Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.2% → 69.2%

Time: 22.2s

Alternatives: 9

Speedup: 151.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 24.2% 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: 69.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_1 := \frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right)\\ \mathbf{if}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \frac{M}{d}}{\frac{d}{D \cdot \left(h \cdot M\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_1 (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))))
   (if (<= t_1 INFINITY) t_1 (* 0.25 (/ (* D (/ M d)) (/ d (* D (* h 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));
	double t_1 = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = 0.25 * ((D * (M / d)) / (d / (D * (h * M))));
	}
	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 t_1 = (c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = 0.25 * ((D * (M / d)) / (d / (D * (h * M))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	t_1 = (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))
	tmp = 0
	if t_1 <= math.inf:
		tmp = t_1
	else:
		tmp = 0.25 * ((D * (M / d)) / (d / (D * (h * M))))
	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)))
	t_1 = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(0.25 * Float64(Float64(D * Float64(M / d)) / Float64(d / Float64(D * Float64(h * M)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	t_1 = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	tmp = 0.0;
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = 0.25 * ((D * (M / d)) / (d / (D * (h * M))));
	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]}, Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(0.25 * N[(N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision] / N[(d / N[(D * N[(h * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 72.7%

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

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

   1. Initial program 0.0%

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

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

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

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

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

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

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

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

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

   3. Simplified 1.1%

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

   5. Taylor expanded in c0 around -inf

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

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

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

      2. unpow2 N/A

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

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

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

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

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

   7. Simplified 21.0%

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

   8. Taylor expanded in c0 around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 45.0%

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

   10. Simplified 45.0%

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. times-frac N/A

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

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

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

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

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

       11. *-commutative N/A

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

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

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

       13. *-commutative N/A

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

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

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

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

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

       16. *-lowering-*.f64 71.4%

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

   12. Applied egg-rr 71.4%

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

       1. *-commutative N/A

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

       2. clear-num N/A

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

       3. un-div-inv N/A

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

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

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

       5. associate-/l* N/A

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

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

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

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

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

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

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

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

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

       10. *-lowering-*.f64 73.0%

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

   14. Applied egg-rr 73.0%

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

4. Final simplification 72.9%

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

Alternative 2: 50.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \cdot d \leq 5 \cdot 10^{-182}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \frac{M \cdot \frac{h \cdot M}{d}}{d}\right)\right)\\ \mathbf{elif}\;d \cdot d \leq 5 \cdot 10^{-105}:\\ \;\;\;\;\frac{\frac{\frac{c0}{\frac{w}{\frac{-0.5}{d \cdot d} \cdot \left(\left(M \cdot \left(D \cdot \left(D \cdot M\right)\right)\right) \cdot \frac{w \cdot \left(w \cdot h\right)}{c0}\right) + \frac{2}{D \cdot D} \cdot \frac{c0 \cdot d}{\frac{h}{d}}}}}{w}}{2}\\ \mathbf{elif}\;d \cdot d \leq \infty:\\ \;\;\;\;0.25 \cdot \left(\frac{M \cdot \left(h \cdot D\right)}{d} \cdot \frac{D \cdot M}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{M \cdot \frac{D \cdot \left(D \cdot \left(h \cdot M\right)\right)}{d}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= (* d d) 5e-182)
   (* 0.25 (* D (* D (/ (* M (/ (* h M) d)) d))))
   (if (<= (* d d) 5e-105)
     (/
      (/
       (/
        c0
        (/
         w
         (+
          (* (/ -0.5 (* d d)) (* (* M (* D (* D M))) (/ (* w (* w h)) c0)))
          (* (/ 2.0 (* D D)) (/ (* c0 d) (/ h d))))))
       w)
      2.0)
     (if (<= (* d d) INFINITY)
       (* 0.25 (* (/ (* M (* h D)) d) (/ (* D M) d)))
       (* 0.25 (/ (* M (/ (* D (* D (* h M))) d)) d))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((d * d) <= 5e-182) {
		tmp = 0.25 * (D * (D * ((M * ((h * M) / d)) / d)));
	} else if ((d * d) <= 5e-105) {
		tmp = ((c0 / (w / (((-0.5 / (d * d)) * ((M * (D * (D * M))) * ((w * (w * h)) / c0))) + ((2.0 / (D * D)) * ((c0 * d) / (h / d)))))) / w) / 2.0;
	} else if ((d * d) <= ((double) INFINITY)) {
		tmp = 0.25 * (((M * (h * D)) / d) * ((D * M) / d));
	} else {
		tmp = 0.25 * ((M * ((D * (D * (h * M))) / d)) / d);
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((d * d) <= 5e-182) {
		tmp = 0.25 * (D * (D * ((M * ((h * M) / d)) / d)));
	} else if ((d * d) <= 5e-105) {
		tmp = ((c0 / (w / (((-0.5 / (d * d)) * ((M * (D * (D * M))) * ((w * (w * h)) / c0))) + ((2.0 / (D * D)) * ((c0 * d) / (h / d)))))) / w) / 2.0;
	} else if ((d * d) <= Double.POSITIVE_INFINITY) {
		tmp = 0.25 * (((M * (h * D)) / d) * ((D * M) / d));
	} else {
		tmp = 0.25 * ((M * ((D * (D * (h * M))) / d)) / d);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (d * d) <= 5e-182:
		tmp = 0.25 * (D * (D * ((M * ((h * M) / d)) / d)))
	elif (d * d) <= 5e-105:
		tmp = ((c0 / (w / (((-0.5 / (d * d)) * ((M * (D * (D * M))) * ((w * (w * h)) / c0))) + ((2.0 / (D * D)) * ((c0 * d) / (h / d)))))) / w) / 2.0
	elif (d * d) <= math.inf:
		tmp = 0.25 * (((M * (h * D)) / d) * ((D * M) / d))
	else:
		tmp = 0.25 * ((M * ((D * (D * (h * M))) / d)) / d)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (Float64(d * d) <= 5e-182)
		tmp = Float64(0.25 * Float64(D * Float64(D * Float64(Float64(M * Float64(Float64(h * M) / d)) / d))));
