Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.5% → 56.4%

Time: 21.5s

Alternatives: 12

Speedup: 156.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 24.5% 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: 56.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{\frac{c0 \cdot d}{\left(2 \cdot w\right) \cdot D}}{\frac{w \cdot \left(D \cdot \left(h \cdot 0.5\right)\right)}{c0 \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
        INFINITY)
     (/ (/ (* c0 d) (* (* 2.0 w) D)) (/ (* w (* D (* h 0.5))) (* c0 d)))
     0.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= ((double) INFINITY)) {
		tmp = ((c0 * d) / ((2.0 * w) * D)) / ((w * (D * (h * 0.5))) / (c0 * d));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = ((c0 * d) / ((2.0 * w) * D)) / ((w * (D * (h * 0.5))) / (c0 * d));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))) <= math.inf:
		tmp = ((c0 * d) / ((2.0 * w) * D)) / ((w * (D * (h * 0.5))) / (c0 * d))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M))))) <= Inf)
		tmp = Float64(Float64(Float64(c0 * d) / Float64(Float64(2.0 * w) * D)) / Float64(Float64(w * Float64(D * Float64(h * 0.5))) / Float64(c0 * d)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= Inf)
		tmp = ((c0 * d) / ((2.0 * w) * D)) / ((w * (D * (h * 0.5))) / (c0 * d));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(c0 * d), $MachinePrecision] / N[(N[(2.0 * w), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] / N[(N[(w * N[(D * N[(h * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]

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 75.9%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 70.8

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

   5. Simplified 70.8%

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

      1. associate-/r* N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      12. *-lowering-*.f64 69.7

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

   7. Applied egg-rr 69.7%

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

      1. associate-/r* N/A

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

      2. associate-/l/ N/A

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

      3. associate-*r/ N/A

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

      4. associate-/l/ N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. associate-*l/ N/A

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

      9. *-commutative N/A

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

      10. associate-*l/ N/A

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

      11. times-frac N/A

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

      12. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      17. associate-/l* N/A

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

      18. clear-num N/A

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

      19. un-div-inv N/A

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

      20. /-lowering-/.f64 N/A

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

      21. clear-num N/A

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

   9. Applied egg-rr 73.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\frac{d}{D} \cdot \left(\frac{d}{D} \cdot \frac{c0}{w \cdot \left(h \cdot 0.5\right)}\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

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

       2. associate-*l/ N/A

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

       3. clear-num N/A

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

       4. un-div-inv N/A

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

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

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

       6. frac-times N/A

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

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

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

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

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

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

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

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

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

       11. clear-num N/A

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

       12. associate-*l/ N/A

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

       13. frac-times N/A

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

       14. *-commutative N/A

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

       15. clear-num N/A

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

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

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

   11. Applied egg-rr 78.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. mul-1-neg N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. metadata-eval N/A

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

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

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{\left(-1 + 1\right)}\right) \]

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

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot \left(-1 + 1\right)\right)} \]

      13. metadata-eval 36.1

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

   5. Simplified 36.1%

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

      1. associate-*r* N/A

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

      2. mul0-rgt 41.7

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

   7. Applied egg-rr 41.7%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{\frac{c0 \cdot d}{\left(2 \cdot w\right) \cdot D}}{\frac{w \cdot \left(D \cdot \left(h \cdot 0.5\right)\right)}{c0 \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 41.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 1.2 \cdot 10^{-209}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c0 \cdot d\right) \cdot \frac{c0 \cdot d}{\left(2 \cdot w\right) \cdot D}}{\left(w \cdot D\right) \cdot \left(0 - h \cdot -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= M 1.2e-209)
   0.0
   (/
    (* (* c0 d) (/ (* c0 d) (* (* 2.0 w) D)))
    (* (* w D) (- 0.0 (* h -0.5))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 1.2e-209) {
		tmp = 0.0;
	} else {
		tmp = ((c0 * d) * ((c0 * d) / ((2.0 * w) * D))) / ((w * D) * (0.0 - (h * -0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 1.2d-209) then
        tmp = 0.0d0
    else
        tmp = ((c0 * d_1) * ((c0 * d_1) / ((2.0d0 * w) * d))) / ((w * d) * (0.0d0 - (h * (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 1.2e-209) {
		tmp = 0.0;
	} else {
		tmp = ((c0 * d) * ((c0 * d) / ((2.0 * w) * D))) / ((w * D) * (0.0 - (h * -0.5)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if M <= 1.2e-209:
		tmp = 0.0
	else:
		tmp = ((c0 * d) * ((c0 * d) / ((2.0 * w) * D))) / ((w * D) * (0.0 - (h * -0.5)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (M <= 1.2e-209)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(c0 * d) * Float64(Float64(c0 * d) / Float64(Float64(2.0 * w) * D))) / Float64(Float64(w * D) * Float64(0.0 - Float64(h * -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (M <= 1.2e-209)
		tmp = 0.0;
	else
		tmp = ((c0 * d) * ((c0 * d) / ((2.0 * w) * D))) / ((w * D) * (0.0 - (h * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[M, 1.2e-209], 0.0, N[(N[(N[(c0 * d), $MachinePrecision] * N[(N[(c0 * d), $MachinePrecision] / N[(N[(2.0 * w), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(w * D), $MachinePrecision] * N[(0.0 - N[(h * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.2000000000000001e-209

   1. Initial program 29.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. mul-1-neg N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. metadata-eval N/A

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

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

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{\left(-1 + 1\right)}\right) \]

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

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot \left(-1 + 1\right)\right)} \]

      13. metadata-eval 29.2

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

   5. Simplified 29.2%

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

      1. associate-*r* N/A

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

      2. mul0-rgt 33.3

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

   7. Applied egg-rr 33.3%

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

   if 1.2000000000000001e-209 < M

   1. Initial program 17.6%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 28.7

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

   5. Simplified 28.7%

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

      1. associate-/r* N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      12. *-lowering-*.f64 31.0

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

   7. Applied egg-rr 31.0%

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

      1. associate-/r* N/A

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

      2. associate-/l/ N/A

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

      3. associate-*r/ N/A

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

      4. associate-/l/ N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. associate-*l/ N/A

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

      9. *-commutative N/A

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

      10. associate-*l/ N/A

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

      11. times-frac N/A

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

      12. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      17. associate-/l* N/A

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

      18. clear-num N/A

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

      19. un-div-inv N/A

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

      20. /-lowering-/.f64 N/A

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

      21. clear-num N/A

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

   9. Applied egg-rr 38.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\frac{d}{D} \cdot \left(\frac{d}{D} \cdot \frac{c0}{w \cdot \left(h \cdot 0.5\right)}\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

