Henrywood and Agarwal, Equation (13)

Percentage Accurate: 25.1% → 54.7%

Time: 39.6s

Alternatives: 7

Speedup: 21.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 25.1% 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: 54.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{d}{D}\right)}^{2}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\\ t_2 := \frac{c0}{2 \cdot w} \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-246}:\\ \;\;\;\;c0 \cdot \frac{\frac{1}{\frac{h}{2 \cdot \frac{c0 \cdot t\_0}{w}}}}{2 \cdot w}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;c0 \cdot \frac{\frac{{M}^{2}}{c0 \cdot \frac{t\_0}{w \cdot h} + \frac{\frac{c0}{h} \cdot \frac{{d}^{2}}{w}}{{D}^{2}}}}{2 \cdot w}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;c0 \cdot \frac{2 \cdot \frac{c0 \cdot {d}^{2}}{\left(w \cdot h\right) \cdot {D}^{2}}}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (pow (/ d D) 2.0))
        (t_1 (/ (* c0 (* d d)) (* (* D D) (* w h))))
        (t_2 (* (/ c0 (* 2.0 w)) (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))))
   (if (<= t_2 -1e-246)
     (* c0 (/ (/ 1.0 (/ h (* 2.0 (/ (* c0 t_0) w)))) (* 2.0 w)))
     (if (<= t_2 0.0)
       (*
        c0
        (/
         (/
          (pow M 2.0)
          (+
           (* c0 (/ t_0 (* w h)))
           (/ (* (/ c0 h) (/ (pow d 2.0) w)) (pow D 2.0))))
         (* 2.0 w)))
       (if (<= t_2 INFINITY)
         (*
          c0
          (/ (* 2.0 (/ (* c0 (pow d 2.0)) (* (* w h) (pow D 2.0)))) (* 2.0 w)))
         (* c0 (/ 0.0 (* 2.0 w))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = pow((d / D), 2.0);
	double t_1 = (c0 * (d * d)) / ((D * D) * (w * h));
	double t_2 = (c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= -1e-246) {
		tmp = c0 * ((1.0 / (h / (2.0 * ((c0 * t_0) / w)))) / (2.0 * w));
	} else if (t_2 <= 0.0) {
		tmp = c0 * ((pow(M, 2.0) / ((c0 * (t_0 / (w * h))) + (((c0 / h) * (pow(d, 2.0) / w)) / pow(D, 2.0)))) / (2.0 * w));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = c0 * ((2.0 * ((c0 * pow(d, 2.0)) / ((w * h) * pow(D, 2.0)))) / (2.0 * w));
	} else {
		tmp = c0 * (0.0 / (2.0 * w));
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = Math.pow((d / D), 2.0);
	double t_1 = (c0 * (d * d)) / ((D * D) * (w * h));
	double t_2 = (c0 / (2.0 * w)) * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= -1e-246) {
		tmp = c0 * ((1.0 / (h / (2.0 * ((c0 * t_0) / w)))) / (2.0 * w));
	} else if (t_2 <= 0.0) {
		tmp = c0 * ((Math.pow(M, 2.0) / ((c0 * (t_0 / (w * h))) + (((c0 / h) * (Math.pow(d, 2.0) / w)) / Math.pow(D, 2.0)))) / (2.0 * w));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = c0 * ((2.0 * ((c0 * Math.pow(d, 2.0)) / ((w * h) * Math.pow(D, 2.0)))) / (2.0 * w));
	} else {
		tmp = c0 * (0.0 / (2.0 * w));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = math.pow((d / D), 2.0)
	t_1 = (c0 * (d * d)) / ((D * D) * (w * h))
	t_2 = (c0 / (2.0 * w)) * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))
	tmp = 0
	if t_2 <= -1e-246:
		tmp = c0 * ((1.0 / (h / (2.0 * ((c0 * t_0) / w)))) / (2.0 * w))
	elif t_2 <= 0.0:
		tmp = c0 * ((math.pow(M, 2.0) / ((c0 * (t_0 / (w * h))) + (((c0 / h) * (math.pow(d, 2.0) / w)) / math.pow(D, 2.0)))) / (2.0 * w))
	elif t_2 <= math.inf:
		tmp = c0 * ((2.0 * ((c0 * math.pow(d, 2.0)) / ((w * h) * math.pow(D, 2.0)))) / (2.0 * w))
	else:
		tmp = c0 * (0.0 / (2.0 * w))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(d / D) ^ 2.0
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(D * D) * Float64(w * h)))
	t_2 = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M)))))
	tmp = 0.0
	if (t_2 <= -1e-246)
		tmp = Float64(c0 * Float64(Float64(1.0 / Float64(h / Float64(2.0 * Float64(Float64(c0 * t_0) / w)))) / Float64(2.0 * w)));
	elseif (t_2 <= 0.0)
		tmp = Float64(c0 * Float64(Float64((M ^ 2.0) / Float64(Float64(c0 * Float64(t_0 / Float64(w * h))) + Float64(Float64(Float64(c0 / h) * Float64((d ^ 2.0) / w)) / (D ^ 2.0)))) / Float64(2.0 * w)));
	elseif (t_2 <= Inf)
		tmp = Float64(c0 * Float64(Float64(2.0 * Float64(Float64(c0 * (d ^ 2.0)) / Float64(Float64(w * h) * (D ^ 2.0)))) / Float64(2.0 * w)));
	else
		tmp = Float64(c0 * Float64(0.0 / Float64(2.0 * w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (d / D) ^ 2.0;
	t_1 = (c0 * (d * d)) / ((D * D) * (w * h));
	t_2 = (c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	tmp = 0.0;
	if (t_2 <= -1e-246)
		tmp = c0 * ((1.0 / (h / (2.0 * ((c0 * t_0) / w)))) / (2.0 * w));
	elseif (t_2 <= 0.0)
		tmp = c0 * (((M ^ 2.0) / ((c0 * (t_0 / (w * h))) + (((c0 / h) * ((d ^ 2.0) / w)) / (D ^ 2.0)))) / (2.0 * w));
	elseif (t_2 <= Inf)
		tmp = c0 * ((2.0 * ((c0 * (d ^ 2.0)) / ((w * h) * (D ^ 2.0)))) / (2.0 * w));
	else
		tmp = c0 * (0.0 / (2.0 * w));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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]}, If[LessEqual[t$95$2, -1e-246], N[(c0 * N[(N[(1.0 / N[(h / N[(2.0 * N[(N[(c0 * t$95$0), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(c0 * N[(N[(N[Power[M, 2.0], $MachinePrecision] / N[(N[(c0 * N[(t$95$0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(c0 / h), $MachinePrecision] * N[(N[Power[d, 2.0], $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision] / N[Power[D, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(c0 * N[(N[(2.0 * N[(N[(c0 * N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[Power[D, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(0.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 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))))) < -9.99999999999999956e-247

