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\left(\frac{-0.5 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot w\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)} + 0\right) \cdot \left(-c0\right)\right)}}{2 \cdot w} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. 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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left({D}^{2}\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(D \cdot D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot {M}^{2}\right)\right)\right), \left({d}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

      12. *-lowering-*.f64 38.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   10. Simplified 38.7%

       \[\leadsto \color{blue}{\frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot d}} \]
   11. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{4} \cdot D\right) \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{d}, d\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{4} \cdot D\right), \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{d}, d\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, D\right), \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, D\right), \left(\left(D \cdot h\right) \cdot \left(M \cdot M\right)\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, D\right), \mathsf{*.f64}\left(\left(D \cdot h\right), \left(M \cdot M\right)\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, D\right), \mathsf{*.f64}\left(\left(h \cdot D\right), \left(M \cdot M\right)\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, D\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, D\right), \left(M \cdot M\right)\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

       9. *-lowering-*.f64 55.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, D\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, D\right), \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

   12. Applied egg-rr 55.9%

       \[\leadsto \frac{\color{blue}{\left(0.25 \cdot D\right) \cdot \left(\left(h \cdot D\right) \cdot \left(M \cdot M\right)\right)}}{d \cdot d} \]

   if 4.8000000000000003e160 < d

   1. Initial program 27.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\color{blue}{{D}^{2} \cdot \left(h \cdot w\right)}}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(2 \cdot \left(c0 \cdot {d}^{2}\right)\right), \color{blue}{\left({D}^{2} \cdot \left(h \cdot w\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot {d}^{2}\right)\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left({d}^{2}\right)\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(d \cdot d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(h \cdot w\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{h} \cdot w\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{h} \cdot w\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(w \cdot \color{blue}{h}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 34.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(w, \color{blue}{h}\right)\right)\right)\right) \]

   5. Simplified 34.7%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(D \cdot D\right)} \cdot \left(w \cdot h\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}}\right)\right) \]

      4. frac-times N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \left(\frac{c0 \cdot d}{w \cdot h} \cdot \color{blue}{\frac{d}{D \cdot D}}\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\left(2 \cdot \frac{c0 \cdot d}{w \cdot h}\right) \cdot \color{blue}{\frac{d}{D \cdot D}}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \frac{c0 \cdot d}{w \cdot h}\right), \color{blue}{\left(\frac{d}{D \cdot D}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{c0 \cdot d}{w \cdot h}\right)\right), \left(\frac{\color{blue}{d}}{D \cdot D}\right)\right)\right) \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{\frac{c0 \cdot d}{w}}{h}\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{c0 \cdot d}{w}\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(c0 \cdot \frac{d}{w}\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{d}{w}\right)\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \mathsf{/.f64}\left(d, \color{blue}{\left(D \cdot D\right)}\right)\right)\right) \]

      14. *-lowering-*.f64 41.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(D, \color{blue}{D}\right)\right)\right)\right) \]

   7. Applied egg-rr 41.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot \frac{d}{D \cdot D}\right)} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot \frac{\frac{d}{D}}{\color{blue}{D}}\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot \frac{d}{D}}{\color{blue}{D}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot \frac{d}{D}\right), \color{blue}{D}\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot \frac{1}{\frac{D}{d}}\right), D\right)\right) \]

      5. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}}{\frac{D}{d}}\right), D\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right), \left(\frac{D}{d}\right)\right), D\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{c0 \cdot \frac{d}{w}}{h}\right)\right), \left(\frac{D}{d}\right)\right), D\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot \frac{\frac{d}{w}}{h}\right)\right), \left(\frac{D}{d}\right)\right), D\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(\frac{\frac{d}{w}}{h}\right)\right)\right), \left(\frac{D}{d}\right)\right), D\right)\right) \]

      10. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(\frac{d}{h \cdot w}\right)\right)\right), \left(\frac{D}{d}\right)\right), D\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(\frac{d}{w \cdot h}\right)\right)\right), \left(\frac{D}{d}\right)\right), D\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, \left(w \cdot h\right)\right)\right)\right), \left(\frac{D}{d}\right)\right), D\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(w, h\right)\right)\right)\right), \left(\frac{D}{d}\right)\right), D\right)\right) \]

      14. /-lowering-/.f64 50.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(w, h\right)\right)\right)\right), \mathsf{/.f64}\left(D, d\right)\right), D\right)\right) \]

   9. Applied egg-rr 50.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{\frac{2 \cdot \left(c0 \cdot \frac{d}{w \cdot h}\right)}{\frac{D}{d}}}{D}} \]

   10. Taylor expanded in c0 around 0

       \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{{d}^{2} \cdot {c0}^{2}}{\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)} \]

       2. associate-/l* N/A

          \[\leadsto {d}^{2} \cdot \color{blue}{\frac{{c0}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]

       3. *-commutative N/A

          \[\leadsto \frac{{c0}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \cdot \color{blue}{{d}^{2}} \]

       4. unpow2 N/A

          \[\leadsto \frac{{c0}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \cdot \left(d \cdot \color{blue}{d}\right) \]

       5. associate-*r* N/A

          \[\leadsto \left(\frac{{c0}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \cdot d\right) \cdot \color{blue}{d} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{{c0}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \cdot d\right), \color{blue}{d}\right) \]

   12. Simplified 41.7%

       \[\leadsto \color{blue}{\left(\frac{\frac{\frac{c0 \cdot \frac{c0}{D \cdot D}}{w}}{w}}{h} \cdot d\right) \cdot d} \]
3. Recombined 4 regimes into one program.

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 5.5 \cdot 10^{-54}:\\ \;\;\;\;\frac{\left(h \cdot \left(M \cdot M\right)\right) \cdot \frac{D}{\frac{d}{D}}}{\frac{d}{0.25}}\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{+49}:\\ \;\;\;\;\frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{D \cdot \left(D \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)}\\ \mathbf{elif}\;d \leq 4.8 \cdot 10^{+160}:\\ \;\;\;\;\frac{\left(D \cdot 0.25\right) \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot D\right)\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left(d \cdot \frac{\frac{\frac{c0 \cdot \frac{c0}{D \cdot D}}{w}}{w}}{h}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.8% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w} \cdot \frac{\frac{2 \cdot \frac{\frac{c0}{w}}{\frac{h}{d}}}{\frac{D}{d}}}{D}\\ \mathbf{if}\;w \leq -1.8 \cdot 10^{-223}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;w \leq 3.6 \cdot 10^{-119}:\\ \;\;\;\;\frac{D \cdot \left(D \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot 0.25\right)\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0
         (* (/ c0 (* 2.0 w)) (/ (/ (* 2.0 (/ (/ c0 w) (/ h d))) (/ D d)) D))))
   (if (<= w -1.8e-223)
     t_0
     (if (<= w 3.6e-119) (/ (* D (* D (* (* h (* M M)) 0.25))) (* d d)) t_0))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 / (2.0 * w)) * (((2.0 * ((c0 / w) / (h / d))) / (D / d)) / D);
	double tmp;
	if (w <= -1.8e-223) {
		tmp = t_0;
	} else if (w <= 3.6e-119) {
		tmp = (D * (D * ((h * (M * M)) * 0.25))) / (d * d);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (c0 / (2.0d0 * w)) * (((2.0d0 * ((c0 / w) / (h / d_1))) / (d / d_1)) / d)
    if (w <= (-1.8d-223)) then
        tmp = t_0
    else if (w <= 3.6d-119) then
        tmp = (d * (d * ((h * (m * m)) * 0.25d0))) / (d_1 * d_1)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 / (2.0 * w)) * (((2.0 * ((c0 / w) / (h / d))) / (D / d)) / D);
	double tmp;
	if (w <= -1.8e-223) {
		tmp = t_0;
	} else if (w <= 3.6e-119) {
		tmp = (D * (D * ((h * (M * M)) * 0.25))) / (d * d);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 / (2.0 * w)) * (((2.0 * ((c0 / w) / (h / d))) / (D / d)) / D)
	tmp = 0
	if w <= -1.8e-223:
		tmp = t_0
	elif w <= 3.6e-119:
		tmp = (D * (D * ((h * (M * M)) * 0.25))) / (d * d)
	else:
		tmp = t_0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(Float64(Float64(2.0 * Float64(Float64(c0 / w) / Float64(h / d))) / Float64(D / d)) / D))
	tmp = 0.0
	if (w <= -1.8e-223)
		tmp = t_0;
	elseif (w <= 3.6e-119)
		tmp = Float64(Float64(D * Float64(D * Float64(Float64(h * Float64(M * M)) * 0.25))) / Float64(d * d));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 / (2.0 * w)) * (((2.0 * ((c0 / w) / (h / d))) / (D / d)) / D);
	tmp = 0.0;
	if (w <= -1.8e-223)
		tmp = t_0;
	elseif (w <= 3.6e-119)
		tmp = (D * (D * ((h * (M * M)) * 0.25))) / (d * d);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(2.0 * N[(N[(c0 / w), $MachinePrecision] / N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(D / d), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, -1.8e-223], t$95$0, If[LessEqual[w, 3.6e-119], N[(N[(D * N[(D * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if w < -1.8000000000000002e-223 or 3.6e-119 < w

   1. Initial program 29.2%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\color{blue}{{D}^{2} \cdot \left(h \cdot w\right)}}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(2 \cdot \left(c0 \cdot {d}^{2}\right)\right), \color{blue}{\left({D}^{2} \cdot \left(h \cdot w\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot {d}^{2}\right)\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left({d}^{2}\right)\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(d \cdot d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(h \cdot w\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{h} \cdot w\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{h} \cdot w\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(w \cdot \color{blue}{h}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 36.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(w, \color{blue}{h}\right)\right)\right)\right) \]

   5. Simplified 36.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(D \cdot D\right)} \cdot \left(w \cdot h\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}}\right)\right) \]

      4. frac-times N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \left(\frac{c0 \cdot d}{w \cdot h} \cdot \color{blue}{\frac{d}{D \cdot D}}\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\left(2 \cdot \frac{c0 \cdot d}{w \cdot h}\right) \cdot \color{blue}{\frac{d}{D \cdot D}}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \frac{c0 \cdot d}{w \cdot h}\right), \color{blue}{\left(\frac{d}{D \cdot D}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{c0 \cdot d}{w \cdot h}\right)\right), \left(\frac{\color{blue}{d}}{D \cdot D}\right)\right)\right) \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{\frac{c0 \cdot d}{w}}{h}\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{c0 \cdot d}{w}\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(c0 \cdot \frac{d}{w}\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{d}{w}\right)\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \mathsf{/.f64}\left(d, \color{blue}{\left(D \cdot D\right)}\right)\right)\right) \]

      14. *-lowering-*.f64 42.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(D, \color{blue}{D}\right)\right)\right)\right) \]

   7. Applied egg-rr 42.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot \frac{d}{D \cdot D}\right)} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot \frac{\frac{d}{D}}{\color{blue}{D}}\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot \frac{d}{D}}{\color{blue}{D}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot \frac{d}{D}\right), \color{blue}{D}\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot \frac{1}{\frac{D}{d}}\right), D\right)\right) \]

      5. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}}{\frac{D}{d}}\right), D\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right), \left(\frac{D}{d}\right)\right), D\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{c0 \cdot \frac{d}{w}}{h}\right)\right), \left(\frac{D}{d}\right)\right), D\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot \frac{\frac{d}{w}}{h}\right)\right), \left(\frac{D}{d}\right)\right), D\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(\frac{\frac{d}{w}}{h}\right)\right)\right), \left(\frac{D}{d}\right)\right), D\right)\right) \]

