Henrywood and Agarwal, Equation (13)

Percentage Accurate: 25.6% → 69.2%

Time: 23.4s

Alternatives: 8

Speedup: 151.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 25.6% 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

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

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 d) (* D (* w h))))
        (t_1 (/ (* c0 (* d d)) (* (* D D) (* w h)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))
        INFINITY)
     (* h (fma t_0 t_0 (/ (* D (* (* M (* D M)) -0.25)) (* d d))))
     (* 0.25 (* (* (* D M) (/ (* D M) d)) (/ h d))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * d) / (D * (w * h));
	double t_1 = (c0 * (d * d)) / ((D * D) * (w * h));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = h * fma(t_0, t_0, ((D * ((M * (D * M)) * -0.25)) / (d * d)));
	} else {
		tmp = 0.25 * (((D * M) * ((D * M) / d)) * (h / d));
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * d) / Float64(D * Float64(w * h)))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(D * D) * Float64(w * h)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= Inf)
		tmp = Float64(h * fma(t_0, t_0, Float64(Float64(D * Float64(Float64(M * Float64(D * M)) * -0.25)) / Float64(d * d))));
	else
		tmp = Float64(0.25 * Float64(Float64(Float64(D * M) * Float64(Float64(D * M) / d)) * Float64(h / d)));
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * d), $MachinePrecision] / N[(D * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(h * N[(t$95$0 * t$95$0 + N[(N[(D * N[(N[(M * N[(D * M), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(N[(D * M), $MachinePrecision] * N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 79.1%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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. associate-/r* N/A

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

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

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

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

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

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

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

   3. Simplified 70.0%

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

   5. Taylor expanded in h around 0

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

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

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

   7. Simplified 63.2%

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

   8. Taylor expanded in h around inf

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

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

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

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

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

      3. associate-*r/ N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

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

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

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

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

   10. Simplified 68.5%

       \[\leadsto \color{blue}{h \cdot \left(\frac{-0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right)}{d \cdot d} + \frac{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(\left(w \cdot h\right) \cdot \left(w \cdot h\right)\right)}\right)} \]
   11. Step-by-step derivation

       1. +-commutative N/A

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

       2. unswap-sqr N/A

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

       3. unswap-sqr N/A

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

       4. times-frac N/A

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

       5. fma-define N/A

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

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

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

   12. Applied egg-rr 84.2%

       \[\leadsto h \cdot \color{blue}{\mathsf{fma}\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)}, \frac{d \cdot c0}{D \cdot \left(h \cdot w\right)}, \frac{D \cdot \left(\left(M \cdot \left(D \cdot M\right)\right) \cdot -0.25\right)}{d \cdot d}\right)} \]

