Henrywood and Agarwal, Equation (13)

Percentage Accurate: 25.5% → 55.8%

Time: 35.5s

Alternatives: 8

Speedup: 151.0×

• could not determine a ground truth

  1. c0 = 1.5435394323727404e+268
  2. w = 1.3014711577380434e-208
  3. h = -2.430638006457925e-285
  4. D = -2.9583362653464235e-271
  5. d = 1.6768202422875416e+276
  6. M = 2.4655283701400786e+234

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: 25.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 55.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\\ t_2 := t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right)\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-142}:\\ \;\;\;\;\frac{\frac{2 \cdot {\left(c0 \cdot d\right)}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}{2 \cdot w}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{\frac{c0}{w}}{2} \cdot \left(c0 \cdot 0 + 0.5 \cdot \frac{\left(w \cdot h\right) \cdot \left({D}^{2} \cdot {M}^{2}\right)}{c0 \cdot {d}^{2}}\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_0 \cdot \left(2 \cdot \frac{c0}{h \cdot \left(w \cdot {\left(\frac{d}{D}\right)}^{-2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w)))
        (t_1 (/ (* c0 (* d d)) (* (* D D) (* w h))))
        (t_2 (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))))
   (if (<= t_2 -5e-142)
     (/ (/ (* 2.0 (pow (* c0 d) 2.0)) (* (pow D 2.0) (* w h))) (* 2.0 w))
     (if (<= t_2 0.0)
       (*
        (/ (/ c0 w) 2.0)
        (+
         (* c0 0.0)
         (*
          0.5
          (/ (* (* w h) (* (pow D 2.0) (pow M 2.0))) (* c0 (pow d 2.0))))))
       (if (<= t_2 INFINITY)
         (* t_0 (* 2.0 (/ c0 (* h (* w (pow (/ d D) -2.0))))))
         0.0)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((D * D) * (w * h));
	double t_2 = t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= -5e-142) {
		tmp = ((2.0 * pow((c0 * d), 2.0)) / (pow(D, 2.0) * (w * h))) / (2.0 * w);
	} else if (t_2 <= 0.0) {
		tmp = ((c0 / w) / 2.0) * ((c0 * 0.0) + (0.5 * (((w * h) * (pow(D, 2.0) * pow(M, 2.0))) / (c0 * pow(d, 2.0)))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_0 * (2.0 * (c0 / (h * (w * pow((d / D), -2.0)))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((D * D) * (w * h));
	double t_2 = t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= -5e-142) {
		tmp = ((2.0 * Math.pow((c0 * d), 2.0)) / (Math.pow(D, 2.0) * (w * h))) / (2.0 * w);
	} else if (t_2 <= 0.0) {
		tmp = ((c0 / w) / 2.0) * ((c0 * 0.0) + (0.5 * (((w * h) * (Math.pow(D, 2.0) * Math.pow(M, 2.0))) / (c0 * Math.pow(d, 2.0)))));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * (2.0 * (c0 / (h * (w * Math.pow((d / D), -2.0)))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	t_1 = (c0 * (d * d)) / ((D * D) * (w * h))
	t_2 = t_0 * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))
	tmp = 0
	if t_2 <= -5e-142:
		tmp = ((2.0 * math.pow((c0 * d), 2.0)) / (math.pow(D, 2.0) * (w * h))) / (2.0 * w)
	elif t_2 <= 0.0:
		tmp = ((c0 / w) / 2.0) * ((c0 * 0.0) + (0.5 * (((w * h) * (math.pow(D, 2.0) * math.pow(M, 2.0))) / (c0 * math.pow(d, 2.0)))))
	elif t_2 <= math.inf:
		tmp = t_0 * (2.0 * (c0 / (h * (w * math.pow((d / D), -2.0)))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(D * D) * Float64(w * h)))
	t_2 = Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M)))))
	tmp = 0.0
	if (t_2 <= -5e-142)
		tmp = Float64(Float64(Float64(2.0 * (Float64(c0 * d) ^ 2.0)) / Float64((D ^ 2.0) * Float64(w * h))) / Float64(2.0 * w));
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(Float64(c0 / w) / 2.0) * Float64(Float64(c0 * 0.0) + Float64(0.5 * Float64(Float64(Float64(w * h) * Float64((D ^ 2.0) * (M ^ 2.0))) / Float64(c0 * (d ^ 2.0))))));
	elseif (t_2 <= Inf)
		tmp = Float64(t_0 * Float64(2.0 * Float64(c0 / Float64(h * Float64(w * (Float64(d / D) ^ -2.0))))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	t_1 = (c0 * (d * d)) / ((D * D) * (w * h));
	t_2 = t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	tmp = 0.0;
	if (t_2 <= -5e-142)
		tmp = ((2.0 * ((c0 * d) ^ 2.0)) / ((D ^ 2.0) * (w * h))) / (2.0 * w);
	elseif (t_2 <= 0.0)
		tmp = ((c0 / w) / 2.0) * ((c0 * 0.0) + (0.5 * (((w * h) * ((D ^ 2.0) * (M ^ 2.0))) / (c0 * (d ^ 2.0)))));
	elseif (t_2 <= Inf)
		tmp = t_0 * (2.0 * (c0 / (h * (w * ((d / D) ^ -2.0)))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-142], N[(N[(N[(2.0 * N[Power[N[(c0 * d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[D, 2.0], $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(N[(c0 / w), $MachinePrecision] / 2.0), $MachinePrecision] * N[(N[(c0 * 0.0), $MachinePrecision] + N[(0.5 * N[(N[(N[(w * h), $MachinePrecision] * N[(N[Power[D, 2.0], $MachinePrecision] * N[Power[M, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c0 * N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(t$95$0 * N[(2.0 * N[(c0 / N[(h * N[(w * N[Power[N[(d / D), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (/.f64 c0 (*.f64 2 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))))) < -5.0000000000000002e-142