	elseif (Float64(d * d) <= 5e-105)
		tmp = Float64(Float64(Float64(c0 / Float64(w / Float64(Float64(Float64(-0.5 / Float64(d * d)) * Float64(Float64(M * Float64(D * Float64(D * M))) * Float64(Float64(w * Float64(w * h)) / c0))) + Float64(Float64(2.0 / Float64(D * D)) * Float64(Float64(c0 * d) / Float64(h / d)))))) / w) / 2.0);
	elseif (Float64(d * d) <= Inf)
		tmp = Float64(0.25 * Float64(Float64(Float64(M * Float64(h * D)) / d) * Float64(Float64(D * M) / d)));
	else
		tmp = Float64(0.25 * Float64(Float64(M * Float64(Float64(D * Float64(D * Float64(h * M))) / d)) / d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((d * d) <= 5e-182)
		tmp = 0.25 * (D * (D * ((M * ((h * M) / d)) / d)));
	elseif ((d * d) <= 5e-105)
		tmp = ((c0 / (w / (((-0.5 / (d * d)) * ((M * (D * (D * M))) * ((w * (w * h)) / c0))) + ((2.0 / (D * D)) * ((c0 * d) / (h / d)))))) / w) / 2.0;
	elseif ((d * d) <= Inf)
		tmp = 0.25 * (((M * (h * D)) / d) * ((D * M) / d));
	else
		tmp = 0.25 * ((M * ((D * (D * (h * M))) / d)) / d);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[N[(d * d), $MachinePrecision], 5e-182], N[(0.25 * N[(D * N[(D * N[(N[(M * N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(d * d), $MachinePrecision], 5e-105], N[(N[(N[(c0 / N[(w / N[(N[(N[(-0.5 / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[(N[(M * N[(D * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(w * N[(w * h), $MachinePrecision]), $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(c0 * d), $MachinePrecision] / N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[N[(d * d), $MachinePrecision], Infinity], N[(0.25 * N[(N[(N[(M * N[(h * D), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(M * N[(N[(D * N[(D * N[(h * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 d d) < 5.00000000000000024e-182

   1. Initial program 12.7%

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

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

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

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

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

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

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

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

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

   3. Simplified 14.2%

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

   5. Taylor expanded in c0 around -inf

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

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

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

      2. unpow2 N/A

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

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

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

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

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

   7. Simplified 12.8%

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

   8. Taylor expanded in c0 around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 22.9%

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

   10. Simplified 22.9%

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

       1. associate-/l* N/A

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

       2. associate-*l* N/A

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

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

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

       4. *-commutative N/A

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

       5. associate-/r* N/A

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

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

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

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

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. associate-/l* N/A

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

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

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

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

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

       13. *-commutative N/A

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

       14. *-lowering-*.f64 61.7%

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

   12. Applied egg-rr 61.7%

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

   if 5.00000000000000024e-182 < (*.f64 d d) < 4.99999999999999963e-105

   1. Initial program 46.8%

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

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

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

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

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

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

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

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

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

   3. Simplified 46.2%

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

   5. Taylor expanded in w around 0

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

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

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

   7. Simplified 53.9%

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

   9. Applied egg-rr 70.0%

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

   if 4.99999999999999963e-105 < (*.f64 d d) < +inf.0

   1. Initial program 22.2%

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

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

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

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

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

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

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

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

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

   3. Simplified 21.7%

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

   5. Taylor expanded in c0 around -inf

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

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

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

      2. unpow2 N/A

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

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

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

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

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

   7. Simplified 18.7%

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

   8. Taylor expanded in c0 around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 40.7%

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

   10. Simplified 40.7%

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. times-frac N/A

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

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

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

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

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

       11. *-commutative N/A

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

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

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

       13. *-commutative N/A

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

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

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

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

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

       16. *-lowering-*.f64 60.4%

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

   12. Applied egg-rr 60.4%

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

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

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

       5. *-commutative N/A

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

       6. *-lowering-*.f64 59.8%

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

   14. Applied egg-rr 59.8%

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

   if +inf.0 < (*.f64 d d)