   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. associate-/r* N/A

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

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

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

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

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

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

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

   3. Simplified 0.0%

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

   5. Taylor expanded in c0 around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

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

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

   7. Simplified 19.9%

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

   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 47.7%

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

   10. Simplified 47.7%

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

       1. associate-*r* N/A

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

       2. associate-*r* N/A

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

       3. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

       9. /-lowering-/.f64 61.4%

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

   12. Applied egg-rr 61.4%

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

       1. associate-*r* N/A

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

       2. unswap-sqr N/A

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

       3. associate-/l* N/A

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

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

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

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

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

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

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

       7. *-lowering-*.f64 69.4%

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

   14. Applied egg-rr 69.4%

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

4. Final simplification 73.9%

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

Alternative 2: 47.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := D \cdot \left(w \cdot h\right)\\ \mathbf{if}\;c0 \leq 6.2 \cdot 10^{-127}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(\frac{M}{d} \cdot \left(D \cdot \frac{M}{\frac{d}{h}}\right)\right)\right)\\ \mathbf{elif}\;c0 \leq 6.6 \cdot 10^{+25}:\\ \;\;\;\;h \cdot \left(\frac{D \cdot \left(\left(M \cdot \left(D \cdot M\right)\right) \cdot -0.25\right)}{d \cdot d} + \frac{d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)}{t\_0 \cdot t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(\left(D \cdot M\right) \cdot \frac{D \cdot M}{d}\right) \cdot \frac{h}{d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* D (* w h))))
   (if (<= c0 6.2e-127)
     (* 0.25 (* D (* (/ M d) (* D (/ M (/ d h))))))
     (if (<= c0 6.6e+25)
       (*
        h
        (+
         (/ (* D (* (* M (* D M)) -0.25)) (* d d))
         (/ (* d (* d (* c0 c0))) (* t_0 t_0))))
       (* 0.25 (* (* (* D M) (/ (* D M) d)) (/ h d)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = D * (w * h);
	double tmp;
	if (c0 <= 6.2e-127) {
		tmp = 0.25 * (D * ((M / d) * (D * (M / (d / h)))));
	} else if (c0 <= 6.6e+25) {
		tmp = h * (((D * ((M * (D * M)) * -0.25)) / (d * d)) + ((d * (d * (c0 * c0))) / (t_0 * t_0)));
	} else {
		tmp = 0.25 * (((D * M) * ((D * M) / d)) * (h / d));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d * (w * h)
    if (c0 <= 6.2d-127) then
        tmp = 0.25d0 * (d * ((m / d_1) * (d * (m / (d_1 / h)))))
    else if (c0 <= 6.6d+25) then
        tmp = h * (((d * ((m * (d * m)) * (-0.25d0))) / (d_1 * d_1)) + ((d_1 * (d_1 * (c0 * c0))) / (t_0 * t_0)))
    else
        tmp = 0.25d0 * (((d * m) * ((d * m) / d_1)) * (h / d_1))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = D * (w * h);
	double tmp;
	if (c0 <= 6.2e-127) {
		tmp = 0.25 * (D * ((M / d) * (D * (M / (d / h)))));
	} else if (c0 <= 6.6e+25) {
		tmp = h * (((D * ((M * (D * M)) * -0.25)) / (d * d)) + ((d * (d * (c0 * c0))) / (t_0 * t_0)));
	} else {
		tmp = 0.25 * (((D * M) * ((D * M) / d)) * (h / d));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = D * (w * h)
	tmp = 0
	if c0 <= 6.2e-127:
		tmp = 0.25 * (D * ((M / d) * (D * (M / (d / h)))))
	elif c0 <= 6.6e+25:
		tmp = h * (((D * ((M * (D * M)) * -0.25)) / (d * d)) + ((d * (d * (c0 * c0))) / (t_0 * t_0)))
	else:
		tmp = 0.25 * (((D * M) * ((D * M) / d)) * (h / d))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(D * Float64(w * h))
	tmp = 0.0
	if (c0 <= 6.2e-127)
		tmp = Float64(0.25 * Float64(D * Float64(Float64(M / d) * Float64(D * Float64(M / Float64(d / h))))));
	elseif (c0 <= 6.6e+25)
		tmp = Float64(h * Float64(Float64(Float64(D * Float64(Float64(M * Float64(D * M)) * -0.25)) / Float64(d * d)) + Float64(Float64(d * Float64(d * Float64(c0 * c0))) / Float64(t_0 * t_0))));
	else
		tmp = Float64(0.25 * Float64(Float64(Float64(D * M) * Float64(Float64(D * M) / d)) * Float64(h / d)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = D * (w * h);
	tmp = 0.0;
	if (c0 <= 6.2e-127)
		tmp = 0.25 * (D * ((M / d) * (D * (M / (d / h)))));
	elseif (c0 <= 6.6e+25)
		tmp = h * (((D * ((M * (D * M)) * -0.25)) / (d * d)) + ((d * (d * (c0 * c0))) / (t_0 * t_0)));
	else
		tmp = 0.25 * (((D * M) * ((D * M) / d)) * (h / d));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(D * N[(w * h), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c0, 6.2e-127], N[(0.25 * N[(D * N[(N[(M / d), $MachinePrecision] * N[(D * N[(M / N[(d / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c0, 6.6e+25], N[(h * N[(N[(N[(D * N[(N[(M * N[(D * M), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] + N[(N[(d * N[(d * N[(c0 * c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(N[(D * M), $MachinePrecision] * N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c0 < 6.2e-127

   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. associate-/r* N/A

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

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

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

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

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

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

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

   3. Simplified 20.3%

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

   5. Taylor expanded in c0 around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

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

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

   7. Simplified 16.1%

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

   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 42.4%

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

   10. Simplified 42.4%

       \[\leadsto \color{blue}{0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot h\right)}{d \cdot d}} \]
   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. *-lowering-*.f64 N/A