   1. Initial program 87.5%

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

   3. Simplified 80.9%

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

   5. Taylor expanded in h around 0 16.6%

      \[\leadsto c0 \cdot \frac{\color{blue}{\frac{0.5 \cdot \frac{{D}^{2} \cdot \left(h \cdot \left(w \cdot \left(-1 \cdot \frac{M \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot w} + \frac{M \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot w}\right)\right)\right)}{c0 \cdot {d}^{2}} + 2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot w}}{h}}}{2 \cdot w} \]
   6. Step-by-step derivation

   7. Simplified 87.5%

      \[\leadsto c0 \cdot \frac{\color{blue}{\frac{\mathsf{fma}\left(0.5, {D}^{2} \cdot \left(h \cdot \frac{w \cdot 0}{c0 \cdot {d}^{2}}\right), 2 \cdot \frac{\frac{c0 \cdot {d}^{2}}{{D}^{2}}}{w}\right)}{h}}}{2 \cdot w} \]

   8. Taylor expanded in w around 0 87.5%

      \[\leadsto c0 \cdot \frac{\frac{\mathsf{fma}\left(0.5, {D}^{2} \cdot \left(h \cdot \color{blue}{0}\right), 2 \cdot \frac{\frac{c0 \cdot {d}^{2}}{{D}^{2}}}{w}\right)}{h}}{2 \cdot w} \]
   9. Step-by-step derivation

      1. clear-num 87.6%

         \[\leadsto c0 \cdot \frac{\color{blue}{\frac{1}{\frac{h}{\mathsf{fma}\left(0.5, {D}^{2} \cdot \left(h \cdot 0\right), 2 \cdot \frac{\frac{c0 \cdot {d}^{2}}{{D}^{2}}}{w}\right)}}}}{2 \cdot w} \]

      2. inv-pow 87.6%

         \[\leadsto c0 \cdot \frac{\color{blue}{{\left(\frac{h}{\mathsf{fma}\left(0.5, {D}^{2} \cdot \left(h \cdot 0\right), 2 \cdot \frac{\frac{c0 \cdot {d}^{2}}{{D}^{2}}}{w}\right)}\right)}^{-1}}}{2 \cdot w} \]

   10. Applied egg-rr 90.8%

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

       1. unpow-1 90.8%

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

       2. associate-*l/ 90.8%

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

   12. Simplified 90.8%

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

   if -9.99999999999999956e-247 < (*.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))))) < -0.0

   1. Initial program 33.6%

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

   3. Simplified 12.7%

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

   6. Applied egg-rr 13.1%

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

      1. associate--r- 57.8%

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

      2. +-inverses 57.8%

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

      3. associate-*l/ 57.8%

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

      4. associate-/l* 58.2%

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

      5. *-commutative 58.2%

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

      6. sub-neg 58.2%

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

   8. Simplified 47.8%

      \[\leadsto c0 \cdot \frac{\color{blue}{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w} - \sqrt{\mathsf{fma}\left({\left(\frac{c0}{h \cdot w}\right)}^{2}, {\left(\frac{d}{D}\right)}^{4}, -{M}^{2}\right)}}}}{2 \cdot w} \]

   9. Taylor expanded in c0 around -inf 58.6%

      \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w} - \color{blue}{-1 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}}}{2 \cdot w} \]
   10. Step-by-step derivation

       1. mul-1-neg 58.6%

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

       2. times-frac 58.6%

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

   11. Simplified 58.6%

       \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w} - \color{blue}{\left(-\frac{c0}{{D}^{2}} \cdot \frac{{d}^{2}}{h \cdot w}\right)}}}{2 \cdot w} \]
   12. Step-by-step derivation

       1. associate-*l/ 58.6%

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

       2. *-commutative 58.6%

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

   13. Applied egg-rr 58.6%

       \[\leadsto c0 \cdot \frac{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w} - \left(-\color{blue}{\frac{c0 \cdot \frac{{d}^{2}}{w \cdot h}}{{D}^{2}}}\right)}}{2 \cdot w} \]
   14. Step-by-step derivation