   1. Initial program 21.6%

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

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

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

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

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

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

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

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

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

   3. Simplified 21.5%

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

   5. Taylor expanded in c0 around -inf

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

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

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

      2. unpow2 N/A

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

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

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

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

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

   7. Simplified 17.1%

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

   8. Taylor expanded in c0 around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 36.2%

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

   10. Simplified 36.2%

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. times-frac N/A

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

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

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

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

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

       11. *-commutative N/A

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

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

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

       13. *-commutative N/A

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

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

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

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

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

       16. *-lowering-*.f64 57.3%

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

   12. Applied egg-rr 57.3%

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

       1. frac-times N/A

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

       2. associate-*l* N/A

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

       3. times-frac N/A

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

       4. associate-*r/ N/A

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

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

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

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

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

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

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

       8. *-commutative N/A

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

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

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

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

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

       11. *-lowering-*.f64 54.6%

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

   14. Applied egg-rr 54.6%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \cdot d \leq 5 \cdot 10^{-182}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \frac{M \cdot \frac{h \cdot M}{d}}{d}\right)\right)\\ \mathbf{elif}\;d \cdot d \leq 5 \cdot 10^{-105}:\\ \;\;\;\;\frac{\frac{\frac{c0}{\frac{w}{\frac{-0.5}{d \cdot d} \cdot \left(\left(M \cdot \left(D \cdot \left(D \cdot M\right)\right)\right) \cdot \frac{w \cdot \left(w \cdot h\right)}{c0}\right) + \frac{2}{D \cdot D} \cdot \frac{c0 \cdot d}{\frac{h}{d}}}}}{w}}{2}\\ \mathbf{elif}\;d \cdot d \leq \infty:\\ \;\;\;\;0.25 \cdot \left(\frac{M \cdot \left(h \cdot D\right)}{d} \cdot \frac{D \cdot M}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{M \cdot \frac{D \cdot \left(D \cdot \left(h \cdot M\right)\right)}{d}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.5% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \cdot d \leq 5 \cdot 10^{+182}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(M \cdot \left(D \cdot \frac{\frac{h \cdot M}{d}}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{M \cdot \frac{D \cdot \left(D \cdot \left(h \cdot M\right)\right)}{d}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= (* d d) 5e+182)
   (* 0.25 (* D (* M (* D (/ (/ (* h M) d) d)))))
   (* 0.25 (/ (* M (/ (* D (* D (* h M))) d)) d))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((d * d) <= 5e+182) {
		tmp = 0.25 * (D * (M * (D * (((h * M) / d) / d))));
	} else {
		tmp = 0.25 * ((M * ((D * (D * (h * M))) / d)) / d);
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((d_1 * d_1) <= 5d+182) then
        tmp = 0.25d0 * (d * (m * (d * (((h * m) / d_1) / d_1))))
    else
        tmp = 0.25d0 * ((m * ((d * (d * (h * m))) / d_1)) / d_1)
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((d * d) <= 5e+182) {
		tmp = 0.25 * (D * (M * (D * (((h * M) / d) / d))));
	} else {
		tmp = 0.25 * ((M * ((D * (D * (h * M))) / d)) / d);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (d * d) <= 5e+182:
		tmp = 0.25 * (D * (M * (D * (((h * M) / d) / d))))
	else:
		tmp = 0.25 * ((M * ((D * (D * (h * M))) / d)) / d)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (Float64(d * d) <= 5e+182)
		tmp = Float64(0.25 * Float64(D * Float64(M * Float64(D * Float64(Float64(Float64(h * M) / d) / d)))));
	else
		tmp = Float64(0.25 * Float64(Float64(M * Float64(Float64(D * Float64(D * Float64(h * M))) / d)) / d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((d * d) <= 5e+182)
		tmp = 0.25 * (D * (M * (D * (((h * M) / d) / d))));
	else
		tmp = 0.25 * ((M * ((D * (D * (h * M))) / d)) / d);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[N[(d * d), $MachinePrecision], 5e+182], N[(0.25 * N[(D * N[(M * N[(D * N[(N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(M * N[(N[(D * N[(D * N[(h * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 d d) < 4.99999999999999973e182