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

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

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

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

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

       9. *-commutative N/A

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

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

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

       11. *-lowering-*.f64 50.3%

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

   12. Applied egg-rr 50.3%

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

       1. *-commutative N/A

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

       2. times-frac N/A

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

       3. associate-*l* N/A

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

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

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

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

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

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

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

       7. associate-/l* N/A

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

       8. clear-num N/A

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

       9. un-div-inv N/A

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

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

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

       11. /-lowering-/.f64 59.0%

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

   14. Applied egg-rr 59.0%

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

   if 6.2e-127 < c0 < 6.6000000000000002e25

   1. Initial program 38.8%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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. associate-/r* N/A

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

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

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

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

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

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

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

   3. Simplified 35.1%

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

   5. Taylor expanded in h around 0

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

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

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

   7. Simplified 46.4%

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

   8. Taylor expanded in h around inf

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

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

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

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

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

      3. associate-*r/ N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

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

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

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

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

   10. Simplified 46.3%

       \[\leadsto \color{blue}{h \cdot \left(\frac{-0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right)}{d \cdot d} + \frac{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(\left(w \cdot h\right) \cdot \left(w \cdot h\right)\right)}\right)} \]
   11. Step-by-step derivation

       1. div-inv N/A

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

       2. fma-define N/A

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

       3. frac-2neg N/A

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

       4. distribute-frac-neg N/A

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

       5. fmm-undef N/A

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

       6. div-inv N/A

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

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

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

   12. Applied egg-rr 65.7%

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

   if 6.6000000000000002e25 < c0

   1. Initial program 23.1%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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. associate-/r* N/A

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

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

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

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

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

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

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

   3. Simplified 18.7%

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

   5. Taylor expanded in c0 around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

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

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

   7. Simplified 14.7%

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

   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 33.8%

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

   10. Simplified 33.8%

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

       1. associate-*r* N/A

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

       2. associate-*r* N/A

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

       3. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

       9. /-lowering-/.f64 49.1%

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

   12. Applied egg-rr 49.1%

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

       1. associate-*r* N/A

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

       2. unswap-sqr N/A

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

       3. associate-/l* N/A

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

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

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

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

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

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

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

       7. *-lowering-*.f64 56.7%

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

   14. Applied egg-rr 56.7%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq 6.2 \cdot 10^{-127}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(\frac{M}{d} \cdot \left(D \cdot \frac{M}{\frac{d}{h}}\right)\right)\right)\\ \mathbf{elif}\;c0 \leq 6.6 \cdot 10^{+25}:\\ \;\;\;\;h \cdot \left(\frac{D \cdot \left(\left(M \cdot \left(D \cdot M\right)\right) \cdot -0.25\right)}{d \cdot d} + \frac{d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)}{\left(D \cdot \left(w \cdot h\right)\right) \cdot \left(D \cdot \left(w \cdot h\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(\left(D \cdot M\right) \cdot \frac{D \cdot M}{d}\right) \cdot \frac{h}{d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 43.6% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \cdot M \leq 10^{-267}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \cdot M \leq 5 \cdot 10^{+116}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{d \cdot d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \frac{M \cdot \left(h \cdot M\right)}{d \cdot d}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= (* M M) 1e-267)
   0.0
   (if (<= (* M M) 5e+116)
     (* 0.25 (* D (* D (* (* M M) (/ h (* 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 ((M * M) <= 1e-267) {
		tmp = 0.0;
	} else if ((M * M) <= 5e+116) {
		tmp = 0.25 * (D * (D * ((M * M) * (h / (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 ((m * m) <= 1d-267) then
        tmp = 0.0d0
    else if ((m * m) <= 5d+116) then
        tmp = 0.25d0 * (d * (d * ((m * m) * (h / (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 ((M * M) <= 1e-267) {
		tmp = 0.0;
	} else if ((M * M) <= 5e+116) {
		tmp = 0.25 * (D * (D * ((M * M) * (h / (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 (M * M) <= 1e-267:
		tmp = 0.0
	elif (M * M) <= 5e+116:
		tmp = 0.25 * (D * (D * ((M * M) * (h / (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 (Float64(M * M) <= 1e-267)
		tmp = 0.0;
	elseif (Float64(M * M) <= 5e+116)
		tmp = Float64(0.25 * Float64(D * Float64(D * Float64(Float64(M * M) * Float64(h / Float64(d * d))))));
	else
		tmp = Float64(0.25 * Float64(D * Float64(D * Float64(Float64(M * Float64(h * M)) / Float64(d * d)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((M * M) <= 1e-267)
		tmp = 0.0;
	elseif ((M * M) <= 5e+116)
		tmp = 0.25 * (D * (D * ((M * M) * (h / (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[N[(M * M), $MachinePrecision], 1e-267], 0.0, If[LessEqual[N[(M * M), $MachinePrecision], 5e+116], N[(0.25 * N[(D * N[(D * N[(N[(M * M), $MachinePrecision] * N[(h / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(D * N[(D * N[(N[(M * N[(h * M), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 M M) < 9.9999999999999998e-268