       1. associate-*r/ 58.6%

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

       2. *-commutative 58.6%

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

       3. times-frac 68.2%

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

   15. Simplified 68.2%

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

   if -0.0 < (*.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))))) < +inf.0

   1. Initial program 73.4%

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

   3. Simplified 73.5%

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

   5. Taylor expanded in c0 around inf 78.7%

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

   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%

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

   3. Simplified 18.8%

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

   5. Taylor expanded in c0 around -inf 1.3%

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

      1. distribute-lft-in 0.6%

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

      2. mul-1-neg 0.6%

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

      3. distribute-rgt-neg-in 0.6%

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

      4. associate-/l* 0.1%

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

      5. mul-1-neg 0.1%

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

      6. associate-/l* 0.1%

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

      7. distribute-lft1-in 0.1%

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

      8. metadata-eval 0.1%

         \[\leadsto c0 \cdot \frac{-1 \cdot \left(\color{blue}{0} \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{2 \cdot w} \]

      9. mul0-lft 36.9%

         \[\leadsto c0 \cdot \frac{-1 \cdot \color{blue}{0}}{2 \cdot w} \]

      10. metadata-eval 36.9%

          \[\leadsto c0 \cdot \frac{\color{blue}{0}}{2 \cdot w} \]

   7. Simplified 36.9%

      \[\leadsto c0 \cdot \frac{\color{blue}{0}}{2 \cdot w} \]
3. Recombined 4 regimes into one program.

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq -1 \cdot 10^{-246}:\\ \;\;\;\;c0 \cdot \frac{\frac{1}{\frac{h}{2 \cdot \frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w}}}}{2 \cdot w}\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq 0:\\ \;\;\;\;c0 \cdot \frac{\frac{{M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h} + \frac{\frac{c0}{h} \cdot \frac{{d}^{2}}{w}}{{D}^{2}}}}{2 \cdot w}\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;c0 \cdot \frac{2 \cdot \frac{c0 \cdot {d}^{2}}{\left(w \cdot h\right) \cdot {D}^{2}}}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 54.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot {d}^{2}}{\left(w \cdot h\right) \cdot {D}^{2}}\\ t_1 := {\left(\frac{d}{D}\right)}^{2}\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\\ t_3 := \frac{c0}{2 \cdot w} \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - M \cdot M}\right)\\ \mathbf{if}\;t\_3 \leq -1 \cdot 10^{-246}:\\ \;\;\;\;c0 \cdot \frac{\frac{1}{\frac{h}{2 \cdot \frac{c0 \cdot t\_1}{w}}}}{2 \cdot w}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;c0 \cdot \frac{\frac{{M}^{2}}{t\_0 + t\_1 \cdot \frac{c0}{w \cdot h}}}{2 \cdot w}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;c0 \cdot \frac{2 \cdot t\_0}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (pow d 2.0)) (* (* w h) (pow D 2.0))))
        (t_1 (pow (/ d D) 2.0))
        (t_2 (/ (* c0 (* d d)) (* (* D D) (* w h))))
        (t_3 (* (/ c0 (* 2.0 w)) (+ t_2 (sqrt (- (* t_2 t_2) (* M M)))))))
   (if (<= t_3 -1e-246)
     (* c0 (/ (/ 1.0 (/ h (* 2.0 (/ (* c0 t_1) w)))) (* 2.0 w)))
     (if (<= t_3 0.0)
       (* c0 (/ (/ (pow M 2.0) (+ t_0 (* t_1 (/ c0 (* w h))))) (* 2.0 w)))
       (if (<= t_3 INFINITY)
         (* c0 (/ (* 2.0 t_0) (* 2.0 w)))
         (* c0 (/ 0.0 (* 2.0 w))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * pow(d, 2.0)) / ((w * h) * pow(D, 2.0));
	double t_1 = pow((d / D), 2.0);
	double t_2 = (c0 * (d * d)) / ((D * D) * (w * h));
	double t_3 = (c0 / (2.0 * w)) * (t_2 + sqrt(((t_2 * t_2) - (M * M))));
	double tmp;
	if (t_3 <= -1e-246) {
		tmp = c0 * ((1.0 / (h / (2.0 * ((c0 * t_1) / w)))) / (2.0 * w));
	} else if (t_3 <= 0.0) {
		tmp = c0 * ((pow(M, 2.0) / (t_0 + (t_1 * (c0 / (w * h))))) / (2.0 * w));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = c0 * ((2.0 * t_0) / (2.0 * w));
	} else {
		tmp = c0 * (0.0 / (2.0 * w));
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * Math.pow(d, 2.0)) / ((w * h) * Math.pow(D, 2.0));
	double t_1 = Math.pow((d / D), 2.0);
	double t_2 = (c0 * (d * d)) / ((D * D) * (w * h));
	double t_3 = (c0 / (2.0 * w)) * (t_2 + Math.sqrt(((t_2 * t_2) - (M * M))));
	double tmp;
	if (t_3 <= -1e-246) {
		tmp = c0 * ((1.0 / (h / (2.0 * ((c0 * t_1) / w)))) / (2.0 * w));
	} else if (t_3 <= 0.0) {
		tmp = c0 * ((Math.pow(M, 2.0) / (t_0 + (t_1 * (c0 / (w * h))))) / (2.0 * w));
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = c0 * ((2.0 * t_0) / (2.0 * w));
	} else {
		tmp = c0 * (0.0 / (2.0 * w));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * math.pow(d, 2.0)) / ((w * h) * math.pow(D, 2.0))
	t_1 = math.pow((d / D), 2.0)
	t_2 = (c0 * (d * d)) / ((D * D) * (w * h))
	t_3 = (c0 / (2.0 * w)) * (t_2 + math.sqrt(((t_2 * t_2) - (M * M))))
	tmp = 0
	if t_3 <= -1e-246:
		tmp = c0 * ((1.0 / (h / (2.0 * ((c0 * t_1) / w)))) / (2.0 * w))
	elif t_3 <= 0.0:
		tmp = c0 * ((math.pow(M, 2.0) / (t_0 + (t_1 * (c0 / (w * h))))) / (2.0 * w))
	elif t_3 <= math.inf:
		tmp = c0 * ((2.0 * t_0) / (2.0 * w))
	else:
		tmp = c0 * (0.0 / (2.0 * w))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * (d ^ 2.0)) / Float64(Float64(w * h) * (D ^ 2.0)))
	t_1 = Float64(d / D) ^ 2.0
	t_2 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(D * D) * Float64(w * h)))
	t_3 = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M)))))
	tmp = 0.0
	if (t_3 <= -1e-246)
		tmp = Float64(c0 * Float64(Float64(1.0 / Float64(h / Float64(2.0 * Float64(Float64(c0 * t_1) / w)))) / Float64(2.0 * w)));
	elseif (t_3 <= 0.0)
		tmp = Float64(c0 * Float64(Float64((M ^ 2.0) / Float64(t_0 + Float64(t_1 * Float64(c0 / Float64(w * h))))) / Float64(2.0 * w)));
	elseif (t_3 <= Inf)
		tmp = Float64(c0 * Float64(Float64(2.0 * t_0) / Float64(2.0 * w)));
	else
		tmp = Float64(c0 * Float64(0.0 / Float64(2.0 * w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d ^ 2.0)) / ((w * h) * (D ^ 2.0));
	t_1 = (d / D) ^ 2.0;
	t_2 = (c0 * (d * d)) / ((D * D) * (w * h));
	t_3 = (c0 / (2.0 * w)) * (t_2 + sqrt(((t_2 * t_2) - (M * M))));
	tmp = 0.0;
	if (t_3 <= -1e-246)
		tmp = c0 * ((1.0 / (h / (2.0 * ((c0 * t_1) / w)))) / (2.0 * w));
	elseif (t_3 <= 0.0)
		tmp = c0 * (((M ^ 2.0) / (t_0 + (t_1 * (c0 / (w * h))))) / (2.0 * w));
	elseif (t_3 <= Inf)
		tmp = c0 * ((2.0 * t_0) / (2.0 * w));
	else
		tmp = c0 * (0.0 / (2.0 * w));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[Power[D, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1e-246], N[(c0 * N[(N[(1.0 / N[(h / N[(2.0 * N[(N[(c0 * t$95$1), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(c0 * N[(N[(N[Power[M, 2.0], $MachinePrecision] / N[(t$95$0 + N[(t$95$1 * N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(c0 * N[(N[(2.0 * t$95$0), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(0.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 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))))) < -9.99999999999999956e-247