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

       2. frac-times N/A

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

       3. associate-*r/ N/A

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

       4. *-commutative N/A

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

       5. frac-2neg N/A

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

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

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

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

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

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

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

       9. frac-times N/A

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

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

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

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

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

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

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

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

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

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

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

   11. Applied egg-rr 43.3%

       \[\leadsto \color{blue}{\frac{-\frac{c0 \cdot d}{\left(2 \cdot w\right) \cdot D} \cdot \left(c0 \cdot d\right)}{\left(w \cdot D\right) \cdot \left(h \cdot -0.5\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.2 \cdot 10^{-209}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c0 \cdot d\right) \cdot \frac{c0 \cdot d}{\left(2 \cdot w\right) \cdot D}}{\left(w \cdot D\right) \cdot \left(0 - h \cdot -0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 40.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 1.8 \cdot 10^{-209}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c0 \cdot d\right) \cdot \frac{c0 \cdot d}{\left(2 \cdot w\right) \cdot D}}{w \cdot \left(D \cdot \left(h \cdot 0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= M 1.8e-209)
   0.0
   (/ (* (* c0 d) (/ (* c0 d) (* (* 2.0 w) D))) (* w (* D (* h 0.5))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 1.8e-209) {
		tmp = 0.0;
	} else {
		tmp = ((c0 * d) * ((c0 * d) / ((2.0 * w) * D))) / (w * (D * (h * 0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 1.8d-209) then
        tmp = 0.0d0
    else
        tmp = ((c0 * d_1) * ((c0 * d_1) / ((2.0d0 * w) * d))) / (w * (d * (h * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 1.8e-209) {
		tmp = 0.0;
	} else {
		tmp = ((c0 * d) * ((c0 * d) / ((2.0 * w) * D))) / (w * (D * (h * 0.5)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if M <= 1.8e-209:
		tmp = 0.0
	else:
		tmp = ((c0 * d) * ((c0 * d) / ((2.0 * w) * D))) / (w * (D * (h * 0.5)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (M <= 1.8e-209)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(c0 * d) * Float64(Float64(c0 * d) / Float64(Float64(2.0 * w) * D))) / Float64(w * Float64(D * Float64(h * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (M <= 1.8e-209)
		tmp = 0.0;
	else
		tmp = ((c0 * d) * ((c0 * d) / ((2.0 * w) * D))) / (w * (D * (h * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[M, 1.8e-209], 0.0, N[(N[(N[(c0 * d), $MachinePrecision] * N[(N[(c0 * d), $MachinePrecision] / N[(N[(2.0 * w), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(w * N[(D * N[(h * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.80000000000000008e-209

   1. Initial program 29.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. mul-1-neg N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. metadata-eval N/A

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

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

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{\left(-1 + 1\right)}\right) \]

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

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot \left(-1 + 1\right)\right)} \]

      13. metadata-eval 29.2

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

   5. Simplified 29.2%

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

      1. associate-*r* N/A

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

      2. mul0-rgt 33.3

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

   7. Applied egg-rr 33.3%

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

   if 1.80000000000000008e-209 < M

   1. Initial program 17.6%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 28.7

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

   5. Simplified 28.7%

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

      1. associate-/r* N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      12. *-lowering-*.f64 31.0

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

   7. Applied egg-rr 31.0%

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

      1. associate-/r* N/A

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

      2. associate-/l/ N/A

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

      3. associate-*r/ N/A

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

      4. associate-/l/ N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. associate-*l/ N/A

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

      9. *-commutative N/A

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

      10. associate-*l/ N/A

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

      11. times-frac N/A

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

      12. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      17. associate-/l* N/A

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

      18. clear-num N/A

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

      19. un-div-inv N/A

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

      20. /-lowering-/.f64 N/A

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

      21. clear-num N/A

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

   9. Applied egg-rr 38.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\frac{d}{D} \cdot \left(\frac{d}{D} \cdot \frac{c0}{w \cdot \left(h \cdot 0.5\right)}\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

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

       2. frac-times N/A

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

       3. associate-*r/ N/A

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

       4. *-commutative N/A

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

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

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

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

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

       7. frac-times N/A

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

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

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

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

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

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

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

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

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

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

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

       13. *-commutative N/A

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

       14. associate-*l* N/A

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

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

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

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

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

       17. *-lowering-*.f64 42.4

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

   11. Applied egg-rr 42.4%

       \[\leadsto \color{blue}{\frac{\frac{c0 \cdot d}{\left(2 \cdot w\right) \cdot D} \cdot \left(c0 \cdot d\right)}{w \cdot \left(\left(h \cdot 0.5\right) \cdot D\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.8 \cdot 10^{-209}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c0 \cdot d\right) \cdot \frac{c0 \cdot d}{\left(2 \cdot w\right) \cdot D}}{w \cdot \left(D \cdot \left(h \cdot 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 39.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 5.1 \cdot 10^{-219}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot d}{D \cdot \left(w \cdot \left(D \cdot \left(h \cdot 0.5\right)\right)\right)} \cdot \left(d \cdot \frac{c0 \cdot 0.5}{w}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= M 5.1e-219)
   0.0
   (* (/ (* c0 d) (* D (* w (* D (* h 0.5))))) (* d (/ (* c0 0.5) w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 5.1e-219) {
		tmp = 0.0;
	} else {
		tmp = ((c0 * d) / (D * (w * (D * (h * 0.5))))) * (d * ((c0 * 0.5) / w));
	}
	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 <= 5.1d-219) then
        tmp = 0.0d0
    else
        tmp = ((c0 * d_1) / (d * (w * (d * (h * 0.5d0))))) * (d_1 * ((c0 * 0.5d0) / w))
    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 <= 5.1e-219) {
		tmp = 0.0;
	} else {
		tmp = ((c0 * d) / (D * (w * (D * (h * 0.5))))) * (d * ((c0 * 0.5) / w));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if M <= 5.1e-219:
		tmp = 0.0
	else:
		tmp = ((c0 * d) / (D * (w * (D * (h * 0.5))))) * (d * ((c0 * 0.5) / w))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (M <= 5.1e-219)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(c0 * d) / Float64(D * Float64(w * Float64(D * Float64(h * 0.5))))) * Float64(d * Float64(Float64(c0 * 0.5) / w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (M <= 5.1e-219)
		tmp = 0.0;
	else
		tmp = ((c0 * d) / (D * (w * (D * (h * 0.5))))) * (d * ((c0 * 0.5) / w));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[M, 5.1e-219], 0.0, N[(N[(N[(c0 * d), $MachinePrecision] / N[(D * N[(w * N[(D * N[(h * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d * N[(N[(c0 * 0.5), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 5.0999999999999998e-219

   1. Initial program 28.3%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. mul-1-neg N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. metadata-eval N/A

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

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

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{\left(-1 + 1\right)}\right) \]

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

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot \left(-1 + 1\right)\right)} \]

      13. metadata-eval 29.9

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

   5. Simplified 29.9%

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

      1. associate-*r* N/A

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

      2. mul0-rgt 34.1

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

   7. Applied egg-rr 34.1%

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

   if 5.0999999999999998e-219 < M

   1. Initial program 19.0%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 29.6

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

   5. Simplified 29.6%

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

      1. associate-/r* N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      12. *-lowering-*.f64 31.8

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

   7. Applied egg-rr 31.8%

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

      1. associate-/r* N/A

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

      2. associate-/l/ N/A

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

      3. associate-*r/ N/A

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

      4. associate-/l/ N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. associate-*l/ N/A

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

      9. *-commutative N/A

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

      10. associate-*l/ N/A

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

      11. times-frac N/A

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

      12. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      17. associate-/l* N/A

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

      18. clear-num N/A

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

      19. un-div-inv N/A

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

      20. /-lowering-/.f64 N/A

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

      21. clear-num N/A

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

   9. Applied egg-rr 38.7%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\frac{d}{D} \cdot \left(\frac{d}{D} \cdot \frac{c0}{w \cdot \left(h \cdot 0.5\right)}\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. associate-*l/ N/A