      10. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(\frac{d}{h \cdot w}\right)\right)\right), \left(\frac{D}{d}\right)\right), D\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(\frac{d}{w \cdot h}\right)\right)\right), \left(\frac{D}{d}\right)\right), D\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, \left(w \cdot h\right)\right)\right)\right), \left(\frac{D}{d}\right)\right), D\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(w, h\right)\right)\right)\right), \left(\frac{D}{d}\right)\right), D\right)\right) \]

      14. /-lowering-/.f64 52.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(w, h\right)\right)\right)\right), \mathsf{/.f64}\left(D, d\right)\right), D\right)\right) \]

   9. Applied egg-rr 52.3%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{\frac{2 \cdot \left(c0 \cdot \frac{d}{w \cdot h}\right)}{\frac{D}{d}}}{D}} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{c0 \cdot d}{w \cdot h}\right)\right), \mathsf{/.f64}\left(D, d\right)\right), D\right)\right) \]

       2. times-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{c0}{w} \cdot \frac{d}{h}\right)\right), \mathsf{/.f64}\left(D, d\right)\right), D\right)\right) \]

       3. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{c0}{w} \cdot \frac{1}{\frac{h}{d}}\right)\right), \mathsf{/.f64}\left(D, d\right)\right), D\right)\right) \]

       4. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{\frac{c0}{w}}{\frac{h}{d}}\right)\right), \mathsf{/.f64}\left(D, d\right)\right), D\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{c0}{w}\right), \left(\frac{h}{d}\right)\right)\right), \mathsf{/.f64}\left(D, d\right)\right), D\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, w\right), \left(\frac{h}{d}\right)\right)\right), \mathsf{/.f64}\left(D, d\right)\right), D\right)\right) \]

       7. /-lowering-/.f64 56.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, w\right), \mathsf{/.f64}\left(h, d\right)\right)\right), \mathsf{/.f64}\left(D, d\right)\right), D\right)\right) \]

   11. Applied egg-rr 56.5%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\frac{2 \cdot \color{blue}{\frac{\frac{c0}{w}}{\frac{h}{d}}}}{\frac{D}{d}}}{D} \]

   if -1.8000000000000002e-223 < w < 3.6e-119

   1. Initial program 13.8%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}{\color{blue}{2 \cdot w}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right), \color{blue}{\left(2 \cdot w\right)}\right) \]

   3. Simplified 14.0%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot \frac{\frac{\frac{d \cdot d}{w} \cdot c0}{h} \cdot c0}{D \cdot D}}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(\mathsf{neg}\left(c0\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 18.7%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\left(\frac{-0.5 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot w\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)} + 0\right) \cdot \left(-c0\right)\right)}}{2 \cdot w} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. 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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left({D}^{2}\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(D \cdot D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot {M}^{2}\right)\right)\right), \left({d}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

      12. *-lowering-*.f64 47.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   10. Simplified 47.6%

       \[\leadsto \color{blue}{\frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot d}} \]

   11. Taylor expanded in D around 0

       \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   12. 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. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right) \cdot \frac{1}{4}\right), \left({\color{blue}{d}}^{2}\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right) \cdot \frac{1}{4}\right), \left({d}^{2}\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \left(h \cdot \frac{1}{4}\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(D \cdot D\right) \cdot {M}^{2}\right) \cdot \left(h \cdot \frac{1}{4}\right)\right), \left({d}^{2}\right)\right) \]

       7. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(D \cdot \left(D \cdot {M}^{2}\right)\right) \cdot \left(h \cdot \frac{1}{4}\right)\right), \left({d}^{2}\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(D \cdot \left(\left(D \cdot {M}^{2}\right) \cdot \left(h \cdot \frac{1}{4}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(\left(D \cdot {M}^{2}\right) \cdot \left(h \cdot \frac{1}{4}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

       10. associate-*r* N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(D \cdot \left({M}^{2} \cdot \left(h \cdot \frac{1}{4}\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(D \cdot \left(\left({M}^{2} \cdot h\right) \cdot \frac{1}{4}\right)\right)\right), \left({d}^{2}\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(D \cdot \left(\frac{1}{4} \cdot \left({M}^{2} \cdot h\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(\frac{1}{4} \cdot \left({M}^{2} \cdot h\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \left({M}^{2} \cdot h\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \left(h \cdot {M}^{2}\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       18. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       19. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

       20. *-lowering-*.f64 55.9%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   13. Simplified 55.9%

       \[\leadsto \color{blue}{\frac{D \cdot \left(D \cdot \left(0.25 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot d}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -1.8 \cdot 10^{-223}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \frac{\frac{2 \cdot \frac{\frac{c0}{w}}{\frac{h}{d}}}{\frac{D}{d}}}{D}\\ \mathbf{elif}\;w \leq 3.6 \cdot 10^{-119}:\\ \;\;\;\;\frac{D \cdot \left(D \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot 0.25\right)\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \frac{\frac{2 \cdot \frac{\frac{c0}{w}}{\frac{h}{d}}}{\frac{D}{d}}}{D}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 39.0% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := w \cdot \left(h \cdot D\right)\\ t_1 := \frac{d \cdot \left(c0 \cdot d\right)}{w}\\ \mathbf{if}\;w \leq -3.5 \cdot 10^{-223}:\\ \;\;\;\;\frac{t\_1}{t\_0} \cdot \frac{c0}{D}\\ \mathbf{elif}\;w \leq 6.6 \cdot 10^{-119}:\\ \;\;\;\;\frac{D \cdot \left(D \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot 0.25\right)\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{D} \cdot \frac{c0}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* w (* h D))) (t_1 (/ (* d (* c0 d)) w)))
   (if (<= w -3.5e-223)
     (* (/ t_1 t_0) (/ c0 D))
     (if (<= w 6.6e-119)
       (/ (* D (* D (* (* h (* M M)) 0.25))) (* d d))
       (* (/ t_1 D) (/ c0 t_0))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = w * (h * D);
	double t_1 = (d * (c0 * d)) / w;
	double tmp;
	if (w <= -3.5e-223) {
		tmp = (t_1 / t_0) * (c0 / D);
	} else if (w <= 6.6e-119) {
		tmp = (D * (D * ((h * (M * M)) * 0.25))) / (d * d);
	} else {
		tmp = (t_1 / D) * (c0 / t_0);
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = w * (h * d)
    t_1 = (d_1 * (c0 * d_1)) / w
    if (w <= (-3.5d-223)) then
        tmp = (t_1 / t_0) * (c0 / d)
    else if (w <= 6.6d-119) then
        tmp = (d * (d * ((h * (m * m)) * 0.25d0))) / (d_1 * d_1)
    else
        tmp = (t_1 / d) * (c0 / t_0)
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = w * (h * D);
	double t_1 = (d * (c0 * d)) / w;
	double tmp;
	if (w <= -3.5e-223) {
		tmp = (t_1 / t_0) * (c0 / D);
	} else if (w <= 6.6e-119) {
		tmp = (D * (D * ((h * (M * M)) * 0.25))) / (d * d);
	} else {
		tmp = (t_1 / D) * (c0 / t_0);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = w * (h * D)
	t_1 = (d * (c0 * d)) / w
	tmp = 0
	if w <= -3.5e-223:
		tmp = (t_1 / t_0) * (c0 / D)
	elif w <= 6.6e-119:
		tmp = (D * (D * ((h * (M * M)) * 0.25))) / (d * d)
	else:
		tmp = (t_1 / D) * (c0 / t_0)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(w * Float64(h * D))
	t_1 = Float64(Float64(d * Float64(c0 * d)) / w)
	tmp = 0.0
	if (w <= -3.5e-223)
		tmp = Float64(Float64(t_1 / t_0) * Float64(c0 / D));
	elseif (w <= 6.6e-119)
		tmp = Float64(Float64(D * Float64(D * Float64(Float64(h * Float64(M * M)) * 0.25))) / Float64(d * d));
	else
		tmp = Float64(Float64(t_1 / D) * Float64(c0 / t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = w * (h * D);
	t_1 = (d * (c0 * d)) / w;
	tmp = 0.0;
	if (w <= -3.5e-223)
		tmp = (t_1 / t_0) * (c0 / D);
	elseif (w <= 6.6e-119)
		tmp = (D * (D * ((h * (M * M)) * 0.25))) / (d * d);
	else
		tmp = (t_1 / D) * (c0 / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(w * N[(h * D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(d * N[(c0 * d), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]}, If[LessEqual[w, -3.5e-223], N[(N[(t$95$1 / t$95$0), $MachinePrecision] * N[(c0 / D), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 6.6e-119], N[(N[(D * N[(D * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / D), $MachinePrecision] * N[(c0 / t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if w < -3.50000000000000009e-223

   1. Initial program 31.3%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\color{blue}{{D}^{2} \cdot \left(h \cdot w\right)}}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(2 \cdot \left(c0 \cdot {d}^{2}\right)\right), \color{blue}{\left({D}^{2} \cdot \left(h \cdot w\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot {d}^{2}\right)\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left({d}^{2}\right)\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(d \cdot d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(h \cdot w\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{h} \cdot w\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{h} \cdot w\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(w \cdot \color{blue}{h}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 40.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(w, \color{blue}{h}\right)\right)\right)\right) \]

   5. Simplified 40.3%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(D \cdot D\right)} \cdot \left(w \cdot h\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}}\right)\right) \]

      4. frac-times N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \left(\frac{c0 \cdot d}{w \cdot h} \cdot \color{blue}{\frac{d}{D \cdot D}}\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\left(2 \cdot \frac{c0 \cdot d}{w \cdot h}\right) \cdot \color{blue}{\frac{d}{D \cdot D}}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \frac{c0 \cdot d}{w \cdot h}\right), \color{blue}{\left(\frac{d}{D \cdot D}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{c0 \cdot d}{w \cdot h}\right)\right), \left(\frac{\color{blue}{d}}{D \cdot D}\right)\right)\right) \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{\frac{c0 \cdot d}{w}}{h}\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{c0 \cdot d}{w}\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(c0 \cdot \frac{d}{w}\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{d}{w}\right)\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \mathsf{/.f64}\left(d, \color{blue}{\left(D \cdot D\right)}\right)\right)\right) \]

      14. *-lowering-*.f64 44.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(D, \color{blue}{D}\right)\right)\right)\right) \]

   7. Applied egg-rr 44.8%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot \frac{d}{D \cdot D}\right)} \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{c0 \cdot \frac{d}{w}}{h} \cdot \frac{d}{D \cdot D}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\frac{c0 \cdot d}{w}}{h} \cdot \frac{d}{D \cdot D}\right)\right) \]

      3. associate-/l/ N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0 \cdot d}{h \cdot w} \cdot \frac{\color{blue}{d}}{D \cdot D}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0 \cdot d}{w \cdot h} \cdot \frac{d}{D \cdot D}\right)\right) \]

      5. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}}\right) \]

      6. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(w \cdot h\right)} \cdot \left(D \cdot D\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \color{blue}{\left(w \cdot h\right)}}\right) \]

      8. associate-/l* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}} \]

      9. clear-num N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{1}{\color{blue}{\frac{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}}} \]

      10. un-div-inv N/A

          \[\leadsto \frac{\frac{c0}{2 \cdot w}}{\color{blue}{\frac{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}}} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{c0}{2 \cdot w}\right), \color{blue}{\left(\frac{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}\right)}\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \left(2 \cdot w\right)\right), \left(\frac{\color{blue}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{\left(D \cdot D\right) \cdot \color{blue}{\left(w \cdot h\right)}}{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}\right)\right) \]

   9. Applied egg-rr 52.9%

      \[\leadsto \color{blue}{\frac{\frac{c0}{2 \cdot w}}{\frac{D}{2 \cdot \left(c0 \cdot d\right)} \cdot \frac{D \cdot \left(w \cdot h\right)}{d}}} \]