   1. Initial program 87.5%

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

   3. Simplified 80.9%

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

   5. Taylor expanded in h around 0 16.6%

      \[\leadsto c0 \cdot \frac{\color{blue}{\frac{0.5 \cdot \frac{{D}^{2} \cdot \left(h \cdot \left(w \cdot \left(-1 \cdot \frac{M \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot w} + \frac{M \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot w}\right)\right)\right)}{c0 \cdot {d}^{2}} + 2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot w}}{h}}}{2 \cdot w} \]
   6. Step-by-step derivation

   7. Simplified 87.5%

      \[\leadsto c0 \cdot \frac{\color{blue}{\frac{\mathsf{fma}\left(0.5, {D}^{2} \cdot \left(h \cdot \frac{w \cdot 0}{c0 \cdot {d}^{2}}\right), 2 \cdot \frac{\frac{c0 \cdot {d}^{2}}{{D}^{2}}}{w}\right)}{h}}}{2 \cdot w} \]

   8. Taylor expanded in w around 0 87.5%

      \[\leadsto c0 \cdot \frac{\frac{\mathsf{fma}\left(0.5, {D}^{2} \cdot \left(h \cdot \color{blue}{0}\right), 2 \cdot \frac{\frac{c0 \cdot {d}^{2}}{{D}^{2}}}{w}\right)}{h}}{2 \cdot w} \]
   9. Step-by-step derivation

      1. clear-num 87.6%

         \[\leadsto c0 \cdot \frac{\color{blue}{\frac{1}{\frac{h}{\mathsf{fma}\left(0.5, {D}^{2} \cdot \left(h \cdot 0\right), 2 \cdot \frac{\frac{c0 \cdot {d}^{2}}{{D}^{2}}}{w}\right)}}}}{2 \cdot w} \]

      2. inv-pow 87.6%

         \[\leadsto c0 \cdot \frac{\color{blue}{{\left(\frac{h}{\mathsf{fma}\left(0.5, {D}^{2} \cdot \left(h \cdot 0\right), 2 \cdot \frac{\frac{c0 \cdot {d}^{2}}{{D}^{2}}}{w}\right)}\right)}^{-1}}}{2 \cdot w} \]

   10. Applied egg-rr 90.8%

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

       1. unpow-1 90.8%

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

       2. associate-*l/ 90.8%

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

   12. Simplified 90.8%

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

   if -9.99999999999999956e-247 < (*.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))))) < -0.0

   1. Initial program 33.6%

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

   3. Simplified 12.7%

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

   6. Applied egg-rr 13.1%

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

      1. associate--r- 57.8%

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

      2. +-inverses 57.8%

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

      3. associate-*l/ 57.8%

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

      4. associate-/l* 58.2%

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

      5. *-commutative 58.2%

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

      6. sub-neg 58.2%

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

   8. Simplified 47.8%

      \[\leadsto c0 \cdot \frac{\color{blue}{\frac{0 + {M}^{2}}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w} - \sqrt{\mathsf{fma}\left({\left(\frac{c0}{h \cdot w}\right)}^{2}, {\left(\frac{d}{D}\right)}^{4}, -{M}^{2}\right)}}}}{2 \cdot w} \]
   9. Step-by-step derivation