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

       3. *-commutative N/A

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

       4. *-commutative N/A

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

       5. associate-*l/ N/A

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

       6. associate-*l* N/A

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

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

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

   11. Applied egg-rr 41.0%

       \[\leadsto \color{blue}{\frac{c0 \cdot d}{D \cdot \left(w \cdot \left(\left(h \cdot 0.5\right) \cdot D\right)\right)} \cdot \left(d \cdot \frac{c0 \cdot 0.5}{w}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 5.1 \cdot 10^{-219}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot d}{D \cdot \left(w \cdot \left(D \cdot \left(h \cdot 0.5\right)\right)\right)} \cdot \left(d \cdot \frac{c0 \cdot 0.5}{w}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 39.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 7.6 \cdot 10^{-218}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(d \cdot \left(c0 \cdot \frac{c0 \cdot d}{D \cdot \left(w \cdot \left(D \cdot \left(h \cdot 0.5\right)\right)\right)}\right)\right) \cdot \frac{0.5}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= M 7.6e-218)
   0.0
   (* (* d (* c0 (/ (* c0 d) (* D (* w (* D (* h 0.5))))))) (/ 0.5 w))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 7.6e-218) {
		tmp = 0.0;
	} else {
		tmp = (d * (c0 * ((c0 * d) / (D * (w * (D * (h * 0.5))))))) * (0.5 / w);
	}
	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 <= 7.6d-218) then
        tmp = 0.0d0
    else
        tmp = (d_1 * (c0 * ((c0 * d_1) / (d * (w * (d * (h * 0.5d0))))))) * (0.5d0 / w)
    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 <= 7.6e-218) {
		tmp = 0.0;
	} else {
		tmp = (d * (c0 * ((c0 * d) / (D * (w * (D * (h * 0.5))))))) * (0.5 / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if M <= 7.6e-218:
		tmp = 0.0
	else:
		tmp = (d * (c0 * ((c0 * d) / (D * (w * (D * (h * 0.5))))))) * (0.5 / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (M <= 7.6e-218)
		tmp = 0.0;
	else
		tmp = Float64(Float64(d * Float64(c0 * Float64(Float64(c0 * d) / Float64(D * Float64(w * Float64(D * Float64(h * 0.5))))))) * Float64(0.5 / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (M <= 7.6e-218)
		tmp = 0.0;
	else
		tmp = (d * (c0 * ((c0 * d) / (D * (w * (D * (h * 0.5))))))) * (0.5 / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[M, 7.6e-218], 0.0, N[(N[(d * N[(c0 * N[(N[(c0 * d), $MachinePrecision] / N[(D * N[(w * N[(D * N[(h * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 / w), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 7.5999999999999997e-218

   1. Initial program 28.3%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. mul-1-neg N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. metadata-eval N/A

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

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

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{\left(-1 + 1\right)}\right) \]

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

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot \left(-1 + 1\right)\right)} \]

      13. metadata-eval 29.9

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

   5. Simplified 29.9%

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

      1. associate-*r* N/A

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

      2. mul0-rgt 34.1

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

   7. Applied egg-rr 34.1%

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

   if 7.5999999999999997e-218 < M

   1. Initial program 19.0%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 29.6

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

   5. Simplified 29.6%

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

      1. associate-/r* N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      12. *-lowering-*.f64 31.8

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

   7. Applied egg-rr 31.8%

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

      1. associate-/r* N/A

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

      2. associate-/l/ N/A

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

      3. associate-*r/ N/A

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

      4. associate-/l/ N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. associate-*l/ N/A

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

      9. *-commutative N/A

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

      10. associate-*l/ N/A

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

      11. times-frac N/A

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

      12. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      17. associate-/l* N/A

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

      18. clear-num N/A

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

      19. un-div-inv N/A

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

      20. /-lowering-/.f64 N/A

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

      21. clear-num N/A

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

   9. Applied egg-rr 38.7%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\frac{d}{D} \cdot \left(\frac{d}{D} \cdot \frac{c0}{w \cdot \left(h \cdot 0.5\right)}\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*l/ N/A

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

       2. *-commutative N/A

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

       3. *-commutative N/A

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

       4. clear-num N/A

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

       5. div-inv N/A

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

       6. div-inv N/A

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

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

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

   11. Applied egg-rr 42.7%

       \[\leadsto \color{blue}{\left(\left(c0 \cdot \frac{c0 \cdot d}{D \cdot \left(w \cdot \left(\left(h \cdot 0.5\right) \cdot D\right)\right)}\right) \cdot d\right) \cdot \frac{0.5}{w}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 7.6 \cdot 10^{-218}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(d \cdot \left(c0 \cdot \frac{c0 \cdot d}{D \cdot \left(w \cdot \left(D \cdot \left(h \cdot 0.5\right)\right)\right)}\right)\right) \cdot \frac{0.5}{w}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 38.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 9.5 \cdot 10^{-17}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c0 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{w}}{w \cdot D}}{h \cdot D}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= M 9.5e-17) 0.0 (/ (/ (* c0 (/ (* c0 (* d d)) w)) (* w D)) (* h D))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 9.5e-17) {
		tmp = 0.0;
	} else {
		tmp = ((c0 * ((c0 * (d * d)) / w)) / (w * 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) :: tmp
    if (m <= 9.5d-17) then
        tmp = 0.0d0
    else
        tmp = ((c0 * ((c0 * (d_1 * d_1)) / w)) / (w * d)) / (h * d)
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 9.5e-17) {
		tmp = 0.0;
	} else {
		tmp = ((c0 * ((c0 * (d * d)) / w)) / (w * D)) / (h * D);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if M <= 9.5e-17:
		tmp = 0.0
	else:
		tmp = ((c0 * ((c0 * (d * d)) / w)) / (w * D)) / (h * D)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (M <= 9.5e-17)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(c0 * Float64(Float64(c0 * Float64(d * d)) / w)) / Float64(w * D)) / Float64(h * D));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (M <= 9.5e-17)
		tmp = 0.0;
	else
		tmp = ((c0 * ((c0 * (d * d)) / w)) / (w * D)) / (h * D);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[M, 9.5e-17], 0.0, N[(N[(N[(c0 * N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision] / N[(w * D), $MachinePrecision]), $MachinePrecision] / N[(h * D), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 9.50000000000000029e-17

   1. Initial program 27.4%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. mul-1-neg N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. metadata-eval N/A

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

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

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{\left(-1 + 1\right)}\right) \]

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

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot \left(-1 + 1\right)\right)} \]

      13. metadata-eval 30.7

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

   5. Simplified 30.7%

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

      1. associate-*r* N/A

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

      2. mul0-rgt 35.0

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

   7. Applied egg-rr 35.0%

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

   if 9.50000000000000029e-17 < M

   1. Initial program 13.6%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 32.1

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

   5. Simplified 32.1%

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

      1. associate-/r* N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      12. *-lowering-*.f64 36.4