   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. *-lowering-*.f64 N/A

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

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

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

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

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

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

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

   3. Simplified 23.1%

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

   5. Taylor expanded in c0 around -inf

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

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

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

      2. unpow2 N/A

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

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

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

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

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

   7. Simplified 14.7%

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

   8. Taylor expanded in c0 around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 33.1%

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

   10. Simplified 33.1%

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

       1. associate-/l* N/A

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

       2. associate-*l* N/A

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

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

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

       4. *-commutative N/A

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

       5. associate-/r* N/A

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

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

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

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

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. associate-/l* N/A

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

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

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

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

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

       13. *-commutative N/A

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

       14. *-lowering-*.f64 55.0%

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

   12. Applied egg-rr 55.0%

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

       1. *-commutative N/A

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

       2. associate-/l* N/A

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

       3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

       8. *-lowering-*.f64 59.5%

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

   14. Applied egg-rr 59.5%

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

   if 4.99999999999999973e182 < (*.f64 d d)

   1. Initial program 19.2%

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

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

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

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

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

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

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

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

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

   3. Simplified 19.9%

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

   5. Taylor expanded in c0 around -inf

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

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

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

      2. unpow2 N/A

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

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

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

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

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

   7. Simplified 19.3%

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

   8. Taylor expanded in c0 around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 39.1%

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

   10. Simplified 39.1%

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. times-frac N/A

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

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

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

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

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

       11. *-commutative N/A

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

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

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

       13. *-commutative N/A

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

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

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

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

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

       16. *-lowering-*.f64 58.1%

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

   12. Applied egg-rr 58.1%

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

       1. frac-times N/A

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

       2. associate-*l* N/A

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

       3. times-frac N/A

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

       4. associate-*r/ N/A

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

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

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

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

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

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

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

       8. *-commutative N/A

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

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

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

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

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

       11. *-lowering-*.f64 61.1%

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

   14. Applied egg-rr 61.1%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \cdot d \leq 5 \cdot 10^{+182}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(M \cdot \left(D \cdot \frac{\frac{h \cdot M}{d}}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{M \cdot \frac{D \cdot \left(D \cdot \left(h \cdot M\right)\right)}{d}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 50.4% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 5 \cdot 10^{-203}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \frac{M \cdot \frac{h \cdot M}{d}}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{D \cdot M}{d} \cdot \frac{D \cdot \left(h \cdot M\right)}{d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= D 5e-203)
   (* 0.25 (* D (* D (/ (* M (/ (* h M) d)) d))))
   (* 0.25 (* (/ (* D M) d) (/ (* D (* h M)) d)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (D <= 5e-203) {
		tmp = 0.25 * (D * (D * ((M * ((h * M) / d)) / d)));
	} else {
		tmp = 0.25 * (((D * M) / d) * ((D * (h * 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 <= 5d-203) then
        tmp = 0.25d0 * (d * (d * ((m * ((h * m) / d_1)) / d_1)))
    else
        tmp = 0.25d0 * (((d * m) / d_1) * ((d * (h * 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 <= 5e-203) {
		tmp = 0.25 * (D * (D * ((M * ((h * M) / d)) / d)));
	} else {
		tmp = 0.25 * (((D * M) / d) * ((D * (h * M)) / d));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if D <= 5e-203:
		tmp = 0.25 * (D * (D * ((M * ((h * M) / d)) / d)))
	else:
		tmp = 0.25 * (((D * M) / d) * ((D * (h * M)) / d))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (D <= 5e-203)
		tmp = Float64(0.25 * Float64(D * Float64(D * Float64(Float64(M * Float64(Float64(h * M) / d)) / d))));
	else
		tmp = Float64(0.25 * Float64(Float64(Float64(D * M) / d) * Float64(Float64(D * Float64(h * M)) / d)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (D <= 5e-203)
		tmp = 0.25 * (D * (D * ((M * ((h * M) / d)) / d)));
	else
		tmp = 0.25 * (((D * M) / d) * ((D * (h * M)) / d));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[D, 5e-203], N[(0.25 * N[(D * N[(D * N[(N[(M * N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision] * N[(N[(D * N[(h * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 5.0000000000000002e-203