      1. div-inv 47.9%

         \[\leadsto c0 \cdot \frac{\color{blue}{\left(0 + {M}^{2}\right) \cdot \frac{1}{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w} - \sqrt{\mathsf{fma}\left({\left(\frac{c0}{h \cdot w}\right)}^{2}, {\left(\frac{d}{D}\right)}^{4}, -{M}^{2}\right)}}}}{2 \cdot w} \]

      2. +-lft-identity 47.9%

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

      3. associate-*r/ 47.9%

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

      4. *-commutative 47.9%

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

      5. associate-*l/ 47.9%

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

      6. associate-/r* 57.9%

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

   10. Applied egg-rr 57.7%

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

       1. associate-*r/ 57.8%

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

       2. *-rgt-identity 57.8%

          \[\leadsto c0 \cdot \frac{\frac{\color{blue}{{M}^{2}}}{\frac{\frac{c0}{w}}{h} \cdot {\left(\frac{d}{D}\right)}^{2} - \sqrt{\mathsf{fma}\left({\left(\frac{\frac{c0}{w}}{h}\right)}^{2}, {\left(\frac{d}{D}\right)}^{4}, -{M}^{2}\right)}}}{2 \cdot w} \]

       3. associate-/l/ 47.9%

          \[\leadsto c0 \cdot \frac{\frac{{M}^{2}}{\color{blue}{\frac{c0}{h \cdot w}} \cdot {\left(\frac{d}{D}\right)}^{2} - \sqrt{\mathsf{fma}\left({\left(\frac{\frac{c0}{w}}{h}\right)}^{2}, {\left(\frac{d}{D}\right)}^{4}, -{M}^{2}\right)}}}{2 \cdot w} \]

       4. fma-neg 47.9%

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

       5. associate-/l/ 47.8%

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

   12. Simplified 47.8%

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

   13. Taylor expanded in c0 around -inf 68.0%

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

       1. *-commutative 68.0%

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

       2. neg-mul-1 68.0%

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

       3. distribute-neg-frac2 68.0%

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

       4. *-commutative 68.0%

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

   15. Simplified 68.0%

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

   if -0.0 < (*.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))))) < +inf.0

   1. Initial program 73.4%

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

   3. Simplified 73.5%

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

   5. Taylor expanded in c0 around inf 78.7%

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

   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%

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

   3. Simplified 18.8%

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

   5. Taylor expanded in c0 around -inf 1.3%

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

      1. distribute-lft-in 0.6%

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

      2. mul-1-neg 0.6%

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

      3. distribute-rgt-neg-in 0.6%

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

      4. associate-/l* 0.1%

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

      5. mul-1-neg 0.1%

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

      6. associate-/l* 0.1%

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

      7. distribute-lft1-in 0.1%

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

      8. metadata-eval 0.1%

         \[\leadsto c0 \cdot \frac{-1 \cdot \left(\color{blue}{0} \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{2 \cdot w} \]

      9. mul0-lft 36.9%

         \[\leadsto c0 \cdot \frac{-1 \cdot \color{blue}{0}}{2 \cdot w} \]

      10. metadata-eval 36.9%

          \[\leadsto c0 \cdot \frac{\color{blue}{0}}{2 \cdot w} \]

   7. Simplified 36.9%

      \[\leadsto c0 \cdot \frac{\color{blue}{0}}{2 \cdot w} \]
3. Recombined 4 regimes into one program.

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq -1 \cdot 10^{-246}:\\ \;\;\;\;c0 \cdot \frac{\frac{1}{\frac{h}{2 \cdot \frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w}}}}{2 \cdot w}\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq 0:\\ \;\;\;\;c0 \cdot \frac{\frac{{M}^{2}}{\frac{c0 \cdot {d}^{2}}{\left(w \cdot h\right) \cdot {D}^{2}} + {\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h}}}{2 \cdot w}\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;c0 \cdot \frac{2 \cdot \frac{c0 \cdot {d}^{2}}{\left(w \cdot h\right) \cdot {D}^{2}}}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 53.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= Inf)
		tmp = c0 * ((2.0 * ((c0 * (d ^ 2.0)) / ((w * h) * (D ^ 2.0)))) / (2.0 * w));
	else
		tmp = c0 * (0.0 / (2.0 * w));
	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[(D * D), $MachinePrecision] * N[(w * h), $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[(2.0 * N[(N[(c0 * N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[Power[D, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(0.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 73.8%

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

   3. Simplified 68.4%

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

   5. Taylor expanded in c0 around inf 74.9%

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

   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%

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

   3. Simplified 18.8%

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

   5. Taylor expanded in c0 around -inf 1.3%

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

      1. distribute-lft-in 0.6%

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

      2. mul-1-neg 0.6%

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

      3. distribute-rgt-neg-in 0.6%

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

      4. associate-/l* 0.1%

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

      5. mul-1-neg 0.1%

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

      6. associate-/l* 0.1%

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

      7. distribute-lft1-in 0.1%

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

      8. metadata-eval 0.1%

         \[\leadsto c0 \cdot \frac{-1 \cdot \left(\color{blue}{0} \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{2 \cdot w} \]

      9. mul0-lft 36.9%

         \[\leadsto c0 \cdot \frac{-1 \cdot \color{blue}{0}}{2 \cdot w} \]

      10. metadata-eval 36.9%

          \[\leadsto c0 \cdot \frac{\color{blue}{0}}{2 \cdot w} \]

   7. Simplified 36.9%

      \[\leadsto c0 \cdot \frac{\color{blue}{0}}{2 \cdot w} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.2%

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

Alternative 4: 53.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= Inf)
		tmp = c0 * ((2.0 * ((c0 * ((d / D) ^ 2.0)) / (w * h))) / (2.0 * w));
	else
		tmp = c0 * (0.0 / (2.0 * w));
	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[(D * D), $MachinePrecision] * N[(w * h), $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[(2.0 * N[(N[(c0 * N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(0.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 73.8%