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

   7. Applied egg-rr 36.4%

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

   8. Taylor expanded in c0 around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

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

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

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

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

      17. unpow2 N/A

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

      18. associate-*l* N/A

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

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

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

      20. *-lowering-*.f64 36.0

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

   10. Simplified 36.0%

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

       1. associate-/r* N/A

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

       2. associate-*r* N/A

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

       3. associate-/r* N/A

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

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

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

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

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

       6. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

       12. *-lowering-*.f64 38.5

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

   12. Applied egg-rr 38.5%

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

4. Final simplification 35.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 9.5 \cdot 10^{-17}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c0 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{w}}{w \cdot D}}{h \cdot D}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 38.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 1.8 \cdot 10^{-15}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h \cdot \left(w \cdot D\right)}}{w \cdot D}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= M 1.8e-15) 0.0 (/ (/ (* c0 (* c0 (* d d))) (* h (* w D))) (* w D))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 1.8e-15) {
		tmp = 0.0;
	} else {
		tmp = ((c0 * (c0 * (d * d))) / (h * (w * D))) / (w * D);
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 1.8d-15) then
        tmp = 0.0d0
    else
        tmp = ((c0 * (c0 * (d_1 * d_1))) / (h * (w * d))) / (w * d)
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 1.8e-15) {
		tmp = 0.0;
	} else {
		tmp = ((c0 * (c0 * (d * d))) / (h * (w * D))) / (w * D);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if M <= 1.8e-15:
		tmp = 0.0
	else:
		tmp = ((c0 * (c0 * (d * d))) / (h * (w * D))) / (w * D)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (M <= 1.8e-15)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(c0 * Float64(c0 * Float64(d * d))) / Float64(h * Float64(w * D))) / Float64(w * D));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (M <= 1.8e-15)
		tmp = 0.0;
	else
		tmp = ((c0 * (c0 * (d * d))) / (h * (w * D))) / (w * D);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[M, 1.8e-15], 0.0, N[(N[(N[(c0 * N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(h * N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(w * D), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.8000000000000001e-15

   1. Initial program 27.4%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. mul-1-neg N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. metadata-eval N/A

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

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

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{\left(-1 + 1\right)}\right) \]

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

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot \left(-1 + 1\right)\right)} \]

      13. metadata-eval 30.7

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

   5. Simplified 30.7%

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

      1. associate-*r* N/A

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

      2. mul0-rgt 35.0

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

   7. Applied egg-rr 35.0%

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

   if 1.8000000000000001e-15 < M

   1. Initial program 13.6%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 32.1

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

   5. Simplified 32.1%

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

      1. associate-/r* N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      12. *-lowering-*.f64 36.4

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

   7. Applied egg-rr 36.4%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. frac-times N/A

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

      4. *-commutative N/A

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

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

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

   9. Applied egg-rr 38.5%

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

   10. Taylor expanded in c0 around 0

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

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

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

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

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

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

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

       6. unpow2 N/A

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

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

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

       8. *-commutative N/A

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

       9. associate-*l* N/A

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

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

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

       11. *-commutative N/A

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

       12. *-lowering-*.f64 38.5

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

   12. Simplified 38.5%

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

4. Final simplification 35.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.8 \cdot 10^{-15}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h \cdot \left(w \cdot D\right)}}{w \cdot D}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 38.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 1.5 \cdot 10^{-16}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c0 \cdot d\right) \cdot \left(c0 \cdot d\right)}{w \cdot \left(w \cdot \left(D \cdot \left(h \cdot D\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= M 1.5e-16) 0.0 (/ (* (* c0 d) (* c0 d)) (* w (* w (* D (* h D)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 1.5e-16) {
		tmp = 0.0;
	} else {
		tmp = ((c0 * d) * (c0 * d)) / (w * (w * (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) :: tmp
    if (m <= 1.5d-16) then
        tmp = 0.0d0
    else
        tmp = ((c0 * d_1) * (c0 * d_1)) / (w * (w * (d * (h * d))))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 1.5e-16) {
		tmp = 0.0;
	} else {
		tmp = ((c0 * d) * (c0 * d)) / (w * (w * (D * (h * D))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if M <= 1.5e-16:
		tmp = 0.0
	else:
		tmp = ((c0 * d) * (c0 * d)) / (w * (w * (D * (h * D))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (M <= 1.5e-16)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(c0 * d) * Float64(c0 * d)) / Float64(w * Float64(w * Float64(D * Float64(h * D)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (M <= 1.5e-16)
		tmp = 0.0;
	else
		tmp = ((c0 * d) * (c0 * d)) / (w * (w * (D * (h * D))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[M, 1.5e-16], 0.0, N[(N[(N[(c0 * d), $MachinePrecision] * N[(c0 * d), $MachinePrecision]), $MachinePrecision] / N[(w * N[(w * N[(D * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.49999999999999997e-16

   1. Initial program 27.4%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. mul-1-neg N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. metadata-eval N/A

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

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

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{\left(-1 + 1\right)}\right) \]

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

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot \left(-1 + 1\right)\right)} \]

      13. metadata-eval 30.7

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

   5. Simplified 30.7%

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

      1. associate-*r* N/A

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

      2. mul0-rgt 35.0

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

   7. Applied egg-rr 35.0%

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

   if 1.49999999999999997e-16 < M

   1. Initial program 13.6%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 32.1

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

   5. Simplified 32.1%

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

      1. associate-/r* N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      12. *-lowering-*.f64 36.4

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

   7. Applied egg-rr 36.4%

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

   8. Taylor expanded in c0 around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

         \[\leadsto \frac{c0 \cdot \color{blue}{\left(c0 \cdot {d}^{2}\right)}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \]

      6. unpow2 N/A

         \[\leadsto \frac{c0 \cdot \left(c0 \cdot \color{blue}{\left(d \cdot d\right)}\right)}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{c0 \cdot \left(c0 \cdot \color{blue}{\left(d \cdot d\right)}\right)}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \]

      8. associate-*r* N/A

         \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{\left({D}^{2} \cdot h\right) \cdot {w}^{2}}} \]

      9. unpow2 N/A

         \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left({D}^{2} \cdot h\right) \cdot \color{blue}{\left(w \cdot w\right)}} \]

      10. associate-*r* N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{\left(\left({D}^{2} \cdot h\right) \cdot w\right) \cdot w}} \]

      11. associate-*r* N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{\left({D}^{2} \cdot \left(h \cdot w\right)\right)} \cdot w} \]

      12. *-commutative N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{w \cdot \left({D}^{2} \cdot \left(h \cdot w\right)\right)}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{w \cdot \left({D}^{2} \cdot \left(h \cdot w\right)\right)}} \]

      14. associate-*r* N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \color{blue}{\left(\left({D}^{2} \cdot h\right) \cdot w\right)}} \]

      15. *-commutative N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \color{blue}{\left(w \cdot \left({D}^{2} \cdot h\right)\right)}} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \color{blue}{\left(w \cdot \left({D}^{2} \cdot h\right)\right)}} \]

      17. unpow2 N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \left(w \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot h\right)\right)} \]

      18. associate-*l* N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \left(w \cdot \color{blue}{\left(D \cdot \left(D \cdot h\right)\right)}\right)} \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \left(w \cdot \color{blue}{\left(D \cdot \left(D \cdot h\right)\right)}\right)} \]

      20. *-lowering-*.f64 36.0

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \left(w \cdot \left(D \cdot \color{blue}{\left(D \cdot h\right)}\right)\right)} \]

   10. Simplified 36.0%

       \[\leadsto \color{blue}{\frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \left(w \cdot \left(D \cdot \left(D \cdot h\right)\right)\right)}} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}}{w \cdot \left(w \cdot \left(D \cdot \left(D \cdot h\right)\right)\right)} \]