   1. Initial program 19.3%

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

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

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

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

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

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

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

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

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

   3. Simplified 18.1%

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

   5. Taylor expanded in c0 around -inf

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

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

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

      2. unpow2 N/A

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

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

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

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

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

   7. Simplified 19.0%

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

   8. Taylor expanded in c0 around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 38.9%

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

   10. Simplified 38.9%

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

       1. associate-/l* N/A

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

       2. associate-*l* N/A

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

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

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

       4. *-commutative N/A

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

       5. associate-/r* N/A

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

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

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

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

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. associate-/l* N/A

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

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

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

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

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

       13. *-commutative N/A

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

       14. *-lowering-*.f64 60.5%

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

   12. Applied egg-rr 60.5%

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

   if 5.0000000000000002e-203 < D

   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. Step-by-step derivation

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

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

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

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

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

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

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

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

   3. Simplified 27.9%

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

   5. Taylor expanded in c0 around -inf

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

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

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

      2. unpow2 N/A

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

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

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

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

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

   7. Simplified 13.4%

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

   8. Taylor expanded in c0 around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 31.1%

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

   10. Simplified 31.1%

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. times-frac N/A

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

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

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

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

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

       11. *-commutative N/A

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

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

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

       13. *-commutative N/A

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

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

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

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

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

       16. *-lowering-*.f64 55.6%

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

   12. Applied egg-rr 55.6%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 5 \cdot 10^{-203}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \frac{M \cdot \frac{h \cdot M}{d}}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{D \cdot M}{d} \cdot \frac{D \cdot \left(h \cdot M\right)}{d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.4% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq -3 \cdot 10^{+65}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(M \cdot \left(D \cdot \frac{\frac{h \cdot M}{d}}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \frac{M \cdot \left(h \cdot \frac{M}{d}\right)}{d}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= h -3e+65)
   (* 0.25 (* D (* M (* D (/ (/ (* h M) d) d)))))
   (* 0.25 (* D (* D (/ (* M (* h (/ M d))) d))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (h <= -3e+65) {
		tmp = 0.25 * (D * (M * (D * (((h * M) / d) / d))));
	} else {
		tmp = 0.25 * (D * (D * ((M * (h * (M / d))) / d)));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (h <= (-3d+65)) then
        tmp = 0.25d0 * (d * (m * (d * (((h * m) / d_1) / d_1))))
    else
        tmp = 0.25d0 * (d * (d * ((m * (h * (m / d_1))) / d_1)))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (h <= -3e+65) {
		tmp = 0.25 * (D * (M * (D * (((h * M) / d) / d))));
	} else {
		tmp = 0.25 * (D * (D * ((M * (h * (M / d))) / d)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if h <= -3e+65:
		tmp = 0.25 * (D * (M * (D * (((h * M) / d) / d))))
	else:
		tmp = 0.25 * (D * (D * ((M * (h * (M / d))) / d)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (h <= -3e+65)
		tmp = Float64(0.25 * Float64(D * Float64(M * Float64(D * Float64(Float64(Float64(h * M) / d) / d)))));
	else
		tmp = Float64(0.25 * Float64(D * Float64(D * Float64(Float64(M * Float64(h * Float64(M / d))) / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (h <= -3e+65)
		tmp = 0.25 * (D * (M * (D * (((h * M) / d) / d))));
	else
		tmp = 0.25 * (D * (D * ((M * (h * (M / d))) / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[h, -3e+65], N[(0.25 * N[(D * N[(M * N[(D * N[(N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(D * N[(D * N[(N[(M * N[(h * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -3.0000000000000002e65

   1. Initial program 18.1%

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

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

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

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

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

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

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

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

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

   3. Simplified 23.7%

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

   5. Taylor expanded in c0 around -inf

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

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

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

      2. unpow2 N/A

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

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

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

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

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

   7. Simplified 18.5%

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

   8. Taylor expanded in c0 around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 36.6%

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

   10. Simplified 36.6%

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

       1. associate-/l* N/A

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

       2. associate-*l* N/A

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

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

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

       4. *-commutative N/A

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

       5. associate-/r* N/A

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

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

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

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

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. associate-/l* N/A

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

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

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

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

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

       13. *-commutative N/A

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

       14. *-lowering-*.f64 48.5%

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

   12. Applied egg-rr 48.5%

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

       1. *-commutative N/A

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

       2. associate-/l* N/A

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

       3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

       8. *-lowering-*.f64 62.5%

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

   14. Applied egg-rr 62.5%

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

   if -3.0000000000000002e65 < h

   1. Initial program 22.1%

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

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

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

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

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

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

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

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

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

   3. Simplified 21.1%

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

   5. Taylor expanded in c0 around -inf

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

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

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

      2. unpow2 N/A

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

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

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

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

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

   7. Simplified 16.8%

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

   8. Taylor expanded in c0 around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 36.2%

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

   10. Simplified 36.2%

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

       1. associate-/l* N/A

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

       2. associate-*l* N/A

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

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

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

       4. *-commutative N/A

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

       5. associate-/r* N/A

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

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

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

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

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. associate-/l* N/A

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

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

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

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

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

       13. *-commutative N/A

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

       14. *-lowering-*.f64 56.6%

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

   12. Applied egg-rr 56.6%

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

       1. *-commutative N/A

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

       2. associate-/l* N/A

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

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

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

       4. /-lowering-/.f64 57.5%

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

   14. Applied egg-rr 57.5%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -3 \cdot 10^{+65}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(M \cdot \left(D \cdot \frac{\frac{h \cdot M}{d}}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \frac{M \cdot \left(h \cdot \frac{M}{d}\right)}{d}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.0% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ 0.25 \cdot \frac{D \cdot \frac{M}{d}}{\frac{d}{D \cdot \left(h \cdot M\right)}} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (* 0.25 (/ (* D (/ M d)) (/ d (* D (* h M))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return 0.25 * ((D * (M / d)) / (d / (D * (h * 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
    code = 0.25d0 * ((d * (m / d_1)) / (d_1 / (d * (h * m))))
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return 0.25 * ((D * (M / d)) / (d / (D * (h * M))));
}