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

   3. Simplified 68.4%

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

   5. Taylor expanded in h around 0 25.1%

      \[\leadsto c0 \cdot \frac{\color{blue}{\frac{0.5 \cdot \frac{{D}^{2} \cdot \left(h \cdot \left(w \cdot \left(-1 \cdot \frac{M \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot w} + \frac{M \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot w}\right)\right)\right)}{c0 \cdot {d}^{2}} + 2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot w}}{h}}}{2 \cdot w} \]
   6. Step-by-step derivation

   7. Simplified 74.7%

      \[\leadsto c0 \cdot \frac{\color{blue}{\frac{\mathsf{fma}\left(0.5, {D}^{2} \cdot \left(h \cdot \frac{w \cdot 0}{c0 \cdot {d}^{2}}\right), 2 \cdot \frac{\frac{c0 \cdot {d}^{2}}{{D}^{2}}}{w}\right)}{h}}}{2 \cdot w} \]

   8. Taylor expanded in w around 0 74.7%

      \[\leadsto c0 \cdot \frac{\frac{\mathsf{fma}\left(0.5, {D}^{2} \cdot \left(h \cdot \color{blue}{0}\right), 2 \cdot \frac{\frac{c0 \cdot {d}^{2}}{{D}^{2}}}{w}\right)}{h}}{2 \cdot w} \]
   9. Step-by-step derivation

      1. div-inv 74.7%

         \[\leadsto c0 \cdot \frac{\color{blue}{\mathsf{fma}\left(0.5, {D}^{2} \cdot \left(h \cdot 0\right), 2 \cdot \frac{\frac{c0 \cdot {d}^{2}}{{D}^{2}}}{w}\right) \cdot \frac{1}{h}}}{2 \cdot w} \]

   10. Applied egg-rr 74.7%

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

       1. associate-*r/ 74.7%

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

       2. *-rgt-identity 74.7%

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

       3. associate-/l* 74.7%

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

       4. *-commutative 74.7%

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

       5. associate-*r/ 74.6%

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

       6. *-rgt-identity 74.6%

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

       7. associate-*r/ 74.6%

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

       8. associate-*l/ 69.3%

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

       9. associate-*r/ 69.3%

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

       10. *-rgt-identity 69.3%

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

       11. associate-/r* 74.6%

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

       12. *-commutative 74.6%

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

       13. associate-*l/ 74.9%

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

       14. times-frac 69.3%

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

   12. Simplified 69.3%

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

       1. frac-times 74.9%

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

       2. *-commutative 74.9%

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

   14. Applied egg-rr 74.9%

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

   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%

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

   3. Simplified 18.8%

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

   5. Taylor expanded in c0 around -inf 1.3%

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

      1. distribute-lft-in 0.6%

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

      2. mul-1-neg 0.6%

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

      3. distribute-rgt-neg-in 0.6%

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

      4. associate-/l* 0.1%

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

      5. mul-1-neg 0.1%

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

      6. associate-/l* 0.1%

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

      7. distribute-lft1-in 0.1%

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

      8. metadata-eval 0.1%

         \[\leadsto c0 \cdot \frac{-1 \cdot \left(\color{blue}{0} \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{2 \cdot w} \]

      9. mul0-lft 36.9%

         \[\leadsto c0 \cdot \frac{-1 \cdot \color{blue}{0}}{2 \cdot w} \]

      10. metadata-eval 36.9%

          \[\leadsto c0 \cdot \frac{\color{blue}{0}}{2 \cdot w} \]

   7. Simplified 36.9%

      \[\leadsto c0 \cdot \frac{\color{blue}{0}}{2 \cdot w} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.2%

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

Alternative 5: 42.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 1.16 \cdot 10^{-32}:\\ \;\;\;\;c0 \cdot \left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{\frac{c0}{w \cdot h}}{w}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= w 1.16e-32)
   (* c0 (* (pow (/ d D) 2.0) (/ (/ c0 (* w h)) w)))
   (* c0 (/ 0.0 (* 2.0 w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (w <= 1.16e-32) {
		tmp = c0 * (pow((d / D), 2.0) * ((c0 / (w * h)) / w));
	} else {
		tmp = c0 * (0.0 / (2.0 * w));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (w <= 1.16d-32) then
        tmp = c0 * (((d_1 / d) ** 2.0d0) * ((c0 / (w * h)) / w))
    else
        tmp = c0 * (0.0d0 / (2.0d0 * w))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (w <= 1.16e-32) {
		tmp = c0 * (Math.pow((d / D), 2.0) * ((c0 / (w * h)) / w));
	} else {
		tmp = c0 * (0.0 / (2.0 * w));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if w <= 1.16e-32:
		tmp = c0 * (math.pow((d / D), 2.0) * ((c0 / (w * h)) / w))
	else:
		tmp = c0 * (0.0 / (2.0 * w))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (w <= 1.16e-32)
		tmp = Float64(c0 * Float64((Float64(d / D) ^ 2.0) * Float64(Float64(c0 / Float64(w * h)) / w)));
	else
		tmp = Float64(c0 * Float64(0.0 / Float64(2.0 * w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (w <= 1.16e-32)
		tmp = c0 * (((d / D) ^ 2.0) * ((c0 / (w * h)) / w));
	else
		tmp = c0 * (0.0 / (2.0 * w));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[w, 1.16e-32], N[(c0 * N[(N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(0.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 1.16000000000000001e-32