       2. unswap-sqr N/A

          \[\leadsto \frac{\color{blue}{\left(c0 \cdot d\right) \cdot \left(c0 \cdot d\right)}}{w \cdot \left(w \cdot \left(D \cdot \left(D \cdot h\right)\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(c0 \cdot d\right) \cdot \left(c0 \cdot d\right)}}{w \cdot \left(w \cdot \left(D \cdot \left(D \cdot h\right)\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(c0 \cdot d\right)} \cdot \left(c0 \cdot d\right)}{w \cdot \left(w \cdot \left(D \cdot \left(D \cdot h\right)\right)\right)} \]

       5. *-lowering-*.f64 38.4

          \[\leadsto \frac{\left(c0 \cdot d\right) \cdot \color{blue}{\left(c0 \cdot d\right)}}{w \cdot \left(w \cdot \left(D \cdot \left(D \cdot h\right)\right)\right)} \]

   12. Applied egg-rr 38.4%

       \[\leadsto \frac{\color{blue}{\left(c0 \cdot d\right) \cdot \left(c0 \cdot d\right)}}{w \cdot \left(w \cdot \left(D \cdot \left(D \cdot h\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.5 \cdot 10^{-16}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c0 \cdot d\right) \cdot \left(c0 \cdot d\right)}{w \cdot \left(w \cdot \left(D \cdot \left(h \cdot D\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 37.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 9 \cdot 10^{-16}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \left(\left(w \cdot D\right) \cdot \left(h \cdot D\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= M 9e-16) 0.0 (/ (* c0 (* c0 (* d d))) (* w (* (* w D) (* h D))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 9e-16) {
		tmp = 0.0;
	} else {
		tmp = (c0 * (c0 * (d * d))) / (w * ((w * 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) :: tmp
    if (m <= 9d-16) then
        tmp = 0.0d0
    else
        tmp = (c0 * (c0 * (d_1 * d_1))) / (w * ((w * d) * (h * d)))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 9e-16) {
		tmp = 0.0;
	} else {
		tmp = (c0 * (c0 * (d * d))) / (w * ((w * D) * (h * D)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if M <= 9e-16:
		tmp = 0.0
	else:
		tmp = (c0 * (c0 * (d * d))) / (w * ((w * D) * (h * D)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (M <= 9e-16)
		tmp = 0.0;
	else
		tmp = Float64(Float64(c0 * Float64(c0 * Float64(d * d))) / Float64(w * Float64(Float64(w * D) * Float64(h * D))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (M <= 9e-16)
		tmp = 0.0;
	else
		tmp = (c0 * (c0 * (d * d))) / (w * ((w * D) * (h * D)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[M, 9e-16], 0.0, N[(N[(c0 * N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(w * N[(N[(w * D), $MachinePrecision] * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 9.0000000000000003e-16

   1. Initial program 27.4%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around -inf

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\left(-1 \cdot c0\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)} \]

      2. distribute-lft1-in N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot c0\right) \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right) \]

      3. mul-1-neg N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(c0\right)\right)} \cdot \left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \]

      5. mul0-lft N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \color{blue}{0}\right) \]

      6. metadata-eval N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \color{blue}{\left(-1 + 1\right)}\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\mathsf{neg}\left(c0 \cdot \left(-1 + 1\right)\right)\right)} \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot \left(\mathsf{neg}\left(\left(-1 + 1\right)\right)\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \left(\mathsf{neg}\left(\color{blue}{0}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{0}\right) \]

      11. metadata-eval N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{\left(-1 + 1\right)}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot \left(-1 + 1\right)\right)} \]

      13. metadata-eval 30.7

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{0}\right) \]

   5. Simplified 30.7%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{c0}{2 \cdot w} \cdot c0\right) \cdot 0} \]

      2. mul0-rgt 35.0

         \[\leadsto \color{blue}{0} \]

   7. Applied egg-rr 35.0%

      \[\leadsto \color{blue}{0} \]

   if 9.0000000000000003e-16 < M

   1. Initial program 13.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot \left(h \cdot w\right)}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot \left(h \cdot w\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\color{blue}{2 \cdot \left(c0 \cdot {d}^{2}\right)}}{{D}^{2} \cdot \left(h \cdot w\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \color{blue}{\left(c0 \cdot {d}^{2}\right)}}{{D}^{2} \cdot \left(h \cdot w\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \color{blue}{\left(d \cdot d\right)}\right)}{{D}^{2} \cdot \left(h \cdot w\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \color{blue}{\left(d \cdot d\right)}\right)}{{D}^{2} \cdot \left(h \cdot w\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{\left(h \cdot w\right) \cdot {D}^{2}}} \]

      8. associate-*l* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{h \cdot \left(w \cdot {D}^{2}\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h \cdot \color{blue}{\left({D}^{2} \cdot w\right)}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{h \cdot \left({D}^{2} \cdot w\right)}} \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h \cdot \color{blue}{\left(w \cdot {D}^{2}\right)}} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h \cdot \color{blue}{\left(w \cdot {D}^{2}\right)}} \]

      13. unpow2 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h \cdot \left(w \cdot \color{blue}{\left(D \cdot D\right)}\right)} \]

      14. *-lowering-*.f64 32.1

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h \cdot \left(w \cdot \color{blue}{\left(D \cdot D\right)}\right)} \]

   5. Simplified 32.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h}}{w \cdot \left(D \cdot D\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h}}{\color{blue}{\left(w \cdot D\right) \cdot D}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{\frac{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h}}{w \cdot D}}{D}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{\frac{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h}}{w \cdot D}}{D}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\color{blue}{\frac{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h}}{w \cdot D}}}{D} \]

      6. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\frac{\frac{\color{blue}{\left(c0 \cdot \left(d \cdot d\right)\right) \cdot 2}}{h}}{w \cdot D}}{D} \]

      7. associate-/l* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\frac{\color{blue}{\left(c0 \cdot \left(d \cdot d\right)\right) \cdot \frac{2}{h}}}{w \cdot D}}{D} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\frac{\color{blue}{\left(c0 \cdot \left(d \cdot d\right)\right) \cdot \frac{2}{h}}}{w \cdot D}}{D} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\frac{\color{blue}{\left(c0 \cdot \left(d \cdot d\right)\right)} \cdot \frac{2}{h}}{w \cdot D}}{D} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\frac{\left(c0 \cdot \color{blue}{\left(d \cdot d\right)}\right) \cdot \frac{2}{h}}{w \cdot D}}{D} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\frac{\left(c0 \cdot \left(d \cdot d\right)\right) \cdot \color{blue}{\frac{2}{h}}}{w \cdot D}}{D} \]

      12. *-lowering-*.f64 36.4

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\frac{\left(c0 \cdot \left(d \cdot d\right)\right) \cdot \frac{2}{h}}{\color{blue}{w \cdot D}}}{D} \]

   7. Applied egg-rr 36.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{\frac{\left(c0 \cdot \left(d \cdot d\right)\right) \cdot \frac{2}{h}}{w \cdot D}}{D}} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\left(c0 \cdot c0\right)} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{c0 \cdot \left(c0 \cdot {d}^{2}\right)}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{c0 \cdot \left(c0 \cdot {d}^{2}\right)}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{c0 \cdot \color{blue}{\left(c0 \cdot {d}^{2}\right)}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \]