   11. Applied egg-rr 46.2%

       \[\leadsto \color{blue}{\frac{-\frac{d \cdot \left(c0 \cdot d\right)}{w}}{w \cdot \left(h \cdot \left(0 - D\right)\right)} \cdot \frac{c0}{D}} \]

   if -3.50000000000000009e-223 < w < 6.60000000000000017e-119

   1. Initial program 13.8%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}{\color{blue}{2 \cdot w}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right), \color{blue}{\left(2 \cdot w\right)}\right) \]

   3. Simplified 14.0%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot \frac{\frac{\frac{d \cdot d}{w} \cdot c0}{h} \cdot c0}{D \cdot D}}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(\mathsf{neg}\left(c0\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 18.7%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\left(\frac{-0.5 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot w\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)} + 0\right) \cdot \left(-c0\right)\right)}}{2 \cdot w} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. 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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left({D}^{2}\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(D \cdot D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot {M}^{2}\right)\right)\right), \left({d}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

      12. *-lowering-*.f64 47.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   10. Simplified 47.6%

       \[\leadsto \color{blue}{\frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot d}} \]

   11. Taylor expanded in D around 0

       \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   12. 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. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right) \cdot \frac{1}{4}\right), \left({\color{blue}{d}}^{2}\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right) \cdot \frac{1}{4}\right), \left({d}^{2}\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \left(h \cdot \frac{1}{4}\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(D \cdot D\right) \cdot {M}^{2}\right) \cdot \left(h \cdot \frac{1}{4}\right)\right), \left({d}^{2}\right)\right) \]

       7. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(D \cdot \left(D \cdot {M}^{2}\right)\right) \cdot \left(h \cdot \frac{1}{4}\right)\right), \left({d}^{2}\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(D \cdot \left(\left(D \cdot {M}^{2}\right) \cdot \left(h \cdot \frac{1}{4}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(\left(D \cdot {M}^{2}\right) \cdot \left(h \cdot \frac{1}{4}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

       10. associate-*r* N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(D \cdot \left({M}^{2} \cdot \left(h \cdot \frac{1}{4}\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(D \cdot \left(\left({M}^{2} \cdot h\right) \cdot \frac{1}{4}\right)\right)\right), \left({d}^{2}\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(D \cdot \left(\frac{1}{4} \cdot \left({M}^{2} \cdot h\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(\frac{1}{4} \cdot \left({M}^{2} \cdot h\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \left({M}^{2} \cdot h\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \left(h \cdot {M}^{2}\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       18. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       19. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

       20. *-lowering-*.f64 55.9%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   13. Simplified 55.9%

       \[\leadsto \color{blue}{\frac{D \cdot \left(D \cdot \left(0.25 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot d}} \]

   if 6.60000000000000017e-119 < w

   1. Initial program 26.4%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\color{blue}{{D}^{2} \cdot \left(h \cdot w\right)}}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(2 \cdot \left(c0 \cdot {d}^{2}\right)\right), \color{blue}{\left({D}^{2} \cdot \left(h \cdot w\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot {d}^{2}\right)\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left({d}^{2}\right)\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(d \cdot d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(h \cdot w\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{h} \cdot w\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{h} \cdot w\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(w \cdot \color{blue}{h}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 30.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(w, \color{blue}{h}\right)\right)\right)\right) \]

   5. Simplified 30.7%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(D \cdot D\right)} \cdot \left(w \cdot h\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}}\right)\right) \]

      4. frac-times N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \left(\frac{c0 \cdot d}{w \cdot h} \cdot \color{blue}{\frac{d}{D \cdot D}}\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\left(2 \cdot \frac{c0 \cdot d}{w \cdot h}\right) \cdot \color{blue}{\frac{d}{D \cdot D}}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \frac{c0 \cdot d}{w \cdot h}\right), \color{blue}{\left(\frac{d}{D \cdot D}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{c0 \cdot d}{w \cdot h}\right)\right), \left(\frac{\color{blue}{d}}{D \cdot D}\right)\right)\right) \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{\frac{c0 \cdot d}{w}}{h}\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{c0 \cdot d}{w}\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(c0 \cdot \frac{d}{w}\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{d}{w}\right)\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \mathsf{/.f64}\left(d, \color{blue}{\left(D \cdot D\right)}\right)\right)\right) \]

      14. *-lowering-*.f64 39.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(D, \color{blue}{D}\right)\right)\right)\right) \]

   7. Applied egg-rr 39.8%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot \frac{d}{D \cdot D}\right)} \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{c0 \cdot \frac{d}{w}}{h} \cdot \frac{d}{D \cdot D}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\frac{c0 \cdot d}{w}}{h} \cdot \frac{d}{D \cdot D}\right)\right) \]

      3. associate-/l/ N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0 \cdot d}{h \cdot w} \cdot \frac{\color{blue}{d}}{D \cdot D}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0 \cdot d}{w \cdot h} \cdot \frac{d}{D \cdot D}\right)\right) \]

      5. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}}\right) \]

      6. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(w \cdot h\right)} \cdot \left(D \cdot D\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \color{blue}{\left(w \cdot h\right)}}\right) \]

      8. associate-/l* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}} \]

      9. clear-num N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{1}{\color{blue}{\frac{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}}} \]

      10. un-div-inv N/A

          \[\leadsto \frac{\frac{c0}{2 \cdot w}}{\color{blue}{\frac{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}}} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{c0}{2 \cdot w}\right), \color{blue}{\left(\frac{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}\right)}\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \left(2 \cdot w\right)\right), \left(\frac{\color{blue}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{\left(D \cdot D\right) \cdot \color{blue}{\left(w \cdot h\right)}}{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}\right)\right) \]

   9. Applied egg-rr 47.5%

      \[\leadsto \color{blue}{\frac{\frac{c0}{2 \cdot w}}{\frac{D}{2 \cdot \left(c0 \cdot d\right)} \cdot \frac{D \cdot \left(w \cdot h\right)}{d}}} \]

   11. Applied egg-rr 46.4%

       \[\leadsto \color{blue}{\frac{-\frac{d \cdot \left(c0 \cdot d\right)}{w}}{D} \cdot \frac{c0}{w \cdot \left(h \cdot \left(0 - D\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -3.5 \cdot 10^{-223}:\\ \;\;\;\;\frac{\frac{d \cdot \left(c0 \cdot d\right)}{w}}{w \cdot \left(h \cdot D\right)} \cdot \frac{c0}{D}\\ \mathbf{elif}\;w \leq 6.6 \cdot 10^{-119}:\\ \;\;\;\;\frac{D \cdot \left(D \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot 0.25\right)\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d \cdot \left(c0 \cdot d\right)}{w}}{D} \cdot \frac{c0}{w \cdot \left(h \cdot D\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 39.5% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{d \cdot \left(c0 \cdot d\right)}{w}}{D} \cdot \frac{c0}{w \cdot \left(h \cdot D\right)}\\ \mathbf{if}\;w \leq -3.6 \cdot 10^{-227}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;w \leq 6.3 \cdot 10^{-119}:\\ \;\;\;\;\frac{D \cdot \left(D \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot 0.25\right)\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ (/ (* d (* c0 d)) w) D) (/ c0 (* w (* h D))))))
   (if (<= w -3.6e-227)
     t_0
     (if (<= w 6.3e-119) (/ (* D (* D (* (* h (* M M)) 0.25))) (* d d)) t_0))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (((d * (c0 * d)) / w) / D) * (c0 / (w * (h * D)));
	double tmp;
	if (w <= -3.6e-227) {
		tmp = t_0;
	} else if (w <= 6.3e-119) {
		tmp = (D * (D * ((h * (M * M)) * 0.25))) / (d * d);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((d_1 * (c0 * d_1)) / w) / d) * (c0 / (w * (h * d)))
    if (w <= (-3.6d-227)) then
        tmp = t_0
    else if (w <= 6.3d-119) then
        tmp = (d * (d * ((h * (m * m)) * 0.25d0))) / (d_1 * d_1)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (((d * (c0 * d)) / w) / D) * (c0 / (w * (h * D)));
	double tmp;
	if (w <= -3.6e-227) {
		tmp = t_0;
	} else if (w <= 6.3e-119) {
		tmp = (D * (D * ((h * (M * M)) * 0.25))) / (d * d);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (((d * (c0 * d)) / w) / D) * (c0 / (w * (h * D)))
	tmp = 0
	if w <= -3.6e-227:
		tmp = t_0
	elif w <= 6.3e-119:
		tmp = (D * (D * ((h * (M * M)) * 0.25))) / (d * d)
	else:
		tmp = t_0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(Float64(Float64(d * Float64(c0 * d)) / w) / D) * Float64(c0 / Float64(w * Float64(h * D))))
	tmp = 0.0
	if (w <= -3.6e-227)
		tmp = t_0;
	elseif (w <= 6.3e-119)
		tmp = Float64(Float64(D * Float64(D * Float64(Float64(h * Float64(M * M)) * 0.25))) / Float64(d * d));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (((d * (c0 * d)) / w) / D) * (c0 / (w * (h * D)));
	tmp = 0.0;
	if (w <= -3.6e-227)
		tmp = t_0;
	elseif (w <= 6.3e-119)
		tmp = (D * (D * ((h * (M * M)) * 0.25))) / (d * d);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(N[(N[(d * N[(c0 * d), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision] / D), $MachinePrecision] * N[(c0 / N[(w * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, -3.6e-227], t$95$0, If[LessEqual[w, 6.3e-119], N[(N[(D * N[(D * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if w < -3.5999999999999999e-227 or 6.3e-119 < w

   1. Initial program 29.2%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\color{blue}{{D}^{2} \cdot \left(h \cdot w\right)}}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(2 \cdot \left(c0 \cdot {d}^{2}\right)\right), \color{blue}{\left({D}^{2} \cdot \left(h \cdot w\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot {d}^{2}\right)\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left({d}^{2}\right)\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(d \cdot d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(h \cdot w\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{h} \cdot w\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{h} \cdot w\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(w \cdot \color{blue}{h}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 36.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(w, \color{blue}{h}\right)\right)\right)\right) \]

   5. Simplified 36.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(D \cdot D\right)} \cdot \left(w \cdot h\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}}\right)\right) \]

      4. frac-times N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \left(\frac{c0 \cdot d}{w \cdot h} \cdot \color{blue}{\frac{d}{D \cdot D}}\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\left(2 \cdot \frac{c0 \cdot d}{w \cdot h}\right) \cdot \color{blue}{\frac{d}{D \cdot D}}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \frac{c0 \cdot d}{w \cdot h}\right), \color{blue}{\left(\frac{d}{D \cdot D}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{c0 \cdot d}{w \cdot h}\right)\right), \left(\frac{\color{blue}{d}}{D \cdot D}\right)\right)\right) \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{\frac{c0 \cdot d}{w}}{h}\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{c0 \cdot d}{w}\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(c0 \cdot \frac{d}{w}\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{d}{w}\right)\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \mathsf{/.f64}\left(d, \color{blue}{\left(D \cdot D\right)}\right)\right)\right) \]

      14. *-lowering-*.f64 42.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(D, \color{blue}{D}\right)\right)\right)\right) \]

   7. Applied egg-rr 42.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot \frac{d}{D \cdot D}\right)} \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{c0 \cdot \frac{d}{w}}{h} \cdot \frac{d}{D \cdot D}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\frac{c0 \cdot d}{w}}{h} \cdot \frac{d}{D \cdot D}\right)\right) \]

      3. associate-/l/ N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0 \cdot d}{h \cdot w} \cdot \frac{\color{blue}{d}}{D \cdot D}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0 \cdot d}{w \cdot h} \cdot \frac{d}{D \cdot D}\right)\right) \]

      5. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}}\right) \]

      6. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(w \cdot h\right)} \cdot \left(D \cdot D\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \color{blue}{\left(w \cdot h\right)}}\right) \]

