Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.5% → 54.9%

Time: 19.1s

Alternatives: 11

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: 54.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_0 \cdot \left(\left(2 \cdot \frac{c0 \cdot d}{w \cdot \left(h \cdot D\right)}\right) \cdot \frac{d}{D}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w))) (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M))))) INFINITY)
     (* t_0 (* (* 2.0 (/ (* c0 d) (* w (* h D)))) (/ d D)))
     0.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_0 * ((2.0 * ((c0 * d) / (w * (h * D)))) * (d / 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 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * ((2.0 * ((c0 * d) / (w * (h * D)))) * (d / D));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= Inf)
		tmp = t_0 * ((2.0 * ((c0 * d) / (w * (h * D)))) * (d / D));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(N[(2.0 * N[(N[(c0 * d), $MachinePrecision] / N[(w * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / 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 76.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 77.6

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

   5. Simplified 77.6%

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. frac-times N/A

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

      6. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

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

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

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

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

      16. /-lowering-/.f64 80.9

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

   7. Applied egg-rr 80.9%

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

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

   1. Initial program 0.0%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. 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 43.5

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

   5. Simplified 43.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]
   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 49.6

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

   7. Applied egg-rr 49.6%

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

Alternative 2: 54.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_0 \cdot \left(\frac{d}{D} \cdot \left(2 \cdot \left(d \cdot \frac{c0}{\left(w \cdot h\right) \cdot D}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w))) (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M))))) INFINITY)
     (* t_0 (* (/ d D) (* 2.0 (* d (/ c0 (* (* w h) D))))))
     0.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_0 * ((d / D) * (2.0 * (d * (c0 / ((w * h) * 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 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * ((d / D) * (2.0 * (d * (c0 / ((w * h) * D)))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= Inf)
		tmp = t_0 * ((d / D) * (2.0 * (d * (c0 / ((w * h) * D)))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(N[(d / D), $MachinePrecision] * N[(2.0 * N[(d * N[(c0 / N[(N[(w * h), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 76.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 77.6

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

   5. Simplified 77.6%

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. frac-times N/A

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

      6. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

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

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

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

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

      16. /-lowering-/.f64 80.9

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

   7. Applied egg-rr 80.9%

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

   8. Taylor expanded in c0 around 0

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

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

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

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

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

      7. *-lowering-*.f64 79.9

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

   10. Simplified 79.9%

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

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

   1. Initial program 0.0%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. 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 43.5

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

   5. Simplified 43.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]
   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 49.6

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

   7. Applied egg-rr 49.6%

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

4. Final simplification 59.8%

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

Alternative 3: 54.7% accurate, 0.7× speedup

Math

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

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) (* (* w h) D)) (* 2.0 d)) (* c0 0.5)) (* w 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) / ((w * h) * D)) * (2.0 * d)) * (c0 * 0.5)) / (w * 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) / ((w * h) * D)) * (2.0 * d)) * (c0 * 0.5)) / (w * 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) / ((w * h) * D)) * (2.0 * d)) * (c0 * 0.5)) / (w * 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(Float64(Float64(c0 * d) / Float64(Float64(w * h) * D)) * Float64(2.0 * d)) * Float64(c0 * 0.5)) / Float64(w * 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) / ((w * h) * D)) * (2.0 * d)) * (c0 * 0.5)) / (w * 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[(N[(N[(c0 * d), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * N[(2.0 * d), $MachinePrecision]), $MachinePrecision] * N[(c0 * 0.5), $MachinePrecision]), $MachinePrecision] / N[(w * D), $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 76.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 77.6

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

   5. Simplified 77.6%

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. frac-times N/A

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

      6. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

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

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

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

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

      16. /-lowering-/.f64 80.9

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

   7. Applied egg-rr 80.9%

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

      1. times-frac N/A

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

      2. associate-*r/ N/A

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

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

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

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

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

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

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

      6. *-lowering-*.f64 80.1

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

   9. Applied egg-rr 80.1%

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

       1. *-commutative N/A

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

       2. associate-*r/ N/A

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

       3. associate-/r* N/A

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

       4. frac-times N/A

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

       5. *-commutative N/A

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

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

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

   11. Applied egg-rr 79.9%

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

   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 43.5

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

   5. Simplified 43.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]
   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 49.6