      6. unpow2 N/A

         \[\leadsto \frac{c0 \cdot \left(c0 \cdot \color{blue}{\left(d \cdot d\right)}\right)}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{c0 \cdot \left(c0 \cdot \color{blue}{\left(d \cdot d\right)}\right)}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \]

      8. associate-*r* N/A

         \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{\left({D}^{2} \cdot h\right) \cdot {w}^{2}}} \]

      9. unpow2 N/A

         \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left({D}^{2} \cdot h\right) \cdot \color{blue}{\left(w \cdot w\right)}} \]

      10. associate-*r* N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{\left(\left({D}^{2} \cdot h\right) \cdot w\right) \cdot w}} \]

      11. associate-*r* N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{\left({D}^{2} \cdot \left(h \cdot w\right)\right)} \cdot w} \]

      12. *-commutative N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{w \cdot \left({D}^{2} \cdot \left(h \cdot w\right)\right)}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{w \cdot \left({D}^{2} \cdot \left(h \cdot w\right)\right)}} \]

      14. associate-*r* N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \color{blue}{\left(\left({D}^{2} \cdot h\right) \cdot w\right)}} \]

      15. *-commutative N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \color{blue}{\left(w \cdot \left({D}^{2} \cdot h\right)\right)}} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \color{blue}{\left(w \cdot \left({D}^{2} \cdot h\right)\right)}} \]

      17. unpow2 N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \left(w \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot h\right)\right)} \]

      18. associate-*l* N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \left(w \cdot \color{blue}{\left(D \cdot \left(D \cdot h\right)\right)}\right)} \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \left(w \cdot \color{blue}{\left(D \cdot \left(D \cdot h\right)\right)}\right)} \]

      20. *-lowering-*.f64 36.0

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \left(w \cdot \left(D \cdot \color{blue}{\left(D \cdot h\right)}\right)\right)} \]

   10. Simplified 36.0%

       \[\leadsto \color{blue}{\frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \left(w \cdot \left(D \cdot \left(D \cdot h\right)\right)\right)}} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \color{blue}{\left(\left(w \cdot D\right) \cdot \left(D \cdot h\right)\right)}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \color{blue}{\left(\left(w \cdot D\right) \cdot \left(D \cdot h\right)\right)}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \left(\color{blue}{\left(w \cdot D\right)} \cdot \left(D \cdot h\right)\right)} \]

       4. *-lowering-*.f64 37.9

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \left(\left(w \cdot D\right) \cdot \color{blue}{\left(D \cdot h\right)}\right)} \]

   12. Applied egg-rr 37.9%

       \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \color{blue}{\left(\left(w \cdot D\right) \cdot \left(D \cdot h\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 9 \cdot 10^{-16}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \left(\left(w \cdot D\right) \cdot \left(h \cdot D\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 37.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 2.9 \cdot 10^{-15}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \left(w \cdot \left(D \cdot \left(h \cdot D\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= M 2.9e-15) 0.0 (/ (* c0 (* c0 (* d d))) (* w (* w (* D (* h D)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 2.9e-15) {
		tmp = 0.0;
	} else {
		tmp = (c0 * (c0 * (d * d))) / (w * (w * (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) :: tmp
    if (m <= 2.9d-15) then
        tmp = 0.0d0
    else
        tmp = (c0 * (c0 * (d_1 * d_1))) / (w * (w * (d * (h * d))))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 2.9e-15) {
		tmp = 0.0;
	} else {
		tmp = (c0 * (c0 * (d * d))) / (w * (w * (D * (h * D))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if M <= 2.9e-15:
		tmp = 0.0
	else:
		tmp = (c0 * (c0 * (d * d))) / (w * (w * (D * (h * D))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (M <= 2.9e-15)
		tmp = 0.0;
	else
		tmp = Float64(Float64(c0 * Float64(c0 * Float64(d * d))) / Float64(w * Float64(w * Float64(D * Float64(h * D)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (M <= 2.9e-15)
		tmp = 0.0;
	else
		tmp = (c0 * (c0 * (d * d))) / (w * (w * (D * (h * D))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[M, 2.9e-15], 0.0, N[(N[(c0 * N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(w * N[(w * N[(D * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.90000000000000019e-15

   1. Initial program 27.4%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around -inf

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\left(-1 \cdot c0\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)} \]

      2. distribute-lft1-in N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot c0\right) \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right) \]

      3. mul-1-neg N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(c0\right)\right)} \cdot \left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \]

      5. mul0-lft N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \color{blue}{0}\right) \]

      6. metadata-eval N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \color{blue}{\left(-1 + 1\right)}\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\mathsf{neg}\left(c0 \cdot \left(-1 + 1\right)\right)\right)} \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot \left(\mathsf{neg}\left(\left(-1 + 1\right)\right)\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \left(\mathsf{neg}\left(\color{blue}{0}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{0}\right) \]

      11. metadata-eval N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{\left(-1 + 1\right)}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot \left(-1 + 1\right)\right)} \]

      13. metadata-eval 30.7

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{0}\right) \]

   5. Simplified 30.7%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{c0}{2 \cdot w} \cdot c0\right) \cdot 0} \]

      2. mul0-rgt 35.0

         \[\leadsto \color{blue}{0} \]

   7. Applied egg-rr 35.0%

      \[\leadsto \color{blue}{0} \]

   if 2.90000000000000019e-15 < M

   1. Initial program 13.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot \left(h \cdot w\right)}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot \left(h \cdot w\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\color{blue}{2 \cdot \left(c0 \cdot {d}^{2}\right)}}{{D}^{2} \cdot \left(h \cdot w\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \color{blue}{\left(c0 \cdot {d}^{2}\right)}}{{D}^{2} \cdot \left(h \cdot w\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \color{blue}{\left(d \cdot d\right)}\right)}{{D}^{2} \cdot \left(h \cdot w\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \color{blue}{\left(d \cdot d\right)}\right)}{{D}^{2} \cdot \left(h \cdot w\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{\left(h \cdot w\right) \cdot {D}^{2}}} \]

      8. associate-*l* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{h \cdot \left(w \cdot {D}^{2}\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h \cdot \color{blue}{\left({D}^{2} \cdot w\right)}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{h \cdot \left({D}^{2} \cdot w\right)}} \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h \cdot \color{blue}{\left(w \cdot {D}^{2}\right)}} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h \cdot \color{blue}{\left(w \cdot {D}^{2}\right)}} \]

      13. unpow2 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h \cdot \left(w \cdot \color{blue}{\left(D \cdot D\right)}\right)} \]

      14. *-lowering-*.f64 32.1

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h \cdot \left(w \cdot \color{blue}{\left(D \cdot D\right)}\right)} \]

   5. Simplified 32.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h}}{w \cdot \left(D \cdot D\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h}}{\color{blue}{\left(w \cdot D\right) \cdot D}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{\frac{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h}}{w \cdot D}}{D}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{\frac{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h}}{w \cdot D}}{D}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\color{blue}{\frac{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h}}{w \cdot D}}}{D} \]

      6. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\frac{\frac{\color{blue}{\left(c0 \cdot \left(d \cdot d\right)\right) \cdot 2}}{h}}{w \cdot D}}{D} \]