      8. associate-/l* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}} \]

      9. clear-num N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{1}{\color{blue}{\frac{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}}} \]

      10. un-div-inv N/A

          \[\leadsto \frac{\frac{c0}{2 \cdot w}}{\color{blue}{\frac{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}}} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{c0}{2 \cdot w}\right), \color{blue}{\left(\frac{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}\right)}\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \left(2 \cdot w\right)\right), \left(\frac{\color{blue}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{\left(D \cdot D\right) \cdot \color{blue}{\left(w \cdot h\right)}}{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}\right)\right) \]

   9. Applied egg-rr 50.6%

      \[\leadsto \color{blue}{\frac{\frac{c0}{2 \cdot w}}{\frac{D}{2 \cdot \left(c0 \cdot d\right)} \cdot \frac{D \cdot \left(w \cdot h\right)}{d}}} \]

   11. Applied egg-rr 46.2%

       \[\leadsto \color{blue}{\frac{-\frac{d \cdot \left(c0 \cdot d\right)}{w}}{D} \cdot \frac{c0}{w \cdot \left(h \cdot \left(0 - D\right)\right)}} \]

   if -3.5999999999999999e-227 < w < 6.3e-119

   1. Initial program 13.8%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}{\color{blue}{2 \cdot w}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right), \color{blue}{\left(2 \cdot w\right)}\right) \]

   3. Simplified 14.0%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot \frac{\frac{\frac{d \cdot d}{w} \cdot c0}{h} \cdot c0}{D \cdot D}}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(\mathsf{neg}\left(c0\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 18.7%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\left(\frac{-0.5 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot w\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)} + 0\right) \cdot \left(-c0\right)\right)}}{2 \cdot w} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. 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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left({D}^{2}\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(D \cdot D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot {M}^{2}\right)\right)\right), \left({d}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

      12. *-lowering-*.f64 47.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   10. Simplified 47.6%

       \[\leadsto \color{blue}{\frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot d}} \]

   11. Taylor expanded in D around 0

       \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   12. 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. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right) \cdot \frac{1}{4}\right), \left({\color{blue}{d}}^{2}\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right) \cdot \frac{1}{4}\right), \left({d}^{2}\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \left(h \cdot \frac{1}{4}\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(D \cdot D\right) \cdot {M}^{2}\right) \cdot \left(h \cdot \frac{1}{4}\right)\right), \left({d}^{2}\right)\right) \]

       7. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(D \cdot \left(D \cdot {M}^{2}\right)\right) \cdot \left(h \cdot \frac{1}{4}\right)\right), \left({d}^{2}\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(D \cdot \left(\left(D \cdot {M}^{2}\right) \cdot \left(h \cdot \frac{1}{4}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(\left(D \cdot {M}^{2}\right) \cdot \left(h \cdot \frac{1}{4}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

       10. associate-*r* N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(D \cdot \left({M}^{2} \cdot \left(h \cdot \frac{1}{4}\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(D \cdot \left(\left({M}^{2} \cdot h\right) \cdot \frac{1}{4}\right)\right)\right), \left({d}^{2}\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(D \cdot \left(\frac{1}{4} \cdot \left({M}^{2} \cdot h\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(\frac{1}{4} \cdot \left({M}^{2} \cdot h\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \left({M}^{2} \cdot h\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \left(h \cdot {M}^{2}\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       18. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       19. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

       20. *-lowering-*.f64 55.9%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   13. Simplified 55.9%

       \[\leadsto \color{blue}{\frac{D \cdot \left(D \cdot \left(0.25 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot d}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -3.6 \cdot 10^{-227}:\\ \;\;\;\;\frac{\frac{d \cdot \left(c0 \cdot d\right)}{w}}{D} \cdot \frac{c0}{w \cdot \left(h \cdot D\right)}\\ \mathbf{elif}\;w \leq 6.3 \cdot 10^{-119}:\\ \;\;\;\;\frac{D \cdot \left(D \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot 0.25\right)\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d \cdot \left(c0 \cdot d\right)}{w}}{D} \cdot \frac{c0}{w \cdot \left(h \cdot D\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 43.5% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := h \cdot \left(M \cdot M\right)\\ \mathbf{if}\;d \leq 1.35 \cdot 10^{-163}:\\ \;\;\;\;\frac{t\_0 \cdot \frac{D}{\frac{d}{D}}}{\frac{d}{0.25}}\\ \mathbf{elif}\;d \leq 9.5 \cdot 10^{+157}:\\ \;\;\;\;\frac{\left(D \cdot 0.25\right) \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot D\right)\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(D \cdot D\right) \cdot t\_0}{\frac{d}{0.25}}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* h (* M M))))
   (if (<= d 1.35e-163)
     (/ (* t_0 (/ D (/ d D))) (/ d 0.25))
     (if (<= d 9.5e+157)
       (/ (* (* D 0.25) (* (* M M) (* h D))) (* d d))
       (/ (/ (* (* D D) t_0) (/ d 0.25)) d)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = h * (M * M);
	double tmp;
	if (d <= 1.35e-163) {
		tmp = (t_0 * (D / (d / D))) / (d / 0.25);
	} else if (d <= 9.5e+157) {
		tmp = ((D * 0.25) * ((M * M) * (h * D))) / (d * d);
	} else {
		tmp = (((D * D) * t_0) / (d / 0.25)) / d;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = h * (m * m)
    if (d_1 <= 1.35d-163) then
        tmp = (t_0 * (d / (d_1 / d))) / (d_1 / 0.25d0)
    else if (d_1 <= 9.5d+157) then
        tmp = ((d * 0.25d0) * ((m * m) * (h * d))) / (d_1 * d_1)
    else
        tmp = (((d * d) * t_0) / (d_1 / 0.25d0)) / d_1
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = h * (M * M);
	double tmp;
	if (d <= 1.35e-163) {
		tmp = (t_0 * (D / (d / D))) / (d / 0.25);
	} else if (d <= 9.5e+157) {
		tmp = ((D * 0.25) * ((M * M) * (h * D))) / (d * d);
	} else {
		tmp = (((D * D) * t_0) / (d / 0.25)) / d;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = h * (M * M)
	tmp = 0
	if d <= 1.35e-163:
		tmp = (t_0 * (D / (d / D))) / (d / 0.25)
	elif d <= 9.5e+157:
		tmp = ((D * 0.25) * ((M * M) * (h * D))) / (d * d)
	else:
		tmp = (((D * D) * t_0) / (d / 0.25)) / d
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(h * Float64(M * M))
	tmp = 0.0
	if (d <= 1.35e-163)
		tmp = Float64(Float64(t_0 * Float64(D / Float64(d / D))) / Float64(d / 0.25));
	elseif (d <= 9.5e+157)
		tmp = Float64(Float64(Float64(D * 0.25) * Float64(Float64(M * M) * Float64(h * D))) / Float64(d * d));
	else
		tmp = Float64(Float64(Float64(Float64(D * D) * t_0) / Float64(d / 0.25)) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = h * (M * M);
	tmp = 0.0;
	if (d <= 1.35e-163)
		tmp = (t_0 * (D / (d / D))) / (d / 0.25);
	elseif (d <= 9.5e+157)
		tmp = ((D * 0.25) * ((M * M) * (h * D))) / (d * d);
	else
		tmp = (((D * D) * t_0) / (d / 0.25)) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, 1.35e-163], N[(N[(t$95$0 * N[(D / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d / 0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 9.5e+157], N[(N[(N[(D * 0.25), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(D * D), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(d / 0.25), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < 1.35000000000000007e-163

   1. Initial program 24.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}{\color{blue}{2 \cdot w}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right), \color{blue}{\left(2 \cdot w\right)}\right) \]

   3. Simplified 22.2%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot \frac{\frac{\frac{d \cdot d}{w} \cdot c0}{h} \cdot c0}{D \cdot D}}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(\mathsf{neg}\left(c0\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 13.4%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\left(\frac{-0.5 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot w\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)} + 0\right) \cdot \left(-c0\right)\right)}}{2 \cdot w} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. 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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left({D}^{2}\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(D \cdot D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot {M}^{2}\right)\right)\right), \left({d}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

      12. *-lowering-*.f64 30.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   10. Simplified 30.1%

       \[\leadsto \color{blue}{\frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot d}} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{1}{4}}{\color{blue}{d} \cdot d} \]

       2. times-frac N/A

          \[\leadsto \frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d} \cdot \color{blue}{\frac{\frac{1}{4}}{d}} \]

       3. clear-num N/A

          \[\leadsto \frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d} \cdot \frac{1}{\color{blue}{\frac{d}{\frac{1}{4}}}} \]

       4. un-div-inv N/A

          \[\leadsto \frac{\frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d}}{\color{blue}{\frac{d}{\frac{1}{4}}}} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d}\right), \color{blue}{\left(\frac{d}{\frac{1}{4}}\right)}\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(h \cdot \left(M \cdot M\right)\right) \cdot \left(D \cdot D\right)}{d}\right), \left(\frac{d}{\frac{1}{4}}\right)\right) \]

       7. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \frac{D \cdot D}{d}\right), \left(\frac{\color{blue}{d}}{\frac{1}{4}}\right)\right) \]

       8. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \left(D \cdot \frac{D}{d}\right)\right), \left(\frac{d}{\frac{1}{4}}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(h \cdot \left(M \cdot M\right)\right), \left(D \cdot \frac{D}{d}\right)\right), \left(\frac{\color{blue}{d}}{\frac{1}{4}}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \left(M \cdot M\right)\right), \left(D \cdot \frac{D}{d}\right)\right), \left(\frac{d}{\frac{1}{4}}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), \left(D \cdot \frac{D}{d}\right)\right), \left(\frac{d}{\frac{1}{4}}\right)\right) \]

       12. clear-num N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), \left(D \cdot \frac{1}{\frac{d}{D}}\right)\right), \left(\frac{d}{\frac{1}{4}}\right)\right) \]

       13. un-div-inv N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), \left(\frac{D}{\frac{d}{D}}\right)\right), \left(\frac{d}{\frac{1}{4}}\right)\right) \]

       14. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), \mathsf{/.f64}\left(D, \left(\frac{d}{D}\right)\right)\right), \left(\frac{d}{\frac{1}{4}}\right)\right) \]

       15. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), \mathsf{/.f64}\left(D, \mathsf{/.f64}\left(d, D\right)\right)\right), \left(\frac{d}{\frac{1}{4}}\right)\right) \]

       16. /-lowering-/.f64 38.6%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), \mathsf{/.f64}\left(D, \mathsf{/.f64}\left(d, D\right)\right)\right), \mathsf{/.f64}\left(d, \color{blue}{\frac{1}{4}}\right)\right) \]

   12. Applied egg-rr 38.6%

       \[\leadsto \color{blue}{\frac{\left(h \cdot \left(M \cdot M\right)\right) \cdot \frac{D}{\frac{d}{D}}}{\frac{d}{0.25}}} \]

   if 1.35000000000000007e-163 < d < 9.4999999999999996e157

   1. Initial program 23.2%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}{\color{blue}{2 \cdot w}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right), \color{blue}{\left(2 \cdot w\right)}\right) \]

   3. Simplified 21.8%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot \frac{\frac{\frac{d \cdot d}{w} \cdot c0}{h} \cdot c0}{D \cdot D}}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(\mathsf{neg}\left(c0\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 15.3%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\left(\frac{-0.5 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot w\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)} + 0\right) \cdot \left(-c0\right)\right)}}{2 \cdot w} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. 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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left({D}^{2}\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(D \cdot D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot {M}^{2}\right)\right)\right), \left({d}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