   1. Initial program 31.7%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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. associate-/r* N/A

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

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

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

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

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

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

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

   3. Simplified 25.0%

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

   5. Taylor expanded in c0 around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

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

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

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

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

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 44.1%

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

   7. Simplified 44.1%

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

      1. associate-*r* N/A

         \[\leadsto \left(\frac{\frac{c0}{2}}{w} \cdot c0\right) \cdot \color{blue}{0} \]

      2. mul0-rgt 54.5%

         \[\leadsto 0 \]

   9. Applied egg-rr 54.5%

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

   if 9.9999999999999998e-268 < (*.f64 M M) < 5.00000000000000025e116

   1. Initial program 26.5%

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

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

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

      2. associate-/r* N/A

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

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

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

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

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

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

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

   3. Simplified 25.1%

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

   5. Taylor expanded in c0 around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

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

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

   7. Simplified 20.7%

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

   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 47.0%

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

   10. Simplified 47.0%

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

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

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

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

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

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

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

       9. *-commutative N/A

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

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

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

       11. *-lowering-*.f64 48.6%

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

   12. Applied egg-rr 48.6%

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

       1. associate-*r* N/A

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

       2. associate-/l* N/A

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

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

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

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

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

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

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

       6. *-lowering-*.f64 55.2%

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

   14. Applied egg-rr 55.2%

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

   if 5.00000000000000025e116 < (*.f64 M M)

   1. Initial program 13.4%

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

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

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

      2. associate-/r* N/A

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

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

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

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

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

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

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

   3. Simplified 13.4%

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

   5. Taylor expanded in c0 around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

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

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

   7. Simplified 7.8%

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

   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 21.6%

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

   10. Simplified 21.6%

       \[\leadsto \color{blue}{0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot h\right)}{d \cdot d}} \]
   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. *-lowering-*.f64 N/A

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

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

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

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

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

       9. *-commutative N/A

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

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

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

       11. *-lowering-*.f64 42.1%

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

   12. Applied egg-rr 42.1%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot M \leq 10^{-267}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \cdot M \leq 5 \cdot 10^{+116}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{d \cdot d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \frac{M \cdot \left(h \cdot M\right)}{d \cdot d}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 42.0% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \cdot M \leq 1.2 \cdot 10^{-266}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{d \cdot d}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= (* M M) 1.2e-266) 0.0 (* 0.25 (* D (* D (* (* M M) (/ h (* d d))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((M * M) <= 1.2e-266) {
		tmp = 0.0;
	} else {
		tmp = 0.25 * (D * (D * ((M * M) * (h / (d * d)))));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((m * m) <= 1.2d-266) then
        tmp = 0.0d0
    else
        tmp = 0.25d0 * (d * (d * ((m * m) * (h / (d_1 * d_1)))))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((M * M) <= 1.2e-266) {
		tmp = 0.0;
	} else {
		tmp = 0.25 * (D * (D * ((M * M) * (h / (d * d)))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (M * M) <= 1.2e-266:
		tmp = 0.0
	else:
		tmp = 0.25 * (D * (D * ((M * M) * (h / (d * d)))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (Float64(M * M) <= 1.2e-266)
		tmp = 0.0;
	else
		tmp = Float64(0.25 * Float64(D * Float64(D * Float64(Float64(M * M) * Float64(h / Float64(d * d))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((M * M) <= 1.2e-266)
		tmp = 0.0;
	else
		tmp = 0.25 * (D * (D * ((M * M) * (h / (d * d)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[N[(M * M), $MachinePrecision], 1.2e-266], 0.0, N[(0.25 * N[(D * N[(D * N[(N[(M * M), $MachinePrecision] * N[(h / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M M) < 1.2e-266