      7. associate-/l* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\frac{\color{blue}{\left(c0 \cdot \left(d \cdot d\right)\right) \cdot \frac{2}{h}}}{w \cdot D}}{D} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\frac{\color{blue}{\left(c0 \cdot \left(d \cdot d\right)\right) \cdot \frac{2}{h}}}{w \cdot D}}{D} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\frac{\color{blue}{\left(c0 \cdot \left(d \cdot d\right)\right)} \cdot \frac{2}{h}}{w \cdot D}}{D} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\frac{\left(c0 \cdot \color{blue}{\left(d \cdot d\right)}\right) \cdot \frac{2}{h}}{w \cdot D}}{D} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\frac{\left(c0 \cdot \left(d \cdot d\right)\right) \cdot \color{blue}{\frac{2}{h}}}{w \cdot D}}{D} \]

      12. *-lowering-*.f64 36.4

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\frac{\left(c0 \cdot \left(d \cdot d\right)\right) \cdot \frac{2}{h}}{\color{blue}{w \cdot D}}}{D} \]

   7. Applied egg-rr 36.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{\frac{\left(c0 \cdot \left(d \cdot d\right)\right) \cdot \frac{2}{h}}{w \cdot D}}{D}} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\left(c0 \cdot c0\right)} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{c0 \cdot \left(c0 \cdot {d}^{2}\right)}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{c0 \cdot \left(c0 \cdot {d}^{2}\right)}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{c0 \cdot \color{blue}{\left(c0 \cdot {d}^{2}\right)}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \]

      6. unpow2 N/A

         \[\leadsto \frac{c0 \cdot \left(c0 \cdot \color{blue}{\left(d \cdot d\right)}\right)}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{c0 \cdot \left(c0 \cdot \color{blue}{\left(d \cdot d\right)}\right)}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \]

      8. associate-*r* N/A

         \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{\left({D}^{2} \cdot h\right) \cdot {w}^{2}}} \]

      9. unpow2 N/A

         \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left({D}^{2} \cdot h\right) \cdot \color{blue}{\left(w \cdot w\right)}} \]

      10. associate-*r* N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{\left(\left({D}^{2} \cdot h\right) \cdot w\right) \cdot w}} \]

      11. associate-*r* N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{\left({D}^{2} \cdot \left(h \cdot w\right)\right)} \cdot w} \]

      12. *-commutative N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{w \cdot \left({D}^{2} \cdot \left(h \cdot w\right)\right)}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{w \cdot \left({D}^{2} \cdot \left(h \cdot w\right)\right)}} \]

      14. associate-*r* N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \color{blue}{\left(\left({D}^{2} \cdot h\right) \cdot w\right)}} \]

      15. *-commutative N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \color{blue}{\left(w \cdot \left({D}^{2} \cdot h\right)\right)}} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \color{blue}{\left(w \cdot \left({D}^{2} \cdot h\right)\right)}} \]

      17. unpow2 N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \left(w \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot h\right)\right)} \]

      18. associate-*l* N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \left(w \cdot \color{blue}{\left(D \cdot \left(D \cdot h\right)\right)}\right)} \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \left(w \cdot \color{blue}{\left(D \cdot \left(D \cdot h\right)\right)}\right)} \]

      20. *-lowering-*.f64 36.0

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \left(w \cdot \left(D \cdot \color{blue}{\left(D \cdot h\right)}\right)\right)} \]

   10. Simplified 36.0%

       \[\leadsto \color{blue}{\frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \left(w \cdot \left(D \cdot \left(D \cdot h\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.9 \cdot 10^{-15}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \left(w \cdot \left(D \cdot \left(h \cdot D\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 36.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 1.55 \cdot 10^{-15}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \left(h \cdot D\right)\right) \cdot \left(w \cdot w\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= M 1.55e-15) 0.0 (* c0 (/ (* c0 (* d d)) (* (* D (* h D)) (* w w))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 1.55e-15) {
		tmp = 0.0;
	} else {
		tmp = c0 * ((c0 * (d * d)) / ((D * (h * D)) * (w * w)));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 1.55d-15) then
        tmp = 0.0d0
    else
        tmp = c0 * ((c0 * (d_1 * d_1)) / ((d * (h * d)) * (w * w)))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 1.55e-15) {
		tmp = 0.0;
	} else {
		tmp = c0 * ((c0 * (d * d)) / ((D * (h * D)) * (w * w)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if M <= 1.55e-15:
		tmp = 0.0
	else:
		tmp = c0 * ((c0 * (d * d)) / ((D * (h * D)) * (w * w)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (M <= 1.55e-15)
		tmp = 0.0;
	else
		tmp = Float64(c0 * Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(D * Float64(h * D)) * Float64(w * w))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (M <= 1.55e-15)
		tmp = 0.0;
	else
		tmp = c0 * ((c0 * (d * d)) / ((D * (h * D)) * (w * w)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[M, 1.55e-15], 0.0, N[(c0 * N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(D * N[(h * D), $MachinePrecision]), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.5499999999999999e-15

   1. Initial program 27.4%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around -inf

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\left(-1 \cdot c0\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)} \]

      2. distribute-lft1-in N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot c0\right) \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right) \]

      3. mul-1-neg N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(c0\right)\right)} \cdot \left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \]

      5. mul0-lft N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \color{blue}{0}\right) \]

      6. metadata-eval N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \color{blue}{\left(-1 + 1\right)}\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\mathsf{neg}\left(c0 \cdot \left(-1 + 1\right)\right)\right)} \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot \left(\mathsf{neg}\left(\left(-1 + 1\right)\right)\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \left(\mathsf{neg}\left(\color{blue}{0}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{0}\right) \]

      11. metadata-eval N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{\left(-1 + 1\right)}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot \left(-1 + 1\right)\right)} \]

      13. metadata-eval 30.7

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{0}\right) \]

   5. Simplified 30.7%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{c0}{2 \cdot w} \cdot c0\right) \cdot 0} \]

      2. mul0-rgt 35.0

         \[\leadsto \color{blue}{0} \]

   7. Applied egg-rr 35.0%

      \[\leadsto \color{blue}{0} \]

   if 1.5499999999999999e-15 < M

   1. Initial program 13.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot \left(h \cdot w\right)}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot \left(h \cdot w\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\color{blue}{2 \cdot \left(c0 \cdot {d}^{2}\right)}}{{D}^{2} \cdot \left(h \cdot w\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \color{blue}{\left(c0 \cdot {d}^{2}\right)}}{{D}^{2} \cdot \left(h \cdot w\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \color{blue}{\left(d \cdot d\right)}\right)}{{D}^{2} \cdot \left(h \cdot w\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \color{blue}{\left(d \cdot d\right)}\right)}{{D}^{2} \cdot \left(h \cdot w\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{\left(h \cdot w\right) \cdot {D}^{2}}} \]

      8. associate-*l* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{h \cdot \left(w \cdot {D}^{2}\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h \cdot \color{blue}{\left({D}^{2} \cdot w\right)}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{h \cdot \left({D}^{2} \cdot w\right)}} \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h \cdot \color{blue}{\left(w \cdot {D}^{2}\right)}} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h \cdot \color{blue}{\left(w \cdot {D}^{2}\right)}} \]

      13. unpow2 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h \cdot \left(w \cdot \color{blue}{\left(D \cdot D\right)}\right)} \]

      14. *-lowering-*.f64 32.1

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h \cdot \left(w \cdot \color{blue}{\left(D \cdot D\right)}\right)} \]