      12. *-lowering-*.f64 36.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   10. Simplified 36.6%

       \[\leadsto \color{blue}{\frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot d}} \]
   11. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{4} \cdot D\right) \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{d}, d\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{4} \cdot D\right), \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{d}, d\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, D\right), \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, D\right), \left(\left(D \cdot h\right) \cdot \left(M \cdot M\right)\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, D\right), \mathsf{*.f64}\left(\left(D \cdot h\right), \left(M \cdot M\right)\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, D\right), \mathsf{*.f64}\left(\left(h \cdot D\right), \left(M \cdot M\right)\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, D\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, D\right), \left(M \cdot M\right)\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

       9. *-lowering-*.f64 46.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, D\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, D\right), \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

   12. Applied egg-rr 46.3%

       \[\leadsto \frac{\color{blue}{\left(0.25 \cdot D\right) \cdot \left(\left(h \cdot D\right) \cdot \left(M \cdot M\right)\right)}}{d \cdot d} \]

   if 9.4999999999999996e157 < d

   1. Initial program 27.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}{\color{blue}{2 \cdot w}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right), \color{blue}{\left(2 \cdot w\right)}\right) \]

   3. Simplified 28.1%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot \frac{\frac{\frac{d \cdot d}{w} \cdot c0}{h} \cdot c0}{D \cdot D}}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(\mathsf{neg}\left(c0\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 19.6%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\left(\frac{-0.5 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot w\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)} + 0\right) \cdot \left(-c0\right)\right)}}{2 \cdot w} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. 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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left({D}^{2}\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(D \cdot D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot {M}^{2}\right)\right)\right), \left({d}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

      12. *-lowering-*.f64 28.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   10. Simplified 28.2%

       \[\leadsto \color{blue}{\frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot d}} \]
   11. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d}}{\color{blue}{d}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{4} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d}\right), \color{blue}{d}\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{1}{4}}{d}\right), d\right) \]

       4. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right), d\right) \]

       5. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{1}{\frac{d}{\frac{1}{4}}}\right), d\right) \]

       6. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d}{\frac{1}{4}}}\right), d\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right), \left(\frac{d}{\frac{1}{4}}\right)\right), d\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(D \cdot D\right), \left(h \cdot \left(M \cdot M\right)\right)\right), \left(\frac{d}{\frac{1}{4}}\right)\right), d\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot \left(M \cdot M\right)\right)\right), \left(\frac{d}{\frac{1}{4}}\right)\right), d\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right), \left(\frac{d}{\frac{1}{4}}\right)\right), d\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \left(\frac{d}{\frac{1}{4}}\right)\right), d\right) \]

       12. /-lowering-/.f64 39.7%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{/.f64}\left(d, \frac{1}{4}\right)\right), d\right) \]

   12. Applied egg-rr 39.7%

       \[\leadsto \color{blue}{\frac{\frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d}{0.25}}}{d}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.35 \cdot 10^{-163}:\\ \;\;\;\;\frac{\left(h \cdot \left(M \cdot M\right)\right) \cdot \frac{D}{\frac{d}{D}}}{\frac{d}{0.25}}\\ \mathbf{elif}\;d \leq 9.5 \cdot 10^{+157}:\\ \;\;\;\;\frac{\left(D \cdot 0.25\right) \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot D\right)\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d}{0.25}}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 43.3% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(h \cdot \left(M \cdot M\right)\right) \cdot \frac{D}{\frac{d}{D}}}{\frac{d}{0.25}}\\ \mathbf{if}\;d \leq 1.35 \cdot 10^{-163}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 9.4 \cdot 10^{+157}:\\ \;\;\;\;\frac{\left(D \cdot 0.25\right) \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot D\right)\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* (* h (* M M)) (/ D (/ d D))) (/ d 0.25))))
   (if (<= d 1.35e-163)
     t_0
     (if (<= d 9.4e+157) (/ (* (* D 0.25) (* (* M M) (* h D))) (* d d)) t_0))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((h * (M * M)) * (D / (d / D))) / (d / 0.25);
	double tmp;
	if (d <= 1.35e-163) {
		tmp = t_0;
	} else if (d <= 9.4e+157) {
		tmp = ((D * 0.25) * ((M * M) * (h * D))) / (d * d);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((h * (m * m)) * (d / (d_1 / d))) / (d_1 / 0.25d0)
    if (d_1 <= 1.35d-163) then
        tmp = t_0
    else if (d_1 <= 9.4d+157) then
        tmp = ((d * 0.25d0) * ((m * m) * (h * d))) / (d_1 * d_1)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((h * (M * M)) * (D / (d / D))) / (d / 0.25);
	double tmp;
	if (d <= 1.35e-163) {
		tmp = t_0;
	} else if (d <= 9.4e+157) {
		tmp = ((D * 0.25) * ((M * M) * (h * D))) / (d * d);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = ((h * (M * M)) * (D / (d / D))) / (d / 0.25)
	tmp = 0
	if d <= 1.35e-163:
		tmp = t_0
	elif d <= 9.4e+157:
		tmp = ((D * 0.25) * ((M * M) * (h * D))) / (d * d)
	else:
		tmp = t_0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(Float64(h * Float64(M * M)) * Float64(D / Float64(d / D))) / Float64(d / 0.25))
	tmp = 0.0
	if (d <= 1.35e-163)
		tmp = t_0;
	elseif (d <= 9.4e+157)
		tmp = Float64(Float64(Float64(D * 0.25) * Float64(Float64(M * M) * Float64(h * D))) / Float64(d * d));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = ((h * (M * M)) * (D / (d / D))) / (d / 0.25);
	tmp = 0.0;
	if (d <= 1.35e-163)
		tmp = t_0;
	elseif (d <= 9.4e+157)
		tmp = ((D * 0.25) * ((M * M) * (h * D))) / (d * d);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] * N[(D / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d / 0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, 1.35e-163], t$95$0, If[LessEqual[d, 9.4e+157], N[(N[(N[(D * 0.25), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < 1.35000000000000007e-163 or 9.40000000000000061e157 < d

   1. Initial program 25.4%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}{\color{blue}{2 \cdot w}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right), \color{blue}{\left(2 \cdot w\right)}\right) \]

   3. Simplified 23.7%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot \frac{\frac{\frac{d \cdot d}{w} \cdot c0}{h} \cdot c0}{D \cdot D}}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(\mathsf{neg}\left(c0\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 15.0%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\left(\frac{-0.5 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot w\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)} + 0\right) \cdot \left(-c0\right)\right)}}{2 \cdot w} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. 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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left({D}^{2}\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(D \cdot D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot {M}^{2}\right)\right)\right), \left({d}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

      12. *-lowering-*.f64 29.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   10. Simplified 29.6%

       \[\leadsto \color{blue}{\frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot d}} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{1}{4}}{\color{blue}{d} \cdot d} \]

       2. times-frac N/A

          \[\leadsto \frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d} \cdot \color{blue}{\frac{\frac{1}{4}}{d}} \]

       3. clear-num N/A

          \[\leadsto \frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d} \cdot \frac{1}{\color{blue}{\frac{d}{\frac{1}{4}}}} \]

       4. un-div-inv N/A

          \[\leadsto \frac{\frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d}}{\color{blue}{\frac{d}{\frac{1}{4}}}} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d}\right), \color{blue}{\left(\frac{d}{\frac{1}{4}}\right)}\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(h \cdot \left(M \cdot M\right)\right) \cdot \left(D \cdot D\right)}{d}\right), \left(\frac{d}{\frac{1}{4}}\right)\right) \]

       7. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \frac{D \cdot D}{d}\right), \left(\frac{\color{blue}{d}}{\frac{1}{4}}\right)\right) \]

       8. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \left(D \cdot \frac{D}{d}\right)\right), \left(\frac{d}{\frac{1}{4}}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(h \cdot \left(M \cdot M\right)\right), \left(D \cdot \frac{D}{d}\right)\right), \left(\frac{\color{blue}{d}}{\frac{1}{4}}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \left(M \cdot M\right)\right), \left(D \cdot \frac{D}{d}\right)\right), \left(\frac{d}{\frac{1}{4}}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), \left(D \cdot \frac{D}{d}\right)\right), \left(\frac{d}{\frac{1}{4}}\right)\right) \]

       12. clear-num N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), \left(D \cdot \frac{1}{\frac{d}{D}}\right)\right), \left(\frac{d}{\frac{1}{4}}\right)\right) \]

       13. un-div-inv N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), \left(\frac{D}{\frac{d}{D}}\right)\right), \left(\frac{d}{\frac{1}{4}}\right)\right) \]

       14. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), \mathsf{/.f64}\left(D, \left(\frac{d}{D}\right)\right)\right), \left(\frac{d}{\frac{1}{4}}\right)\right) \]

       15. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), \mathsf{/.f64}\left(D, \mathsf{/.f64}\left(d, D\right)\right)\right), \left(\frac{d}{\frac{1}{4}}\right)\right) \]

       16. /-lowering-/.f64 38.4%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), \mathsf{/.f64}\left(D, \mathsf{/.f64}\left(d, D\right)\right)\right), \mathsf{/.f64}\left(d, \color{blue}{\frac{1}{4}}\right)\right) \]

   12. Applied egg-rr 38.4%

       \[\leadsto \color{blue}{\frac{\left(h \cdot \left(M \cdot M\right)\right) \cdot \frac{D}{\frac{d}{D}}}{\frac{d}{0.25}}} \]

   if 1.35000000000000007e-163 < d < 9.40000000000000061e157

   1. Initial program 23.2%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}{\color{blue}{2 \cdot w}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right), \color{blue}{\left(2 \cdot w\right)}\right) \]

   3. Simplified 21.8%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot \frac{\frac{\frac{d \cdot d}{w} \cdot c0}{h} \cdot c0}{D \cdot D}}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(\mathsf{neg}\left(c0\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 15.3%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\left(\frac{-0.5 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot w\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)} + 0\right) \cdot \left(-c0\right)\right)}}{2 \cdot w} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. 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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left({D}^{2}\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(D \cdot D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot {M}^{2}\right)\right)\right), \left({d}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

      12. *-lowering-*.f64 36.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   10. Simplified 36.6%

       \[\leadsto \color{blue}{\frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot d}} \]
   11. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{4} \cdot D\right) \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{d}, d\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{4} \cdot D\right), \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{d}, d\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, D\right), \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, D\right), \left(\left(D \cdot h\right) \cdot \left(M \cdot M\right)\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, D\right), \mathsf{*.f64}\left(\left(D \cdot h\right), \left(M \cdot M\right)\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, D\right), \mathsf{*.f64}\left(\left(h \cdot D\right), \left(M \cdot M\right)\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, D\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, D\right), \left(M \cdot M\right)\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

       9. *-lowering-*.f64 46.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, D\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, D\right), \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