   1. Initial program 31.7%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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. associate-/r* N/A

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

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

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

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

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

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

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

   3. Simplified 25.0%

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

   5. Taylor expanded in c0 around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

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

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

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

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

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 44.1%

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

   7. Simplified 44.1%

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

      1. associate-*r* N/A

         \[\leadsto \left(\frac{\frac{c0}{2}}{w} \cdot c0\right) \cdot \color{blue}{0} \]

      2. mul0-rgt 54.5%

         \[\leadsto 0 \]

   9. Applied egg-rr 54.5%

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

   if 1.2e-266 < (*.f64 M M)

   1. Initial program 20.1%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. 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. associate-/r* N/A

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

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

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

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

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

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

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

   3. Simplified 19.4%

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

   5. Taylor expanded in c0 around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

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

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

   7. Simplified 14.4%

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

   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 34.5%

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

   10. Simplified 34.5%

       \[\leadsto \color{blue}{0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot h\right)}{d \cdot d}} \]
   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. *-lowering-*.f64 N/A

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

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

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

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

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

       9. *-commutative N/A

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

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

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

       11. *-lowering-*.f64 45.4%

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

   12. Applied egg-rr 45.4%

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

       1. associate-*r* N/A

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

       2. associate-/l* N/A

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

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

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

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

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

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

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

       6. *-lowering-*.f64 42.8%

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

   14. Applied egg-rr 42.8%

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

Alternative 5: 49.8% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq 3.3 \cdot 10^{-163}:\\ \;\;\;\;0.25 \cdot \left(\left(\left(D \cdot M\right) \cdot \frac{D \cdot M}{d}\right) \cdot \frac{h}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \frac{\frac{M}{d} \cdot \left(h \cdot M\right)}{d}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= h 3.3e-163)
   (* 0.25 (* (* (* D M) (/ (* D M) d)) (/ h d)))
   (* 0.25 (* D (* D (/ (* (/ M d) (* h M)) d))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (h <= 3.3e-163) {
		tmp = 0.25 * (((D * M) * ((D * M) / d)) * (h / d));
	} else {
		tmp = 0.25 * (D * (D * (((M / 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 (h <= 3.3d-163) then
        tmp = 0.25d0 * (((d * m) * ((d * m) / d_1)) * (h / d_1))
    else
        tmp = 0.25d0 * (d * (d * (((m / d_1) * (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 (h <= 3.3e-163) {
		tmp = 0.25 * (((D * M) * ((D * M) / d)) * (h / d));
	} else {
		tmp = 0.25 * (D * (D * (((M / d) * (h * M)) / d)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if h <= 3.3e-163:
		tmp = 0.25 * (((D * M) * ((D * M) / d)) * (h / d))
	else:
		tmp = 0.25 * (D * (D * (((M / d) * (h * M)) / d)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (h <= 3.3e-163)
		tmp = Float64(0.25 * Float64(Float64(Float64(D * M) * Float64(Float64(D * M) / d)) * Float64(h / d)));
	else
		tmp = Float64(0.25 * Float64(D * Float64(D * Float64(Float64(Float64(M / d) * Float64(h * M)) / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (h <= 3.3e-163)
		tmp = 0.25 * (((D * M) * ((D * M) / d)) * (h / d));
	else
		tmp = 0.25 * (D * (D * (((M / d) * (h * M)) / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[h, 3.3e-163], N[(0.25 * N[(N[(N[(D * M), $MachinePrecision] * N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(D * N[(D * N[(N[(N[(M / d), $MachinePrecision] * N[(h * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 3.30000000000000001e-163