Python

def code(c0, w, h, D, d, M):
	return 0.25 * ((D * (M / d)) / (d / (D * (h * M))))

Julia

function code(c0, w, h, D, d, M)
	return Float64(0.25 * Float64(Float64(D * Float64(M / d)) / Float64(d / Float64(D * Float64(h * M)))))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = 0.25 * ((D * (M / d)) / (d / (D * (h * M))));
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(0.25 * N[(N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision] / N[(d / N[(D * N[(h * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 21.6%

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

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

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

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

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

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

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

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

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

3. Simplified 21.5%

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

5. Taylor expanded in c0 around -inf

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

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

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

   2. unpow2 N/A

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

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

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

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

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

7. Simplified 17.1%

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

8. Taylor expanded in c0 around 0

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

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

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

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

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

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

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

   4. unpow2 N/A

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

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

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

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

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

   7. unpow2 N/A

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

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

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 36.2%

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

10. Simplified 36.2%

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

    1. *-commutative N/A

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

    2. associate-*r* N/A

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

    3. *-commutative N/A

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

    4. associate-*r* N/A

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

    5. associate-*r* N/A

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

    6. *-commutative N/A

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

    7. associate-*r* N/A

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

    8. times-frac N/A

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

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

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

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

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

    11. *-commutative N/A

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

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

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

    13. *-commutative N/A

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

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

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

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

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

    16. *-lowering-*.f64 57.3%

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

12. Applied egg-rr 57.3%

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

    1. *-commutative N/A

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

    2. clear-num N/A

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

    3. un-div-inv N/A

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

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

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

    5. associate-/l* N/A

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

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

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

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

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

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

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

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

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

    10. *-lowering-*.f64 58.6%

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

14. Applied egg-rr 58.6%

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

15. Final simplification 58.6%

    \[\leadsto 0.25 \cdot \frac{D \cdot \frac{M}{d}}{\frac{d}{D \cdot \left(h \cdot M\right)}} \]
16. Add Preprocessing

Alternative 7: 49.4% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ 0.25 \cdot \left(D \cdot \left(D \cdot \frac{M \cdot \left(h \cdot \frac{M}{d}\right)}{d}\right)\right) \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (* 0.25 (* D (* D (/ (* M (* h (/ M d))) d)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return 0.25 * (D * (D * ((M * (h * (M / d))) / d)));
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = 0.25d0 * (d * (d * ((m * (h * (m / d_1))) / d_1)))
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return 0.25 * (D * (D * ((M * (h * (M / d))) / d)));
}

Python

def code(c0, w, h, D, d, M):
	return 0.25 * (D * (D * ((M * (h * (M / d))) / d)))

Julia

function code(c0, w, h, D, d, M)
	return Float64(0.25 * Float64(D * Float64(D * Float64(Float64(M * Float64(h * Float64(M / d))) / d))))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = 0.25 * (D * (D * ((M * (h * (M / d))) / d)));
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(0.25 * N[(D * N[(D * N[(N[(M * N[(h * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 21.6%

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

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

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

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

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

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

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

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

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

3. Simplified 21.5%

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

5. Taylor expanded in c0 around -inf

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

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

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

   2. unpow2 N/A

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

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

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

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

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

7. Simplified 17.1%

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

8. Taylor expanded in c0 around 0

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

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

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

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

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

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

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

   4. unpow2 N/A

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

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

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

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

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\left({M}^{2}\right), h\right)\right), \left({d}^{2}\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\left(M \cdot M\right), h\right)\right), \left({d}^{2}\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right), \left({d}^{2}\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right), \left(d \cdot \color{blue}{d}\right)\right)\right) \]

   10. *-lowering-*.f64 36.2%

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right)\right) \]

10. Simplified 36.2%

    \[\leadsto \color{blue}{0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot h\right)}{d \cdot d}} \]
11. Step-by-step derivation

    1. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left(\left(D \cdot D\right) \cdot \color{blue}{\frac{\left(M \cdot M\right) \cdot h}{d \cdot d}}\right)\right) \]

    2. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \color{blue}{\left(D \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)}\right)\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \color{blue}{\left(D \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)}\right)\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(D \cdot \frac{h \cdot \left(M \cdot M\right)}{\color{blue}{d} \cdot d}\right)\right)\right) \]

    5. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(D \cdot \frac{\frac{h \cdot \left(M \cdot M\right)}{d}}{\color{blue}{d}}\right)\right)\right) \]