   1. Initial program 23.2%

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

   3. Simplified 36.2%

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

   5. Taylor expanded in h around 0 7.7%

      \[\leadsto c0 \cdot \frac{\color{blue}{\frac{0.5 \cdot \frac{{D}^{2} \cdot \left(h \cdot \left(w \cdot \left(-1 \cdot \frac{M \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot w} + \frac{M \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot w}\right)\right)\right)}{c0 \cdot {d}^{2}} + 2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot w}}{h}}}{2 \cdot w} \]
   6. Step-by-step derivation

   7. Simplified 32.5%

      \[\leadsto c0 \cdot \frac{\color{blue}{\frac{\mathsf{fma}\left(0.5, {D}^{2} \cdot \left(h \cdot \frac{w \cdot 0}{c0 \cdot {d}^{2}}\right), 2 \cdot \frac{\frac{c0 \cdot {d}^{2}}{{D}^{2}}}{w}\right)}{h}}}{2 \cdot w} \]

   8. Taylor expanded in w around 0 33.1%

      \[\leadsto c0 \cdot \frac{\frac{\mathsf{fma}\left(0.5, {D}^{2} \cdot \left(h \cdot \color{blue}{0}\right), 2 \cdot \frac{\frac{c0 \cdot {d}^{2}}{{D}^{2}}}{w}\right)}{h}}{2 \cdot w} \]
   9. Step-by-step derivation

      1. div-inv 33.1%

         \[\leadsto c0 \cdot \frac{\color{blue}{\mathsf{fma}\left(0.5, {D}^{2} \cdot \left(h \cdot 0\right), 2 \cdot \frac{\frac{c0 \cdot {d}^{2}}{{D}^{2}}}{w}\right) \cdot \frac{1}{h}}}{2 \cdot w} \]

   10. Applied egg-rr 44.7%

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

       1. associate-*r/ 44.8%

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

       2. *-rgt-identity 44.8%

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

       3. associate-/l* 44.8%

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

       4. *-commutative 44.8%

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

       5. associate-*r/ 44.2%

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

       6. *-rgt-identity 44.2%

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

       7. associate-*r/ 44.2%

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

       8. associate-*l/ 41.8%

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

       9. associate-*r/ 41.8%

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

       10. *-rgt-identity 41.8%

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

       11. associate-/r* 44.5%

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

       12. *-commutative 44.5%

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

       13. associate-*l/ 44.1%

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

       14. times-frac 41.8%

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

   12. Simplified 41.8%

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

   14. Applied egg-rr 44.5%

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

       1. unpow1 44.5%

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

       2. *-lft-identity 44.5%

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

       3. associate-/l* 44.4%

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

       4. *-commutative 44.4%

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

   16. Simplified 44.4%

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

   if 1.16000000000000001e-32 < w

   1. Initial program 15.9%

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

   3. Simplified 21.1%

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

   5. Taylor expanded in c0 around -inf 6.9%

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

      1. distribute-lft-in 4.7%

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

      2. mul-1-neg 4.7%

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

      3. distribute-rgt-neg-in 4.7%

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

      4. associate-/l* 2.5%

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

      5. mul-1-neg 2.5%

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

      6. associate-/l* 2.6%

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

      7. distribute-lft1-in 2.6%

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

      8. metadata-eval 2.6%

         \[\leadsto c0 \cdot \frac{-1 \cdot \left(\color{blue}{0} \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{2 \cdot w} \]

      9. mul0-lft 47.2%

         \[\leadsto c0 \cdot \frac{-1 \cdot \color{blue}{0}}{2 \cdot w} \]

      10. metadata-eval 47.2%

          \[\leadsto c0 \cdot \frac{\color{blue}{0}}{2 \cdot w} \]

   7. Simplified 47.2%

      \[\leadsto c0 \cdot \frac{\color{blue}{0}}{2 \cdot w} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 1.16 \cdot 10^{-32}:\\ \;\;\;\;c0 \cdot \left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{\frac{c0}{w \cdot h}}{w}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 43.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 3.6 \cdot 10^{+162}:\\ \;\;\;\;c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h}}{w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= w 3.6e+162)
   (* c0 (/ (* (pow (/ d D) 2.0) (/ c0 (* w h))) w))
   (* c0 (/ 0.0 (* 2.0 w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (w <= 3.6e+162) {
		tmp = c0 * ((pow((d / D), 2.0) * (c0 / (w * h))) / w);
	} else {
		tmp = c0 * (0.0 / (2.0 * w));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (w <= 3.6d+162) then
        tmp = c0 * ((((d_1 / d) ** 2.0d0) * (c0 / (w * h))) / w)
    else
        tmp = c0 * (0.0d0 / (2.0d0 * w))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (w <= 3.6e+162) {
		tmp = c0 * ((Math.pow((d / D), 2.0) * (c0 / (w * h))) / w);
	} else {
		tmp = c0 * (0.0 / (2.0 * w));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if w <= 3.6e+162:
		tmp = c0 * ((math.pow((d / D), 2.0) * (c0 / (w * h))) / w)
	else:
		tmp = c0 * (0.0 / (2.0 * w))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (w <= 3.6e+162)
		tmp = Float64(c0 * Float64(Float64((Float64(d / D) ^ 2.0) * Float64(c0 / Float64(w * h))) / w));
	else
		tmp = Float64(c0 * Float64(0.0 / Float64(2.0 * w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (w <= 3.6e+162)
		tmp = c0 * ((((d / D) ^ 2.0) * (c0 / (w * h))) / w);
	else
		tmp = c0 * (0.0 / (2.0 * w));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[w, 3.6e+162], N[(c0 * N[(N[(N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision] * N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(0.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 3.59999999999999994e162