   1. Initial program 23.3%

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

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

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

      2. associate-/r* N/A

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

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

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

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

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

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

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

   3. Simplified 20.9%

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

   5. Taylor expanded in c0 around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

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

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

   7. Simplified 14.7%

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

   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 37.9%

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

   10. Simplified 37.9%

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

       1. associate-*r* N/A

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

       2. associate-*r* N/A

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

       3. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

       9. /-lowering-/.f64 49.5%

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

   12. Applied egg-rr 49.5%

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

       1. associate-*r* N/A

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

       2. unswap-sqr N/A

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

       3. associate-/l* N/A

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

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

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

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

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

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

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

       7. *-lowering-*.f64 58.1%

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

   14. Applied egg-rr 58.1%

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

   if 3.30000000000000001e-163 < h

   1. Initial program 25.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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. associate-/r* N/A

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

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

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

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

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

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

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

   3. Simplified 22.2%

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

   5. Taylor expanded in c0 around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

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

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

   7. Simplified 21.2%

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

   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 39.9%

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

   10. Simplified 39.9%

       \[\leadsto \color{blue}{0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot h\right)}{d \cdot d}} \]
   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. *-lowering-*.f64 N/A

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

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

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

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

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

       9. *-commutative N/A

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

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

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

       11. *-lowering-*.f64 48.4%

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

   12. Applied egg-rr 48.4%

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

       1. times-frac N/A

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

       2. associate-*r/ N/A

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

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

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

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

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

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

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

       6. *-lowering-*.f64 55.6%

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

   14. Applied egg-rr 55.6%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 3.3 \cdot 10^{-163}:\\ \;\;\;\;0.25 \cdot \left(\left(\left(D \cdot M\right) \cdot \frac{D \cdot M}{d}\right) \cdot \frac{h}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \frac{\frac{M}{d} \cdot \left(h \cdot M\right)}{d}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.5% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c0 \leq -4 \cdot 10^{-104}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(\frac{M}{d} \cdot \left(D \cdot \frac{M}{\frac{d}{h}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \frac{\frac{M}{d} \cdot \left(h \cdot M\right)}{d}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= c0 -4e-104)
   (* 0.25 (* D (* (/ M d) (* D (/ M (/ d h))))))
   (* 0.25 (* D (* D (/ (* (/ M d) (* h M)) d))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (c0 <= -4e-104) {
		tmp = 0.25 * (D * ((M / d) * (D * (M / (d / h)))));
	} else {
		tmp = 0.25 * (D * (D * (((M / 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 (c0 <= (-4d-104)) then
        tmp = 0.25d0 * (d * ((m / d_1) * (d * (m / (d_1 / h)))))
    else
        tmp = 0.25d0 * (d * (d * (((m / d_1) * (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 (c0 <= -4e-104) {
		tmp = 0.25 * (D * ((M / d) * (D * (M / (d / h)))));
	} else {
		tmp = 0.25 * (D * (D * (((M / d) * (h * M)) / d)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if c0 <= -4e-104:
		tmp = 0.25 * (D * ((M / d) * (D * (M / (d / h)))))
	else:
		tmp = 0.25 * (D * (D * (((M / d) * (h * M)) / d)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (c0 <= -4e-104)
		tmp = Float64(0.25 * Float64(D * Float64(Float64(M / d) * Float64(D * Float64(M / Float64(d / h))))));
	else
		tmp = Float64(0.25 * Float64(D * Float64(D * Float64(Float64(Float64(M / d) * Float64(h * M)) / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (c0 <= -4e-104)
		tmp = 0.25 * (D * ((M / d) * (D * (M / (d / h)))));
	else
		tmp = 0.25 * (D * (D * (((M / d) * (h * M)) / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[c0, -4e-104], N[(0.25 * N[(D * N[(N[(M / d), $MachinePrecision] * N[(D * N[(M / N[(d / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(D * N[(D * N[(N[(N[(M / d), $MachinePrecision] * N[(h * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c0 < -3.99999999999999971e-104