   12. Applied egg-rr 46.3%

       \[\leadsto \frac{\color{blue}{\left(0.25 \cdot D\right) \cdot \left(\left(h \cdot D\right) \cdot \left(M \cdot M\right)\right)}}{d \cdot d} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.35 \cdot 10^{-163}:\\ \;\;\;\;\frac{\left(h \cdot \left(M \cdot M\right)\right) \cdot \frac{D}{\frac{d}{D}}}{\frac{d}{0.25}}\\ \mathbf{elif}\;d \leq 9.4 \cdot 10^{+157}:\\ \;\;\;\;\frac{\left(D \cdot 0.25\right) \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot D\right)\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(h \cdot \left(M \cdot M\right)\right) \cdot \frac{D}{\frac{d}{D}}}{\frac{d}{0.25}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 40.0% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(D \cdot D\right) \cdot 0.25}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\\ \mathbf{if}\;d \leq 6 \cdot 10^{-167}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.55 \cdot 10^{+167}:\\ \;\;\;\;\frac{\left(D \cdot 0.25\right) \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot D\right)\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ (* (* D D) 0.25) d) (/ (* h (* M M)) d))))
   (if (<= d 6e-167)
     t_0
     (if (<= d 1.55e+167)
       (/ (* (* D 0.25) (* (* M M) (* h D))) (* d d))
       t_0))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (((D * D) * 0.25) / d) * ((h * (M * M)) / d);
	double tmp;
	if (d <= 6e-167) {
		tmp = t_0;
	} else if (d <= 1.55e+167) {
		tmp = ((D * 0.25) * ((M * M) * (h * D))) / (d * d);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((d * d) * 0.25d0) / d_1) * ((h * (m * m)) / d_1)
    if (d_1 <= 6d-167) then
        tmp = t_0
    else if (d_1 <= 1.55d+167) then
        tmp = ((d * 0.25d0) * ((m * m) * (h * d))) / (d_1 * d_1)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (((D * D) * 0.25) / d) * ((h * (M * M)) / d);
	double tmp;
	if (d <= 6e-167) {
		tmp = t_0;
	} else if (d <= 1.55e+167) {
		tmp = ((D * 0.25) * ((M * M) * (h * D))) / (d * d);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (((D * D) * 0.25) / d) * ((h * (M * M)) / d)
	tmp = 0
	if d <= 6e-167:
		tmp = t_0
	elif d <= 1.55e+167:
		tmp = ((D * 0.25) * ((M * M) * (h * D))) / (d * d)
	else:
		tmp = t_0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(Float64(Float64(D * D) * 0.25) / d) * Float64(Float64(h * Float64(M * M)) / d))
	tmp = 0.0
	if (d <= 6e-167)
		tmp = t_0;
	elseif (d <= 1.55e+167)
		tmp = Float64(Float64(Float64(D * 0.25) * Float64(Float64(M * M) * Float64(h * D))) / Float64(d * d));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (((D * D) * 0.25) / d) * ((h * (M * M)) / d);
	tmp = 0.0;
	if (d <= 6e-167)
		tmp = t_0;
	elseif (d <= 1.55e+167)
		tmp = ((D * 0.25) * ((M * M) * (h * D))) / (d * d);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(N[(N[(D * D), $MachinePrecision] * 0.25), $MachinePrecision] / d), $MachinePrecision] * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, 6e-167], t$95$0, If[LessEqual[d, 1.55e+167], N[(N[(N[(D * 0.25), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < 5.9999999999999996e-167 or 1.55e167 < d

   1. Initial program 25.7%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}{\color{blue}{2 \cdot w}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right), \color{blue}{\left(2 \cdot w\right)}\right) \]

   3. Simplified 24.0%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot \frac{\frac{\frac{d \cdot d}{w} \cdot c0}{h} \cdot c0}{D \cdot D}}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(\mathsf{neg}\left(c0\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 15.5%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\left(\frac{-0.5 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot w\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)} + 0\right) \cdot \left(-c0\right)\right)}}{2 \cdot w} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. 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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left({D}^{2}\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(D \cdot D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot {M}^{2}\right)\right)\right), \left({d}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

      12. *-lowering-*.f64 30.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   10. Simplified 30.0%

       \[\leadsto \color{blue}{\frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot d}} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\color{blue}{d} \cdot d} \]

       2. times-frac N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(D \cdot D\right)}{d} \cdot \color{blue}{\frac{h \cdot \left(M \cdot M\right)}{d}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{4} \cdot \left(D \cdot D\right)}{d}\right), \color{blue}{\left(\frac{h \cdot \left(M \cdot M\right)}{d}\right)}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right), d\right), \left(\frac{\color{blue}{h \cdot \left(M \cdot M\right)}}{d}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot D\right)\right), d\right), \left(\frac{\color{blue}{h} \cdot \left(M \cdot M\right)}{d}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, D\right)\right), d\right), \left(\frac{h \cdot \left(M \cdot M\right)}{d}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, D\right)\right), d\right), \mathsf{/.f64}\left(\left(h \cdot \left(M \cdot M\right)\right), \color{blue}{d}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, D\right)\right), d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(M \cdot M\right)\right), d\right)\right) \]

       9. *-lowering-*.f64 36.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, D\right)\right), d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), d\right)\right) \]

   12. Applied egg-rr 36.2%

       \[\leadsto \color{blue}{\frac{0.25 \cdot \left(D \cdot D\right)}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}} \]

   if 5.9999999999999996e-167 < d < 1.55e167

   1. Initial program 22.7%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}{\color{blue}{2 \cdot w}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right), \color{blue}{\left(2 \cdot w\right)}\right) \]

   3. Simplified 21.4%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot \frac{\frac{\frac{d \cdot d}{w} \cdot c0}{h} \cdot c0}{D \cdot D}}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(\mathsf{neg}\left(c0\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 14.2%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\left(\frac{-0.5 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot w\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)} + 0\right) \cdot \left(-c0\right)\right)}}{2 \cdot w} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. 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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left({D}^{2}\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(D \cdot D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot {M}^{2}\right)\right)\right), \left({d}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

      12. *-lowering-*.f64 35.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   10. Simplified 35.0%

       \[\leadsto \color{blue}{\frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot d}} \]
   11. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left(D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{4} \cdot D\right) \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{d}, d\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{4} \cdot D\right), \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{d}, d\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, D\right), \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, D\right), \left(\left(D \cdot h\right) \cdot \left(M \cdot M\right)\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, D\right), \mathsf{*.f64}\left(\left(D \cdot h\right), \left(M \cdot M\right)\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, D\right), \mathsf{*.f64}\left(\left(h \cdot D\right), \left(M \cdot M\right)\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, D\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, D\right), \left(M \cdot M\right)\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

       9. *-lowering-*.f64 45.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, D\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, D\right), \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

   12. Applied egg-rr 45.3%

       \[\leadsto \frac{\color{blue}{\left(0.25 \cdot D\right) \cdot \left(\left(h \cdot D\right) \cdot \left(M \cdot M\right)\right)}}{d \cdot d} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 6 \cdot 10^{-167}:\\ \;\;\;\;\frac{\left(D \cdot D\right) \cdot 0.25}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\\ \mathbf{elif}\;d \leq 1.55 \cdot 10^{+167}:\\ \;\;\;\;\frac{\left(D \cdot 0.25\right) \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot D\right)\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(D \cdot D\right) \cdot 0.25}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 40.3% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := h \cdot \left(M \cdot M\right)\\ t_1 := \frac{\left(D \cdot D\right) \cdot 0.25}{d} \cdot \frac{t\_0}{d}\\ \mathbf{if}\;d \leq 1.05 \cdot 10^{-166}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq 6 \cdot 10^{+187}:\\ \;\;\;\;\frac{D \cdot \left(D \cdot \left(t\_0 \cdot 0.25\right)\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* h (* M M))) (t_1 (* (/ (* (* D D) 0.25) d) (/ t_0 d))))
   (if (<= d 1.05e-166)
     t_1
     (if (<= d 6e+187) (/ (* D (* D (* t_0 0.25))) (* d d)) t_1))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = h * (M * M);
	double t_1 = (((D * D) * 0.25) / d) * (t_0 / d);
	double tmp;
	if (d <= 1.05e-166) {
		tmp = t_1;
	} else if (d <= 6e+187) {
		tmp = (D * (D * (t_0 * 0.25))) / (d * d);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = h * (m * m)
    t_1 = (((d * d) * 0.25d0) / d_1) * (t_0 / d_1)
    if (d_1 <= 1.05d-166) then
        tmp = t_1
    else if (d_1 <= 6d+187) then
        tmp = (d * (d * (t_0 * 0.25d0))) / (d_1 * d_1)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = h * (M * M);
	double t_1 = (((D * D) * 0.25) / d) * (t_0 / d);
	double tmp;
	if (d <= 1.05e-166) {
		tmp = t_1;
	} else if (d <= 6e+187) {
		tmp = (D * (D * (t_0 * 0.25))) / (d * d);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = h * (M * M)
	t_1 = (((D * D) * 0.25) / d) * (t_0 / d)
	tmp = 0
	if d <= 1.05e-166:
		tmp = t_1
	elif d <= 6e+187:
		tmp = (D * (D * (t_0 * 0.25))) / (d * d)
	else:
		tmp = t_1
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(h * Float64(M * M))
	t_1 = Float64(Float64(Float64(Float64(D * D) * 0.25) / d) * Float64(t_0 / d))
	tmp = 0.0
	if (d <= 1.05e-166)
		tmp = t_1;
	elseif (d <= 6e+187)
		tmp = Float64(Float64(D * Float64(D * Float64(t_0 * 0.25))) / Float64(d * d));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = h * (M * M);
	t_1 = (((D * D) * 0.25) / d) * (t_0 / d);
	tmp = 0.0;
	if (d <= 1.05e-166)
		tmp = t_1;
	elseif (d <= 6e+187)
		tmp = (D * (D * (t_0 * 0.25))) / (d * d);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(D * D), $MachinePrecision] * 0.25), $MachinePrecision] / d), $MachinePrecision] * N[(t$95$0 / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, 1.05e-166], t$95$1, If[LessEqual[d, 6e+187], N[(N[(D * N[(D * N[(t$95$0 * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if d < 1.05e-166 or 5.9999999999999998e187 < d

   1. Initial program 25.2%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}{\color{blue}{2 \cdot w}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right), \color{blue}{\left(2 \cdot w\right)}\right) \]

   3. Simplified 23.4%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot \frac{\frac{\frac{d \cdot d}{w} \cdot c0}{h} \cdot c0}{D \cdot D}}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(\mathsf{neg}\left(c0\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 15.8%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\left(\frac{-0.5 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot w\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)} + 0\right) \cdot \left(-c0\right)\right)}}{2 \cdot w} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. 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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left({D}^{2}\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(D \cdot D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot {M}^{2}\right)\right)\right), \left({d}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

      12. *-lowering-*.f64 30.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   10. Simplified 30.1%

       \[\leadsto \color{blue}{\frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot d}} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\color{blue}{d} \cdot d} \]

       2. times-frac N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(D \cdot D\right)}{d} \cdot \color{blue}{\frac{h \cdot \left(M \cdot M\right)}{d}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{4} \cdot \left(D \cdot D\right)}{d}\right), \color{blue}{\left(\frac{h \cdot \left(M \cdot M\right)}{d}\right)}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right), d\right), \left(\frac{\color{blue}{h \cdot \left(M \cdot M\right)}}{d}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot D\right)\right), d\right), \left(\frac{\color{blue}{h} \cdot \left(M \cdot M\right)}{d}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, D\right)\right), d\right), \left(\frac{h \cdot \left(M \cdot M\right)}{d}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, D\right)\right), d\right), \mathsf{/.f64}\left(\left(h \cdot \left(M \cdot M\right)\right), \color{blue}{d}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, D\right)\right), d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(M \cdot M\right)\right), d\right)\right) \]

       9. *-lowering-*.f64 36.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, D\right)\right), d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), d\right)\right) \]

   12. Applied egg-rr 36.5%

       \[\leadsto \color{blue}{\frac{0.25 \cdot \left(D \cdot D\right)}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}} \]

   if 1.05e-166 < d < 5.9999999999999998e187

   1. Initial program 24.0%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}{\color{blue}{2 \cdot w}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right), \color{blue}{\left(2 \cdot w\right)}\right) \]

   3. Simplified 22.8%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot \frac{\frac{\frac{d \cdot d}{w} \cdot c0}{h} \cdot c0}{D \cdot D}}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(\mathsf{neg}\left(c0\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 13.5%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\left(\frac{-0.5 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot w\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)} + 0\right) \cdot \left(-c0\right)\right)}}{2 \cdot w} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. 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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left({D}^{2}\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(D \cdot D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot {M}^{2}\right)\right)\right), \left({d}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