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

   7. Applied egg-rr 49.6%

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

4. Final simplification 59.8%

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

Alternative 4: 52.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c0 \cdot \left(d \cdot d\right)\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{t\_0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_1 \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_1 \cdot \frac{2 \cdot t\_0}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* c0 (* d d)))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (/ t_0 (* (* w h) (* D D)))))
   (if (<= (* t_1 (+ t_2 (sqrt (- (* t_2 t_2) (* M M))))) INFINITY)
     (* t_1 (/ (* 2.0 t_0) (* h (* w (* D D)))))
     0.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 * (d * d);
	double t_1 = c0 / (2.0 * w);
	double t_2 = t_0 / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_1 * ((2.0 * t_0) / (h * (w * (D * 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);
	double t_1 = c0 / (2.0 * w);
	double t_2 = t_0 / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_2 + Math.sqrt(((t_2 * t_2) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_1 * ((2.0 * t_0) / (h * (w * (D * D))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 * (d * d)
	t_1 = c0 / (2.0 * w)
	t_2 = t_0 / ((w * h) * (D * D))
	tmp = 0
	if (t_1 * (t_2 + math.sqrt(((t_2 * t_2) - (M * M))))) <= math.inf:
		tmp = t_1 * ((2.0 * t_0) / (h * (w * (D * D))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 * Float64(d * d))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(t_0 / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_1 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_1 * Float64(Float64(2.0 * t_0) / Float64(h * Float64(w * Float64(D * 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);
	t_1 = c0 / (2.0 * w);
	t_2 = t_0 / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= Inf)
		tmp = t_1 * ((2.0 * t_0) / (h * (w * (D * D))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$1 * N[(N[(2.0 * t$95$0), $MachinePrecision] / N[(h * N[(w * N[(D * D), $MachinePrecision]), $MachinePrecision]), $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 76.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 77.6

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

   5. Simplified 77.6%

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

   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 43.5

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

   5. Simplified 43.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]
   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 49.6

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

   7. Applied egg-rr 49.6%

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

Alternative 5: 53.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_0 \cdot \left(\left(2 \cdot d\right) \cdot \frac{c0 \cdot d}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w))) (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M))))) INFINITY)
     (* t_0 (* (* 2.0 d) (/ (* c0 d) (* w (* h (* D D))))))
     0.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_0 * ((2.0 * d) * ((c0 * d) / (w * (h * (D * 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 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * ((2.0 * d) * ((c0 * d) / (w * (h * (D * D)))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= Inf)
		tmp = t_0 * ((2.0 * d) * ((c0 * d) / (w * (h * (D * D)))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(N[(2.0 * d), $MachinePrecision] * N[(N[(c0 * d), $MachinePrecision] / N[(w * N[(h * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 76.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 77.6

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

   5. Simplified 77.6%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. associate-*r/ N/A

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

      9. *-commutative N/A

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

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

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

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

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

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

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

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

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

      17. *-lowering-*.f64 76.5

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

   7. Applied egg-rr 76.5%

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

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

   1. Initial program 0.0%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. 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 43.5

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

   5. Simplified 43.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]
   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 49.6

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

   7. Applied egg-rr 49.6%

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

Alternative 6: 53.4% accurate, 0.7× speedup

Math

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

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 0.5) (/ (/ (* d (* 2.0 (* c0 d))) (* h (* D (* w D)))) w))
     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 * 0.5) * (((d * (2.0 * (c0 * d))) / (h * (D * (w * D)))) / w);
	} 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 * 0.5) * (((d * (2.0 * (c0 * d))) / (h * (D * (w * D)))) / w);
	} 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 * 0.5) * (((d * (2.0 * (c0 * d))) / (h * (D * (w * D)))) / w)
	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(c0 * 0.5) * Float64(Float64(Float64(d * Float64(2.0 * Float64(c0 * d))) / Float64(h * Float64(D * Float64(w * D)))) / w));
	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 * 0.5) * (((d * (2.0 * (c0 * d))) / (h * (D * (w * D)))) / w);
	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[(c0 * 0.5), $MachinePrecision] * N[(N[(N[(d * N[(2.0 * N[(c0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(h * N[(D * N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / w), $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 76.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 77.6

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

   5. Simplified 77.6%

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. frac-times N/A

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

      6. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

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

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

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

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

      16. /-lowering-/.f64 80.9

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

   7. Applied egg-rr 80.9%

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

   9. Applied egg-rr 74.3%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around -inf

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)} \]
   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 43.5

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

   5. Simplified 43.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]
   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 49.6

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

   7. Applied egg-rr 49.6%

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

Alternative 7: 53.4% accurate, 0.7× speedup

Math

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

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 (/ (/ (* c0 (* 2.0 (* d d))) (* w (* h D))) (* 2.0 (* w 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 * (((c0 * (2.0 * (d * d))) / (w * (h * D))) / (2.0 * (w * 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 * (((c0 * (2.0 * (d * d))) / (w * (h * D))) / (2.0 * (w * 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 * (((c0 * (2.0 * (d * d))) / (w * (h * D))) / (2.0 * (w * 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(c0 * Float64(Float64(Float64(c0 * Float64(2.0 * Float64(d * d))) / Float64(w * Float64(h * D))) / Float64(2.0 * Float64(w * 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 * (((c0 * (2.0 * (d * d))) / (w * (h * D))) / (2.0 * (w * 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[(c0 * N[(N[(N[(c0 * N[(2.0 * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(w * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[(w * D), $MachinePrecision]), $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 76.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 77.6