   5. Simplified 32.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h}}{w \cdot \left(D \cdot D\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h}}{\color{blue}{\left(w \cdot D\right) \cdot D}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{\frac{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h}}{w \cdot D}}{D}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{\frac{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h}}{w \cdot D}}{D}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\color{blue}{\frac{\frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{h}}{w \cdot D}}}{D} \]

      6. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\frac{\frac{\color{blue}{\left(c0 \cdot \left(d \cdot d\right)\right) \cdot 2}}{h}}{w \cdot D}}{D} \]

      7. associate-/l* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\frac{\color{blue}{\left(c0 \cdot \left(d \cdot d\right)\right) \cdot \frac{2}{h}}}{w \cdot D}}{D} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\frac{\color{blue}{\left(c0 \cdot \left(d \cdot d\right)\right) \cdot \frac{2}{h}}}{w \cdot D}}{D} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\frac{\color{blue}{\left(c0 \cdot \left(d \cdot d\right)\right)} \cdot \frac{2}{h}}{w \cdot D}}{D} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\frac{\left(c0 \cdot \color{blue}{\left(d \cdot d\right)}\right) \cdot \frac{2}{h}}{w \cdot D}}{D} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\frac{\left(c0 \cdot \left(d \cdot d\right)\right) \cdot \color{blue}{\frac{2}{h}}}{w \cdot D}}{D} \]

      12. *-lowering-*.f64 36.4

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\frac{\left(c0 \cdot \left(d \cdot d\right)\right) \cdot \frac{2}{h}}{\color{blue}{w \cdot D}}}{D} \]

   7. Applied egg-rr 36.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{\frac{\left(c0 \cdot \left(d \cdot d\right)\right) \cdot \frac{2}{h}}{w \cdot D}}{D}} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\left(c0 \cdot c0\right)} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{c0 \cdot \left(c0 \cdot {d}^{2}\right)}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{c0 \cdot \left(c0 \cdot {d}^{2}\right)}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{c0 \cdot \color{blue}{\left(c0 \cdot {d}^{2}\right)}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \]

      6. unpow2 N/A

         \[\leadsto \frac{c0 \cdot \left(c0 \cdot \color{blue}{\left(d \cdot d\right)}\right)}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{c0 \cdot \left(c0 \cdot \color{blue}{\left(d \cdot d\right)}\right)}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \]

      8. associate-*r* N/A

         \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{\left({D}^{2} \cdot h\right) \cdot {w}^{2}}} \]

      9. unpow2 N/A

         \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left({D}^{2} \cdot h\right) \cdot \color{blue}{\left(w \cdot w\right)}} \]

      10. associate-*r* N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{\left(\left({D}^{2} \cdot h\right) \cdot w\right) \cdot w}} \]

      11. associate-*r* N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{\left({D}^{2} \cdot \left(h \cdot w\right)\right)} \cdot w} \]

      12. *-commutative N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{w \cdot \left({D}^{2} \cdot \left(h \cdot w\right)\right)}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{w \cdot \left({D}^{2} \cdot \left(h \cdot w\right)\right)}} \]

      14. associate-*r* N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \color{blue}{\left(\left({D}^{2} \cdot h\right) \cdot w\right)}} \]

      15. *-commutative N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \color{blue}{\left(w \cdot \left({D}^{2} \cdot h\right)\right)}} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \color{blue}{\left(w \cdot \left({D}^{2} \cdot h\right)\right)}} \]

      17. unpow2 N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \left(w \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot h\right)\right)} \]

      18. associate-*l* N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \left(w \cdot \color{blue}{\left(D \cdot \left(D \cdot h\right)\right)}\right)} \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \left(w \cdot \color{blue}{\left(D \cdot \left(D \cdot h\right)\right)}\right)} \]

      20. *-lowering-*.f64 36.0

          \[\leadsto \frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \left(w \cdot \left(D \cdot \color{blue}{\left(D \cdot h\right)}\right)\right)} \]

   10. Simplified 36.0%

       \[\leadsto \color{blue}{\frac{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{w \cdot \left(w \cdot \left(D \cdot \left(D \cdot h\right)\right)\right)}} \]
   11. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \color{blue}{c0 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{w \cdot \left(w \cdot \left(D \cdot \left(D \cdot h\right)\right)\right)}} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{w \cdot \left(w \cdot \left(D \cdot \left(D \cdot h\right)\right)\right)} \cdot c0} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{w \cdot \left(w \cdot \left(D \cdot \left(D \cdot h\right)\right)\right)} \cdot c0} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{w \cdot \left(w \cdot \left(D \cdot \left(D \cdot h\right)\right)\right)}} \cdot c0 \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{w \cdot \left(w \cdot \left(D \cdot \left(D \cdot h\right)\right)\right)} \cdot c0 \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{w \cdot \left(w \cdot \left(D \cdot \left(D \cdot h\right)\right)\right)} \cdot c0 \]

       7. associate-*r* N/A

          \[\leadsto \frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(w \cdot w\right) \cdot \left(D \cdot \left(D \cdot h\right)\right)}} \cdot c0 \]

       8. *-commutative N/A

          \[\leadsto \frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(D \cdot \left(D \cdot h\right)\right) \cdot \left(w \cdot w\right)}} \cdot c0 \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(D \cdot \left(D \cdot h\right)\right) \cdot \left(w \cdot w\right)}} \cdot c0 \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(D \cdot \left(D \cdot h\right)\right)} \cdot \left(w \cdot w\right)} \cdot c0 \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \color{blue}{\left(D \cdot h\right)}\right) \cdot \left(w \cdot w\right)} \cdot c0 \]

       12. *-lowering-*.f64 33.9

           \[\leadsto \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \left(D \cdot h\right)\right) \cdot \color{blue}{\left(w \cdot w\right)}} \cdot c0 \]

   12. Applied egg-rr 33.9%

       \[\leadsto \color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \left(D \cdot h\right)\right) \cdot \left(w \cdot w\right)} \cdot c0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.55 \cdot 10^{-15}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \left(h \cdot D\right)\right) \cdot \left(w \cdot w\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 33.4% accurate, 156.0× 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 24.6%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
2. Add Preprocessing

3. Taylor expanded in c0 around -inf

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\left(-1 \cdot c0\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)} \]

   2. distribute-lft1-in N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot c0\right) \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right) \]

   3. mul-1-neg N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(c0\right)\right)} \cdot \left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \]

   5. mul0-lft N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \color{blue}{0}\right) \]

   6. metadata-eval N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\mathsf{neg}\left(c0\right)\right) \cdot \color{blue}{\left(-1 + 1\right)}\right) \]

   7. distribute-lft-neg-in N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\mathsf{neg}\left(c0 \cdot \left(-1 + 1\right)\right)\right)} \]

   8. distribute-rgt-neg-in N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot \left(\mathsf{neg}\left(\left(-1 + 1\right)\right)\right)\right)} \]

   9. metadata-eval N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \left(\mathsf{neg}\left(\color{blue}{0}\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{0}\right) \]

   11. metadata-eval N/A

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{\left(-1 + 1\right)}\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot \left(-1 + 1\right)\right)} \]

   13. metadata-eval 29.2

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{0}\right) \]

5. Simplified 29.2%

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{c0}{2 \cdot w} \cdot c0\right) \cdot 0} \]

   2. mul0-rgt 33.0

      \[\leadsto \color{blue}{0} \]

7. Applied egg-rr 33.0%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024197 
(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))))))