    6. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \color{blue}{\left(\frac{\frac{h \cdot \left(M \cdot M\right)}{d}}{d}\right)}\right)\right)\right) \]

    7. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\left(\frac{h \cdot \left(M \cdot M\right)}{d}\right), \color{blue}{d}\right)\right)\right)\right) \]

    8. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\left(\frac{\left(h \cdot M\right) \cdot M}{d}\right), d\right)\right)\right)\right) \]

    9. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\left(\frac{M \cdot \left(h \cdot M\right)}{d}\right), d\right)\right)\right)\right) \]

    10. associate-/l* N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\left(M \cdot \frac{h \cdot M}{d}\right), d\right)\right)\right)\right) \]

    11. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \left(\frac{h \cdot M}{d}\right)\right), d\right)\right)\right)\right) \]

    12. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{/.f64}\left(\left(h \cdot M\right), d\right)\right), d\right)\right)\right)\right) \]

    13. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{/.f64}\left(\left(M \cdot h\right), d\right)\right), d\right)\right)\right)\right) \]

    14. *-lowering-*.f64 55.5%

        \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, h\right), d\right)\right), d\right)\right)\right)\right) \]

12. Applied egg-rr 55.5%

    \[\leadsto 0.25 \cdot \color{blue}{\left(D \cdot \left(D \cdot \frac{M \cdot \frac{M \cdot h}{d}}{d}\right)\right)} \]
13. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \left(\frac{h \cdot M}{d}\right)\right), d\right)\right)\right)\right) \]

    2. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \left(h \cdot \frac{M}{d}\right)\right), d\right)\right)\right)\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(h, \left(\frac{M}{d}\right)\right)\right), d\right)\right)\right)\right) \]

    4. /-lowering-/.f64 55.9%

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(h, \mathsf{/.f64}\left(M, d\right)\right)\right), d\right)\right)\right)\right) \]

14. Applied egg-rr 55.9%

    \[\leadsto 0.25 \cdot \left(D \cdot \left(D \cdot \frac{M \cdot \color{blue}{\left(h \cdot \frac{M}{d}\right)}}{d}\right)\right) \]
15. Add Preprocessing

Alternative 8: 47.7% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ 0.25 \cdot \left(D \cdot \left(D \cdot \left(M \cdot \frac{\frac{h \cdot M}{d}}{d}\right)\right)\right) \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (* 0.25 (* D (* D (* M (/ (/ (* h M) d) d))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return 0.25 * (D * (D * (M * (((h * M) / d) / d))));
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = 0.25d0 * (d * (d * (m * (((h * m) / d_1) / d_1))))
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return 0.25 * (D * (D * (M * (((h * M) / d) / d))));
}

Python

def code(c0, w, h, D, d, M):
	return 0.25 * (D * (D * (M * (((h * M) / d) / d))))

Julia

function code(c0, w, h, D, d, M)
	return Float64(0.25 * Float64(D * Float64(D * Float64(M * Float64(Float64(Float64(h * M) / d) / d)))))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = 0.25 * (D * (D * (M * (((h * M) / d) / d))));
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(0.25 * N[(D * N[(D * N[(M * N[(N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 21.6%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
2. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{c0}{2 \cdot w}\right), \color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \left(2 \cdot w\right)\right), \left(\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{+.f64}\left(\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right), \color{blue}{\left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}\right)\right) \]

3. Simplified 21.5%

   \[\leadsto \color{blue}{\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)}}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)} - M \cdot M}\right)} \]
4. Add Preprocessing

5. Taylor expanded in c0 around -inf

   \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({c0}^{2}\right), \color{blue}{\left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(c0 \cdot c0\right), \left(\color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \left(\color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(c0, c0\right), \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right), \color{blue}{\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)}\right)\right) \]

7. Simplified 17.1%

   \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \left(\frac{0}{w} + 0.25 \cdot \frac{\frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}}{c0 \cdot c0}\right)} \]

8. Taylor expanded in c0 around 0

   \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
9. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right)}\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right), \color{blue}{\left({d}^{2}\right)}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({D}^{2}\right), \left({M}^{2} \cdot h\right)\right), \left({\color{blue}{d}}^{2}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(D \cdot D\right), \left({M}^{2} \cdot h\right)\right), \left({d}^{2}\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left({M}^{2} \cdot h\right)\right), \left({d}^{2}\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\left({M}^{2}\right), h\right)\right), \left({d}^{2}\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\left(M \cdot M\right), h\right)\right), \left({d}^{2}\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right), \left({d}^{2}\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right), \left(d \cdot \color{blue}{d}\right)\right)\right) \]

   10. *-lowering-*.f64 36.2%

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right)\right) \]

10. Simplified 36.2%

    \[\leadsto \color{blue}{0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot h\right)}{d \cdot d}} \]
11. Step-by-step derivation