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

   5. Simplified 77.6%

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. frac-times N/A

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

      6. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

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

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

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

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

      16. /-lowering-/.f64 80.9

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

   7. Applied egg-rr 80.9%

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

      1. associate-*r/ N/A

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

      2. frac-times N/A

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

      3. associate-*r/ N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-/l/ N/A

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

      10. associate-/l* N/A

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

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

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

   9. Applied egg-rr 74.0%

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

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

   1. Initial program 0.0%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. 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 43.5

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

   5. Simplified 43.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]
   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 49.6

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

   7. Applied egg-rr 49.6%

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

4. Final simplification 57.8%

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

Alternative 8: 52.5% accurate, 0.7× speedup

Math

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

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 (/ (* c0 (* 2.0 (* d d))) (* (* 2.0 w) (* h (* D (* w 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 * ((c0 * (2.0 * (d * d))) / ((2.0 * w) * (h * (D * (w * 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 * ((c0 * (2.0 * (d * d))) / ((2.0 * w) * (h * (D * (w * 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 * ((c0 * (2.0 * (d * d))) / ((2.0 * w) * (h * (D * (w * 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(c0 * Float64(Float64(c0 * Float64(2.0 * Float64(d * d))) / Float64(Float64(2.0 * w) * Float64(h * Float64(D * Float64(w * 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 * ((c0 * (2.0 * (d * d))) / ((2.0 * w) * (h * (D * (w * 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[(c0 * N[(N[(c0 * N[(2.0 * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * w), $MachinePrecision] * N[(h * N[(D * N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 76.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 77.6

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

   5. Simplified 77.6%

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. frac-times N/A

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

      6. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

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

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

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

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

      16. /-lowering-/.f64 80.9

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

   7. Applied egg-rr 80.9%

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

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. associate-/l* N/A

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

      10. frac-times N/A

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

      11. associate-/l* N/A

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

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

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

   9. Applied egg-rr 73.2%

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

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

   1. Initial program 0.0%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. 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 43.5

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

   5. Simplified 43.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]
   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 49.6

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

   7. Applied egg-rr 49.6%

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

4. Final simplification 57.5%

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

Alternative 9: 52.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c0 \cdot \left(d \cdot d\right)\\ t_1 := \frac{t\_0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;c0 \cdot \frac{t\_0}{D \cdot \left(w \cdot \left(\left(w \cdot h\right) \cdot D\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* c0 (* d d))) (t_1 (/ t_0 (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))
        INFINITY)
     (* c0 (/ t_0 (* D (* w (* (* w h) D)))))
     0.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 * (d * d);
	double t_1 = t_0 / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = c0 * (t_0 / (D * (w * ((w * h) * 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);
	double t_1 = t_0 / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = c0 * (t_0 / (D * (w * ((w * h) * D))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

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

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 * Float64(d * d))
	t_1 = Float64(t_0 / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= Inf)
		tmp = Float64(c0 * Float64(t_0 / Float64(D * Float64(w * Float64(Float64(w * h) * 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);
	t_1 = t_0 / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= Inf)
		tmp = c0 * (t_0 / (D * (w * ((w * h) * D))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / 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$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(c0 * N[(t$95$0 / N[(D * N[(w * N[(N[(w * h), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $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 76.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 77.6

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

   5. Simplified 77.6%

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

   6. Taylor expanded in c0 around 0

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

      1. associate-/l* N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. unpow2 N/A

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 61.1

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

   8. Simplified 61.1%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

      8. associate-*l* N/A

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

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

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

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

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

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

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 72.1

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

   10. Applied egg-rr 72.1%

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

   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 43.5

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

   5. Simplified 43.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]
   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 49.6

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

   7. Applied egg-rr 49.6%

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

4. Final simplification 57.2%

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

Alternative 10: 50.2% accurate, 0.7× speedup

Math

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

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 c0) (* d (/ d (* D (* w (* (* w h) 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 * c0) * (d * (d / (D * (w * ((w * h) * 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 * c0) * (d * (d / (D * (w * ((w * h) * 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 * c0) * (d * (d / (D * (w * ((w * h) * 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(c0 * c0) * Float64(d * Float64(d / Float64(D * Float64(w * Float64(Float64(w * h) * 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 * c0) * (d * (d / (D * (w * ((w * h) * 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[(c0 * c0), $MachinePrecision] * N[(d * N[(d / N[(D * N[(w * N[(N[(w * h), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 76.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 77.6

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

   5. Simplified 77.6%

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

   6. Taylor expanded in c0 around 0

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

      1. associate-/l* N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. unpow2 N/A

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 61.1

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

   8. Simplified 61.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. associate-*l* N/A

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

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

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

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

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 64.7

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

   10. Applied egg-rr 64.7%

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

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

   1. Initial program 0.0%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. 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 43.5

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

   5. Simplified 43.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]
   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 49.6

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

   7. Applied egg-rr 49.6%

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

4. Final simplification 54.7%

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

Alternative 11: 32.5% 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 25.7%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
2. 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 31.8

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

5. Simplified 31.8%

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]
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 36.0

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

7. Applied egg-rr 36.0%

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

Reproduce

herbie shell --seed 2024199 
(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))))))