      12. *-lowering-*.f64 34.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   10. Simplified 34.6%

       \[\leadsto \color{blue}{\frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot d}} \]

   11. Taylor expanded in D around 0

       \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   12. 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. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right) \cdot \frac{1}{4}\right), \left({\color{blue}{d}}^{2}\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right) \cdot \frac{1}{4}\right), \left({d}^{2}\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \left(h \cdot \frac{1}{4}\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(D \cdot D\right) \cdot {M}^{2}\right) \cdot \left(h \cdot \frac{1}{4}\right)\right), \left({d}^{2}\right)\right) \]

       7. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(D \cdot \left(D \cdot {M}^{2}\right)\right) \cdot \left(h \cdot \frac{1}{4}\right)\right), \left({d}^{2}\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(D \cdot \left(\left(D \cdot {M}^{2}\right) \cdot \left(h \cdot \frac{1}{4}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(\left(D \cdot {M}^{2}\right) \cdot \left(h \cdot \frac{1}{4}\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

       10. associate-*r* N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(D \cdot \left({M}^{2} \cdot \left(h \cdot \frac{1}{4}\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(D \cdot \left(\left({M}^{2} \cdot h\right) \cdot \frac{1}{4}\right)\right)\right), \left({d}^{2}\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(D \cdot \left(\frac{1}{4} \cdot \left({M}^{2} \cdot h\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(\frac{1}{4} \cdot \left({M}^{2} \cdot h\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \left({M}^{2} \cdot h\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \left(h \cdot {M}^{2}\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       18. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

       19. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

       20. *-lowering-*.f64 42.2%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   13. Simplified 42.2%

       \[\leadsto \color{blue}{\frac{D \cdot \left(D \cdot \left(0.25 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot d}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.05 \cdot 10^{-166}:\\ \;\;\;\;\frac{\left(D \cdot D\right) \cdot 0.25}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\\ \mathbf{elif}\;d \leq 6 \cdot 10^{+187}:\\ \;\;\;\;\frac{D \cdot \left(D \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot 0.25\right)\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(D \cdot D\right) \cdot 0.25}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 43.3% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \cdot M \leq 4 \cdot 10^{-164}:\\ \;\;\;\;\frac{\left(h \cdot \left(M \cdot M\right)\right) \cdot \frac{D}{\frac{d}{D}}}{\frac{d}{0.25}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left(d \cdot \frac{\frac{\frac{c0 \cdot \frac{c0}{D \cdot D}}{w}}{w}}{h}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= (* M M) 4e-164)
   (/ (* (* h (* M M)) (/ D (/ d D))) (/ d 0.25))
   (* d (* d (/ (/ (/ (* c0 (/ c0 (* D D))) w) w) h)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((M * M) <= 4e-164) {
		tmp = ((h * (M * M)) * (D / (d / D))) / (d / 0.25);
	} else {
		tmp = d * (d * ((((c0 * (c0 / (D * D))) / w) / w) / h));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((m * m) <= 4d-164) then
        tmp = ((h * (m * m)) * (d / (d_1 / d))) / (d_1 / 0.25d0)
    else
        tmp = d_1 * (d_1 * ((((c0 * (c0 / (d * d))) / w) / w) / h))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((M * M) <= 4e-164) {
		tmp = ((h * (M * M)) * (D / (d / D))) / (d / 0.25);
	} else {
		tmp = d * (d * ((((c0 * (c0 / (D * D))) / w) / w) / h));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (M * M) <= 4e-164:
		tmp = ((h * (M * M)) * (D / (d / D))) / (d / 0.25)
	else:
		tmp = d * (d * ((((c0 * (c0 / (D * D))) / w) / w) / h))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (Float64(M * M) <= 4e-164)
		tmp = Float64(Float64(Float64(h * Float64(M * M)) * Float64(D / Float64(d / D))) / Float64(d / 0.25));
	else
		tmp = Float64(d * Float64(d * Float64(Float64(Float64(Float64(c0 * Float64(c0 / Float64(D * D))) / w) / w) / h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((M * M) <= 4e-164)
		tmp = ((h * (M * M)) * (D / (d / D))) / (d / 0.25);
	else
		tmp = d * (d * ((((c0 * (c0 / (D * D))) / w) / w) / h));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[N[(M * M), $MachinePrecision], 4e-164], N[(N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] * N[(D / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d / 0.25), $MachinePrecision]), $MachinePrecision], N[(d * N[(d * N[(N[(N[(N[(c0 * N[(c0 / N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision] / w), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M M) < 3.99999999999999985e-164

   1. Initial program 28.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}{\color{blue}{2 \cdot w}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right), \color{blue}{\left(2 \cdot w\right)}\right) \]

   3. Simplified 25.1%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot \frac{\frac{\frac{d \cdot d}{w} \cdot c0}{h} \cdot c0}{D \cdot D}}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(\mathsf{neg}\left(c0\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 23.7%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\left(\frac{-0.5 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot w\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)} + 0\right) \cdot \left(-c0\right)\right)}}{2 \cdot w} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. 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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left({D}^{2}\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(D \cdot D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot {M}^{2}\right)\right)\right), \left({d}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

      12. *-lowering-*.f64 41.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   10. Simplified 41.6%

       \[\leadsto \color{blue}{\frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot d}} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{1}{4}}{\color{blue}{d} \cdot d} \]

       2. times-frac N/A

          \[\leadsto \frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d} \cdot \color{blue}{\frac{\frac{1}{4}}{d}} \]

       3. clear-num N/A

          \[\leadsto \frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d} \cdot \frac{1}{\color{blue}{\frac{d}{\frac{1}{4}}}} \]

       4. un-div-inv N/A

          \[\leadsto \frac{\frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d}}{\color{blue}{\frac{d}{\frac{1}{4}}}} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d}\right), \color{blue}{\left(\frac{d}{\frac{1}{4}}\right)}\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(h \cdot \left(M \cdot M\right)\right) \cdot \left(D \cdot D\right)}{d}\right), \left(\frac{d}{\frac{1}{4}}\right)\right) \]

       7. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \frac{D \cdot D}{d}\right), \left(\frac{\color{blue}{d}}{\frac{1}{4}}\right)\right) \]

       8. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \left(D \cdot \frac{D}{d}\right)\right), \left(\frac{d}{\frac{1}{4}}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(h \cdot \left(M \cdot M\right)\right), \left(D \cdot \frac{D}{d}\right)\right), \left(\frac{\color{blue}{d}}{\frac{1}{4}}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \left(M \cdot M\right)\right), \left(D \cdot \frac{D}{d}\right)\right), \left(\frac{d}{\frac{1}{4}}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), \left(D \cdot \frac{D}{d}\right)\right), \left(\frac{d}{\frac{1}{4}}\right)\right) \]

       12. clear-num N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), \left(D \cdot \frac{1}{\frac{d}{D}}\right)\right), \left(\frac{d}{\frac{1}{4}}\right)\right) \]

       13. un-div-inv N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), \left(\frac{D}{\frac{d}{D}}\right)\right), \left(\frac{d}{\frac{1}{4}}\right)\right) \]

       14. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), \mathsf{/.f64}\left(D, \left(\frac{d}{D}\right)\right)\right), \left(\frac{d}{\frac{1}{4}}\right)\right) \]

       15. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), \mathsf{/.f64}\left(D, \mathsf{/.f64}\left(d, D\right)\right)\right), \left(\frac{d}{\frac{1}{4}}\right)\right) \]

       16. /-lowering-/.f64 47.4%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), \mathsf{/.f64}\left(D, \mathsf{/.f64}\left(d, D\right)\right)\right), \mathsf{/.f64}\left(d, \color{blue}{\frac{1}{4}}\right)\right) \]

   12. Applied egg-rr 47.4%

       \[\leadsto \color{blue}{\frac{\left(h \cdot \left(M \cdot M\right)\right) \cdot \frac{D}{\frac{d}{D}}}{\frac{d}{0.25}}} \]

   if 3.99999999999999985e-164 < (*.f64 M M)

   1. Initial program 21.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\color{blue}{{D}^{2} \cdot \left(h \cdot w\right)}}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(2 \cdot \left(c0 \cdot {d}^{2}\right)\right), \color{blue}{\left({D}^{2} \cdot \left(h \cdot w\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot {d}^{2}\right)\right), \left(\color{blue}{{D}^{2}} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left({d}^{2}\right)\right)\right), \left({D}^{\color{blue}{2}} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(d \cdot d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \left({D}^{2} \cdot \left(h \cdot w\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(h \cdot w\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{h} \cdot w\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{h} \cdot w\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(w \cdot \color{blue}{h}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 35.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(d, d\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(w, \color{blue}{h}\right)\right)\right)\right) \]

   5. Simplified 35.7%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(D \cdot D\right)} \cdot \left(w \cdot h\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}}\right)\right) \]

      4. frac-times N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(2 \cdot \left(\frac{c0 \cdot d}{w \cdot h} \cdot \color{blue}{\frac{d}{D \cdot D}}\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\left(2 \cdot \frac{c0 \cdot d}{w \cdot h}\right) \cdot \color{blue}{\frac{d}{D \cdot D}}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \frac{c0 \cdot d}{w \cdot h}\right), \color{blue}{\left(\frac{d}{D \cdot D}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{c0 \cdot d}{w \cdot h}\right)\right), \left(\frac{\color{blue}{d}}{D \cdot D}\right)\right)\right) \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{\frac{c0 \cdot d}{w}}{h}\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{c0 \cdot d}{w}\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(c0 \cdot \frac{d}{w}\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\frac{d}{w}\right)\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \left(\frac{d}{D \cdot D}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \mathsf{/.f64}\left(d, \color{blue}{\left(D \cdot D\right)}\right)\right)\right) \]

      14. *-lowering-*.f64 44.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, w\right)\right), h\right)\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(D, \color{blue}{D}\right)\right)\right)\right) \]

   7. Applied egg-rr 44.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot \frac{d}{D \cdot D}\right)} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot \frac{\frac{d}{D}}{\color{blue}{D}}\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot \frac{d}{D}}{\color{blue}{D}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot \frac{d}{D}\right), \color{blue}{D}\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right) \cdot \frac{1}{\frac{D}{d}}\right), D\right)\right) \]

      5. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}}{\frac{D}{d}}\right), D\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \frac{c0 \cdot \frac{d}{w}}{h}\right), \left(\frac{D}{d}\right)\right), D\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{c0 \cdot \frac{d}{w}}{h}\right)\right), \left(\frac{D}{d}\right)\right), D\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(c0 \cdot \frac{\frac{d}{w}}{h}\right)\right), \left(\frac{D}{d}\right)\right), D\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(\frac{\frac{d}{w}}{h}\right)\right)\right), \left(\frac{D}{d}\right)\right), D\right)\right) \]

      10. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(\frac{d}{h \cdot w}\right)\right)\right), \left(\frac{D}{d}\right)\right), D\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \left(\frac{d}{w \cdot h}\right)\right)\right), \left(\frac{D}{d}\right)\right), D\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, \left(w \cdot h\right)\right)\right)\right), \left(\frac{D}{d}\right)\right), D\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(w, h\right)\right)\right)\right), \left(\frac{D}{d}\right)\right), D\right)\right) \]

      14. /-lowering-/.f64 54.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(c0, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(w, h\right)\right)\right)\right), \mathsf{/.f64}\left(D, d\right)\right), D\right)\right) \]

   9. Applied egg-rr 54.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{\frac{2 \cdot \left(c0 \cdot \frac{d}{w \cdot h}\right)}{\frac{D}{d}}}{D}} \]

   10. Taylor expanded in c0 around 0

       \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{{d}^{2} \cdot {c0}^{2}}{\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)} \]

       2. associate-/l* N/A

          \[\leadsto {d}^{2} \cdot \color{blue}{\frac{{c0}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]