    1. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left(\left(D \cdot D\right) \cdot \color{blue}{\frac{\left(M \cdot M\right) \cdot h}{d \cdot d}}\right)\right) \]

    2. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left(D \cdot \color{blue}{\left(D \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)}\right)\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \color{blue}{\left(D \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)}\right)\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(D \cdot \frac{h \cdot \left(M \cdot M\right)}{\color{blue}{d} \cdot d}\right)\right)\right) \]

    5. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \left(D \cdot \frac{\frac{h \cdot \left(M \cdot M\right)}{d}}{\color{blue}{d}}\right)\right)\right) \]

    6. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \color{blue}{\left(\frac{\frac{h \cdot \left(M \cdot M\right)}{d}}{d}\right)}\right)\right)\right) \]

    7. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\left(\frac{h \cdot \left(M \cdot M\right)}{d}\right), \color{blue}{d}\right)\right)\right)\right) \]

    8. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\left(\frac{\left(h \cdot M\right) \cdot M}{d}\right), d\right)\right)\right)\right) \]

    9. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\left(\frac{M \cdot \left(h \cdot M\right)}{d}\right), d\right)\right)\right)\right) \]

    10. associate-/l* N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\left(M \cdot \frac{h \cdot M}{d}\right), d\right)\right)\right)\right) \]

    11. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \left(\frac{h \cdot M}{d}\right)\right), d\right)\right)\right)\right) \]

    12. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{/.f64}\left(\left(h \cdot M\right), d\right)\right), d\right)\right)\right)\right) \]

    13. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{/.f64}\left(\left(M \cdot h\right), d\right)\right), d\right)\right)\right)\right) \]

    14. *-lowering-*.f64 55.5%

        \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, h\right), d\right)\right), d\right)\right)\right)\right) \]

12. Applied egg-rr 55.5%

    \[\leadsto 0.25 \cdot \color{blue}{\left(D \cdot \left(D \cdot \frac{M \cdot \frac{M \cdot h}{d}}{d}\right)\right)} \]
13. Step-by-step derivation

    1. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(M \cdot \color{blue}{\frac{\frac{M \cdot h}{d}}{d}}\right)\right)\right)\right) \]

    2. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(\frac{\frac{M \cdot h}{d}}{d} \cdot \color{blue}{M}\right)\right)\right)\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\left(\frac{\frac{M \cdot h}{d}}{d}\right), \color{blue}{M}\right)\right)\right)\right) \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot h}{d}\right), d\right), M\right)\right)\right)\right) \]

    5. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(M \cdot h\right), d\right), d\right), M\right)\right)\right)\right) \]

    6. *-lowering-*.f64 53.9%

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, h\right), d\right), d\right), M\right)\right)\right)\right) \]

14. Applied egg-rr 53.9%

    \[\leadsto 0.25 \cdot \left(D \cdot \left(D \cdot \color{blue}{\left(\frac{\frac{M \cdot h}{d}}{d} \cdot M\right)}\right)\right) \]

15. Final simplification 53.9%

    \[\leadsto 0.25 \cdot \left(D \cdot \left(D \cdot \left(M \cdot \frac{\frac{h \cdot M}{d}}{d}\right)\right)\right) \]
16. Add Preprocessing

Alternative 9: 34.1% 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 21.6%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
2. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{c0}{2 \cdot w}\right), \color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \left(2 \cdot w\right)\right), \left(\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{+.f64}\left(\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right), \color{blue}{\left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}\right)\right) \]

3. Simplified 21.5%

   \[\leadsto \color{blue}{\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)}}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)} - M \cdot M}\right)} \]
4. Add Preprocessing

5. Taylor expanded in c0 around -inf

   \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)}\right) \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\left(-1 \cdot c0\right) \cdot \color{blue}{\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right)\right) \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(\color{blue}{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right) \]

   3. distribute-lft1-in N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(\left(-1 + 1\right) \cdot \color{blue}{\frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(0 \cdot \frac{\color{blue}{{d}^{2}}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right) \]

   5. mul0-lft N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot 0\right)\right) \]

   6. distribute-lft-neg-in N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(\mathsf{neg}\left(c0 \cdot 0\right)\right)\right) \]

   7. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(c0 \cdot \color{blue}{\left(\mathsf{neg}\left(0\right)\right)}\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \left(c0 \cdot 0\right)\right) \]

   9. *-lowering-*.f64 30.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(c0, \mathsf{*.f64}\left(2, w\right)\right), \mathsf{*.f64}\left(c0, \color{blue}{0}\right)\right) \]

7. Simplified 30.8%

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]
8. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\frac{c0}{2 \cdot w} \cdot c0\right) \cdot \color{blue}{0} \]

   2. mul0-rgt 35.4%

      \[\leadsto 0 \]

9. Applied egg-rr 35.4%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024149 
(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))))))