   1. Initial program 25.5%

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

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

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

      2. associate-/r* N/A

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

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

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

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

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

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

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

   3. Simplified 23.1%

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

   5. Taylor expanded in c0 around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

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

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

   7. Simplified 17.4%

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

   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 32.0%

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

   10. Simplified 32.0%

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

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

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

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

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

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

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

       9. *-commutative N/A

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

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

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

       11. *-lowering-*.f64 42.3%

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

   12. Applied egg-rr 42.3%

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

       1. *-commutative N/A

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

       2. times-frac N/A

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

       3. associate-*l* N/A

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

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

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

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

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

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

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

       7. associate-/l* N/A

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

       8. clear-num N/A

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

       9. un-div-inv N/A

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

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

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

       11. /-lowering-/.f64 54.1%

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

   14. Applied egg-rr 54.1%

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

   if -3.99999999999999971e-104 < c0

   1. Initial program 23.4%

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

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

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

      2. associate-/r* N/A

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

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

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

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

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

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

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

   3. Simplified 20.5%

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

   5. Taylor expanded in c0 around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

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

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

   7. Simplified 16.7%

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

   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 41.8%

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

   10. Simplified 41.8%

       \[\leadsto \color{blue}{0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot h\right)}{d \cdot d}} \]
   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. *-lowering-*.f64 N/A

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

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

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

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

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

       9. *-commutative N/A

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

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

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

       11. *-lowering-*.f64 48.8%

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

   12. Applied egg-rr 48.8%

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

       1. times-frac N/A

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

       2. associate-*r/ N/A

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

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

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

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

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

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

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

       6. *-lowering-*.f64 56.9%

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

   14. Applied egg-rr 56.9%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -4 \cdot 10^{-104}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(\frac{M}{d} \cdot \left(D \cdot \frac{M}{\frac{d}{h}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \frac{\frac{M}{d} \cdot \left(h \cdot M\right)}{d}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.5% accurate, 10.1× speedup

Math

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

FPCore

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

C

double code(double c0, double w, double h, double D, double d, double M) {
	return 0.25 * (D * (D * (((M / d) * (h * M)) / 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 / d_1) * (h * m)) / 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 / d) * (h * M)) / d)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.1%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
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. associate-/r* N/A

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

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

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

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

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

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

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

3. Simplified 21.3%

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

5. Taylor expanded in c0 around -inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

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

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

   4. mul-1-neg N/A

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

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

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

7. Simplified 16.9%

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

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 38.6%

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

10. Simplified 38.6%

    \[\leadsto \color{blue}{0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot h\right)}{d \cdot d}} \]
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. *-lowering-*.f64 N/A

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

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

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

    6. associate-*l* N/A

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

    7. *-commutative N/A

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

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

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

    9. *-commutative N/A

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

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

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

    11. *-lowering-*.f64 46.7%

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

12. Applied egg-rr 46.7%

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

    1. times-frac N/A

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

    2. associate-*r/ N/A

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

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

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

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

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

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

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

    6. *-lowering-*.f64 54.1%

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

14. Applied egg-rr 54.1%

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

15. Final simplification 54.1%

    \[\leadsto 0.25 \cdot \left(D \cdot \left(D \cdot \frac{\frac{M}{d} \cdot \left(h \cdot M\right)}{d}\right)\right) \]
16. Add Preprocessing

Alternative 8: 34.2% accurate, 151.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (c0 w h D d M) :precision binary64 0.0)

C

double code(double c0, double w, double h, double D, double d, double M) {
	return 0.0;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = 0.0d0
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return 0.0;
}

Python

def code(c0, w, h, D, d, M):
	return 0.0

Julia

function code(c0, w, h, D, d, M)
	return 0.0
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = 0.0;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := 0.0

Derivation

1. Initial program 24.1%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
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. associate-/r* N/A

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

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

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

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

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

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

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

3. Simplified 21.3%

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

5. Taylor expanded in c0 around -inf

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. distribute-lft1-in N/A

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

   4. metadata-eval N/A

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

   5. mul0-lft N/A

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

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

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

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

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

   8. metadata-eval N/A

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

   9. *-lowering-*.f64 30.9%

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

7. Simplified 30.9%

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

   1. associate-*r* N/A

      \[\leadsto \left(\frac{\frac{c0}{2}}{w} \cdot c0\right) \cdot \color{blue}{0} \]

   2. mul0-rgt 36.7%

      \[\leadsto 0 \]

9. Applied egg-rr 36.7%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024162 
(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))))))