       3. *-commutative N/A

          \[\leadsto \frac{{c0}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \cdot \color{blue}{{d}^{2}} \]

       4. unpow2 N/A

          \[\leadsto \frac{{c0}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \cdot \left(d \cdot \color{blue}{d}\right) \]

       5. associate-*r* N/A

          \[\leadsto \left(\frac{{c0}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \cdot d\right) \cdot \color{blue}{d} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{{c0}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \cdot d\right), \color{blue}{d}\right) \]

   12. Simplified 44.0%

       \[\leadsto \color{blue}{\left(\frac{\frac{\frac{c0 \cdot \frac{c0}{D \cdot D}}{w}}{w}}{h} \cdot d\right) \cdot d} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot M \leq 4 \cdot 10^{-164}:\\ \;\;\;\;\frac{\left(h \cdot \left(M \cdot M\right)\right) \cdot \frac{D}{\frac{d}{D}}}{\frac{d}{0.25}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left(d \cdot \frac{\frac{\frac{c0 \cdot \frac{c0}{D \cdot D}}{w}}{w}}{h}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 36.6% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \cdot D \leq 5 \cdot 10^{-321}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= (* D D) 5e-321) 0.0 (* (* (* D D) 0.25) (/ (* h (* M M)) (* d d)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((D * D) <= 5e-321) {
		tmp = 0.0;
	} else {
		tmp = ((D * D) * 0.25) * ((h * (M * M)) / (d * d));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((d * d) <= 5d-321) then
        tmp = 0.0d0
    else
        tmp = ((d * d) * 0.25d0) * ((h * (m * m)) / (d_1 * d_1))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((D * D) <= 5e-321) {
		tmp = 0.0;
	} else {
		tmp = ((D * D) * 0.25) * ((h * (M * M)) / (d * d));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (D * D) <= 5e-321:
		tmp = 0.0
	else:
		tmp = ((D * D) * 0.25) * ((h * (M * M)) / (d * d))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (Float64(D * D) <= 5e-321)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(D * D) * 0.25) * Float64(Float64(h * Float64(M * M)) / Float64(d * d)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((D * D) <= 5e-321)
		tmp = 0.0;
	else
		tmp = ((D * D) * 0.25) * ((h * (M * M)) / (d * d));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[N[(D * D), $MachinePrecision], 5e-321], 0.0, N[(N[(N[(D * D), $MachinePrecision] * 0.25), $MachinePrecision] * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 D D) < 4.99994e-321

   1. Initial program 25.4%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}{\color{blue}{2 \cdot w}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right), \color{blue}{\left(2 \cdot w\right)}\right) \]

   3. Simplified 25.2%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot \frac{\frac{\frac{d \cdot d}{w} \cdot c0}{h} \cdot c0}{D \cdot D}}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot c0\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. mul0-lft N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(c0 \cdot \left(-1 + 1\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(\mathsf{neg}\left(\left(-1 + 1\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(\mathsf{neg}\left(0\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      13. metadata-eval 34.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 34.1%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(c0 \cdot 0\right)}}{2 \cdot w} \]
   8. Step-by-step derivation

      1. mul0-rgt N/A

         \[\leadsto \frac{c0 \cdot 0}{2 \cdot w} \]

      2. mul0-rgt N/A

         \[\leadsto \frac{0}{\color{blue}{2} \cdot w} \]

      3. div0 34.1%

         \[\leadsto 0 \]

   9. Applied egg-rr 34.1%

      \[\leadsto \color{blue}{0} \]

   if 4.99994e-321 < (*.f64 D D)

   1. Initial program 24.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}{\color{blue}{2 \cdot w}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right), \color{blue}{\left(2 \cdot w\right)}\right) \]

   3. Simplified 22.0%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot \frac{\frac{\frac{d \cdot d}{w} \cdot c0}{h} \cdot c0}{D \cdot D}}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(\mathsf{neg}\left(c0\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 15.6%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\left(\frac{-0.5 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot w\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)} + 0\right) \cdot \left(-c0\right)\right)}}{2 \cdot w} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. 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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left({D}^{2}\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(D \cdot D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot {M}^{2}\right)\right)\right), \left({d}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

      12. *-lowering-*.f64 35.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   10. Simplified 35.5%

       \[\leadsto \color{blue}{\frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot d}} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\color{blue}{d} \cdot d} \]

       2. associate-/l* N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\frac{h \cdot \left(M \cdot M\right)}{d \cdot d}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right), \color{blue}{\left(\frac{h \cdot \left(M \cdot M\right)}{d \cdot d}\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot D\right)\right), \left(\frac{\color{blue}{h \cdot \left(M \cdot M\right)}}{d \cdot d}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{d \cdot d}\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\left(h \cdot \left(M \cdot M\right)\right), \color{blue}{\left(d \cdot d\right)}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(M \cdot M\right)\right), \left(\color{blue}{d} \cdot d\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), \left(d \cdot d\right)\right)\right) \]

       9. *-lowering-*.f64 36.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right)\right) \]

   12. Applied egg-rr 36.4%

       \[\leadsto \color{blue}{\left(0.25 \cdot \left(D \cdot D\right)\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \cdot D \leq 5 \cdot 10^{-321}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 37.2% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 4.2 \cdot 10^{-160}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(D \cdot D\right) \cdot 0.25}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= D 4.2e-160) 0.0 (* (/ (* (* D D) 0.25) d) (/ (* h (* M M)) d))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (D <= 4.2e-160) {
		tmp = 0.0;
	} else {
		tmp = (((D * D) * 0.25) / d) * ((h * (M * 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) :: tmp
    if (d <= 4.2d-160) then
        tmp = 0.0d0
    else
        tmp = (((d * d) * 0.25d0) / d_1) * ((h * (m * 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 tmp;
	if (D <= 4.2e-160) {
		tmp = 0.0;
	} else {
		tmp = (((D * D) * 0.25) / d) * ((h * (M * M)) / d);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if D <= 4.2e-160:
		tmp = 0.0
	else:
		tmp = (((D * D) * 0.25) / d) * ((h * (M * M)) / d)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (D <= 4.2e-160)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(Float64(D * D) * 0.25) / d) * Float64(Float64(h * Float64(M * M)) / d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (D <= 4.2e-160)
		tmp = 0.0;
	else
		tmp = (((D * D) * 0.25) / d) * ((h * (M * M)) / d);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[D, 4.2e-160], 0.0, N[(N[(N[(N[(D * D), $MachinePrecision] * 0.25), $MachinePrecision] / d), $MachinePrecision] * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 4.2000000000000001e-160

   1. Initial program 24.0%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}{\color{blue}{2 \cdot w}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right), \color{blue}{\left(2 \cdot w\right)}\right) \]

   3. Simplified 23.8%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot \frac{\frac{\frac{d \cdot d}{w} \cdot c0}{h} \cdot c0}{D \cdot D}}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot c0\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. mul0-lft N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(c0 \cdot \left(-1 + 1\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(\mathsf{neg}\left(\left(-1 + 1\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(\mathsf{neg}\left(0\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      13. metadata-eval 27.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 27.2%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(c0 \cdot 0\right)}}{2 \cdot w} \]
   8. Step-by-step derivation

      1. mul0-rgt N/A

         \[\leadsto \frac{c0 \cdot 0}{2 \cdot w} \]

      2. mul0-rgt N/A

         \[\leadsto \frac{0}{\color{blue}{2} \cdot w} \]

      3. div0 27.2%

         \[\leadsto 0 \]

   9. Applied egg-rr 27.2%

      \[\leadsto \color{blue}{0} \]

   if 4.2000000000000001e-160 < D

   1. Initial program 26.2%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}{\color{blue}{2 \cdot w}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right), \color{blue}{\left(2 \cdot w\right)}\right) \]

   3. Simplified 22.2%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot \frac{\frac{\frac{d \cdot d}{w} \cdot c0}{h} \cdot c0}{D \cdot D}}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} - M \cdot M}\right)}{2 \cdot w}} \]
   4. Add Preprocessing

   5. Taylor expanded in c0 around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(\mathsf{neg}\left(c0\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \cdot \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right), \left(-1 \cdot c0\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. Simplified 19.6%

      \[\leadsto \frac{c0 \cdot \color{blue}{\left(\left(\frac{-0.5 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot w\right)\right)}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)} + 0\right) \cdot \left(-c0\right)\right)}}{2 \cdot w} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. 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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({\color{blue}{d}}^{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left({D}^{2}\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(D \cdot D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(h \cdot {M}^{2}\right)\right)\right), \left({d}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left({d}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \left(d \cdot \color{blue}{d}\right)\right) \]

      12. *-lowering-*.f64 37.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right) \]

   10. Simplified 37.2%

       \[\leadsto \color{blue}{\frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot d}} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\color{blue}{d} \cdot d} \]

       2. times-frac N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(D \cdot D\right)}{d} \cdot \color{blue}{\frac{h \cdot \left(M \cdot M\right)}{d}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{4} \cdot \left(D \cdot D\right)}{d}\right), \color{blue}{\left(\frac{h \cdot \left(M \cdot M\right)}{d}\right)}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left(D \cdot D\right)\right), d\right), \left(\frac{\color{blue}{h \cdot \left(M \cdot M\right)}}{d}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot D\right)\right), d\right), \left(\frac{\color{blue}{h} \cdot \left(M \cdot M\right)}{d}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, D\right)\right), d\right), \left(\frac{h \cdot \left(M \cdot M\right)}{d}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, D\right)\right), d\right), \mathsf{/.f64}\left(\left(h \cdot \left(M \cdot M\right)\right), \color{blue}{d}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, D\right)\right), d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(M \cdot M\right)\right), d\right)\right) \]

       9. *-lowering-*.f64 46.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, D\right)\right), d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), d\right)\right) \]

   12. Applied egg-rr 46.0%

       \[\leadsto \color{blue}{\frac{0.25 \cdot \left(D \cdot D\right)}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 4.2 \cdot 10^{-160}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(D \cdot D\right) \cdot 0.25}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 33.3% 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 24.8%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
2. Step-by-step derivation

   1. associate-*l/ N/A

      \[\leadsto \frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}{\color{blue}{2 \cdot w}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right), \color{blue}{\left(2 \cdot w\right)}\right) \]

3. Simplified 23.2%

   \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot \frac{\frac{\frac{d \cdot d}{w} \cdot c0}{h} \cdot c0}{D \cdot D}}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)} - M \cdot M}\right)}{2 \cdot w}} \]
4. Add Preprocessing

5. Taylor expanded in c0 around -inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)}\right), \mathsf{*.f64}\left(2, w\right)\right) \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(-1 \cdot c0\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   3. distribute-lft1-in N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   5. mul0-lft N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   7. distribute-lft-neg-in N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(\mathsf{neg}\left(c0 \cdot \left(-1 + 1\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   8. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(\mathsf{neg}\left(\left(-1 + 1\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(\mathsf{neg}\left(0\right)\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \left(c0 \cdot \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, \left(-1 + 1\right)\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

   13. metadata-eval 29.0%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c0, \mathsf{*.f64}\left(c0, 0\right)\right), \mathsf{*.f64}\left(2, w\right)\right) \]

7. Simplified 29.0%

   \[\leadsto \frac{c0 \cdot \color{blue}{\left(c0 \cdot 0\right)}}{2 \cdot w} \]
8. Step-by-step derivation

   1. mul0-rgt N/A

      \[\leadsto \frac{c0 \cdot 0}{2 \cdot w} \]

   2. mul0-rgt N/A

      \[\leadsto \frac{0}{\color{blue}{2} \cdot w} \]

   3. div0 29.0%

      \[\leadsto 0 \]

9. Applied egg-rr 29.0%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024192 
(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))))))