   1. Initial program 22.4%

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

   3. Simplified 34.2%

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

   5. Taylor expanded in h around 0 7.6%

      \[\leadsto c0 \cdot \frac{\color{blue}{\frac{0.5 \cdot \frac{{D}^{2} \cdot \left(h \cdot \left(w \cdot \left(-1 \cdot \frac{M \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot w} + \frac{M \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot w}\right)\right)\right)}{c0 \cdot {d}^{2}} + 2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot w}}{h}}}{2 \cdot w} \]
   6. Step-by-step derivation

   7. Simplified 31.0%

      \[\leadsto c0 \cdot \frac{\color{blue}{\frac{\mathsf{fma}\left(0.5, {D}^{2} \cdot \left(h \cdot \frac{w \cdot 0}{c0 \cdot {d}^{2}}\right), 2 \cdot \frac{\frac{c0 \cdot {d}^{2}}{{D}^{2}}}{w}\right)}{h}}}{2 \cdot w} \]

   8. Taylor expanded in w around 0 32.0%

      \[\leadsto c0 \cdot \frac{\frac{\mathsf{fma}\left(0.5, {D}^{2} \cdot \left(h \cdot \color{blue}{0}\right), 2 \cdot \frac{\frac{c0 \cdot {d}^{2}}{{D}^{2}}}{w}\right)}{h}}{2 \cdot w} \]
   9. Step-by-step derivation

      1. div-inv 32.0%

         \[\leadsto c0 \cdot \frac{\color{blue}{\mathsf{fma}\left(0.5, {D}^{2} \cdot \left(h \cdot 0\right), 2 \cdot \frac{\frac{c0 \cdot {d}^{2}}{{D}^{2}}}{w}\right) \cdot \frac{1}{h}}}{2 \cdot w} \]

   10. Applied egg-rr 43.7%

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

       1. associate-*r/ 43.7%

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

       2. *-rgt-identity 43.7%

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

       3. associate-/l* 43.7%

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

       4. *-commutative 43.7%

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

       5. associate-*r/ 43.6%

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

       6. *-rgt-identity 43.6%

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

       7. associate-*r/ 43.6%

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

       8. associate-*l/ 41.2%

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

       9. associate-*r/ 41.2%

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

       10. *-rgt-identity 41.2%

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

       11. associate-/r* 43.8%

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

       12. *-commutative 43.8%

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

       13. associate-*l/ 43.1%

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

       14. times-frac 41.2%

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

   12. Simplified 41.2%

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

       1. times-frac 41.2%

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

       2. metadata-eval 41.2%

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

       3. frac-times 43.1%

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

       4. associate-*l/ 43.8%

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

       5. *-commutative 43.8%

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

       6. *-commutative 43.8%

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

   14. Applied egg-rr 43.8%

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

   if 3.59999999999999994e162 < w

   1. Initial program 13.7%

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

   3. Simplified 21.7%

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

   5. Taylor expanded in c0 around -inf 7.3%

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

      1. distribute-lft-in 7.3%

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

      2. mul-1-neg 7.3%

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

      3. distribute-rgt-neg-in 7.3%

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

      4. associate-/l* 7.2%

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

      5. mul-1-neg 7.2%

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

      6. associate-/l* 7.6%

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

      7. distribute-lft1-in 7.6%

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

      8. metadata-eval 7.6%

         \[\leadsto c0 \cdot \frac{-1 \cdot \left(\color{blue}{0} \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{2 \cdot w} \]

      9. mul0-lft 68.0%

         \[\leadsto c0 \cdot \frac{-1 \cdot \color{blue}{0}}{2 \cdot w} \]

      10. metadata-eval 68.0%

          \[\leadsto c0 \cdot \frac{\color{blue}{0}}{2 \cdot w} \]

   7. Simplified 68.0%

      \[\leadsto c0 \cdot \frac{\color{blue}{0}}{2 \cdot w} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 3.6 \cdot 10^{+162}:\\ \;\;\;\;c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h}}{w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 32.8% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ c0 \cdot \frac{0}{2 \cdot w} \end{array} \]

FPCore

(FPCore (c0 w h D d M) :precision binary64 (* c0 (/ 0.0 (* 2.0 w))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return c0 * (0.0 / (2.0 * w));
}

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 = c0 * (0.0d0 / (2.0d0 * w))
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return c0 * (0.0 / (2.0 * w));
}

Python

def code(c0, w, h, D, d, M):
	return c0 * (0.0 / (2.0 * w))

Julia

function code(c0, w, h, D, d, M)
	return Float64(c0 * Float64(0.0 / Float64(2.0 * w)))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = c0 * (0.0 / (2.0 * w));
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(c0 * N[(0.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 21.9%

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

3. Simplified 33.5%

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

5. Taylor expanded in c0 around -inf 2.4%

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

   1. distribute-lft-in 1.9%

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

   2. mul-1-neg 1.9%

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

   3. distribute-rgt-neg-in 1.9%

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

   4. associate-/l* 2.0%

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

   5. mul-1-neg 2.0%

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

   6. associate-/l* 1.6%

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

   7. distribute-lft1-in 1.6%

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

   8. metadata-eval 1.6%

      \[\leadsto c0 \cdot \frac{-1 \cdot \left(\color{blue}{0} \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{2 \cdot w} \]

   9. mul0-lft 27.8%

      \[\leadsto c0 \cdot \frac{-1 \cdot \color{blue}{0}}{2 \cdot w} \]

   10. metadata-eval 27.8%

       \[\leadsto c0 \cdot \frac{\color{blue}{0}}{2 \cdot w} \]

7. Simplified 27.8%

   \[\leadsto c0 \cdot \frac{\color{blue}{0}}{2 \cdot w} \]

8. Final simplification 27.8%

   \[\leadsto c0 \cdot \frac{0}{2 \cdot w} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024079 
(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))))))