   1. Initial program 84.5%

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

   3. Simplified 81.3%

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

   5. Taylor expanded in c0 around inf 87.5%

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

      1. *-commutative 87.5%

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

      2. associate-/l/ 84.4%

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

      3. associate-*l/ 84.4%

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

      4. associate-/l* 85.3%

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

      5. associate-/r* 85.3%

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

      6. associate-/r* 85.4%

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

      7. *-commutative 85.4%

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

      8. associate-/l/ 85.4%

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

      9. *-commutative 85.4%

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

   7. Simplified 85.4%

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

      1. associate-*l/ 85.4%

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

      2. associate-/r* 85.4%

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

      3. clear-num 84.3%

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

      4. unpow2 84.3%

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

      5. unpow2 84.3%

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

      6. frac-times 84.2%

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

      7. pow2 84.2%

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

      8. pow-flip 85.4%

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

      9. metadata-eval 85.4%

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

      10. *-commutative 85.4%

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

   9. Applied egg-rr 85.4%

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

   10. Taylor expanded in c0 around 0 81.0%

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

       1. unpow2 81.0%

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

       2. unpow2 81.0%

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

       3. swap-sqr 87.2%

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

       4. unpow2 87.2%

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

       5. associate-*r/ 87.2%

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

   12. Simplified 87.2%

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

   if -5.0000000000000002e-142 < (*.f64 (/.f64 c0 (*.f64 2 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 44.4%

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

   3. Simplified 17.3%

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

   6. Applied egg-rr 15.1%

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

   7. Taylor expanded in c0 around -inf 75.4%

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

      1. associate-*r* 75.4%

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

      2. mul-1-neg 75.4%

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

      3. distribute-lft1-in 75.4%

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

      4. metadata-eval 75.4%

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

      5. mul0-lft 75.4%

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

      6. distribute-lft-neg-in 75.4%

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

      7. distribute-rgt-neg-in 75.4%

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

      8. metadata-eval 75.4%

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

      9. associate-*r* 78.0%

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

   9. Simplified 78.0%

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

   if 0.0 < (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 79.6%

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

   3. Simplified 73.4%

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

   5. Taylor expanded in c0 around inf 82.1%

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

      1. *-commutative 82.1%

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

      2. associate-/l/ 81.1%

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

      3. associate-*l/ 78.0%

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

      4. associate-/l* 77.1%

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

      5. associate-/r* 77.1%

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

      6. associate-/r* 80.0%

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

      7. *-commutative 80.0%

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

      8. associate-/l/ 77.3%

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

      9. *-commutative 77.3%

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

   7. Simplified 77.3%

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

      1. *-un-lft-identity 77.3%

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

      2. clear-num 76.1%

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

      3. unpow2 76.1%

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

      4. unpow2 76.1%

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

      5. frac-times 76.1%

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

      6. pow2 76.1%

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

      7. pow-flip 77.3%

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

      8. metadata-eval 77.3%

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

   9. Applied egg-rr 77.3%

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

       1. *-lft-identity 77.3%

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

   11. Simplified 77.3%

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

       1. pow1 77.3%

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

       2. associate-*l* 82.1%

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

   13. Applied egg-rr 82.1%

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

       1. unpow1 82.1%

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

       2. *-commutative 82.1%

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

   15. Simplified 82.1%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 2 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 2.2%

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

   5. Taylor expanded in c0 around -inf 3.0%

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

      1. associate-*r* 3.0%

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

      2. neg-mul-1 3.0%

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

      3. distribute-lft1-in 3.0%

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

      4. metadata-eval 3.0%

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

      5. mul0-lft 35.7%

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

      6. distribute-lft-neg-in 35.7%

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

      7. distribute-rgt-neg-in 35.7%

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

      8. metadata-eval 35.7%

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

   7. Simplified 35.7%

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

   8. Taylor expanded in c0 around 0 45.9%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq -5 \cdot 10^{-142}:\\ \;\;\;\;\frac{\frac{2 \cdot {\left(c0 \cdot d\right)}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}{2 \cdot w}\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq 0:\\ \;\;\;\;\frac{\frac{c0}{w}}{2} \cdot \left(c0 \cdot 0 + 0.5 \cdot \frac{\left(w \cdot h\right) \cdot \left({D}^{2} \cdot {M}^{2}\right)}{c0 \cdot {d}^{2}}\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0}{h \cdot \left(w \cdot {\left(\frac{d}{D}\right)}^{-2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 55.4% accurate, 0.5× speedup

Math

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

FPCore

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

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 t_1 = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

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

Python

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 76.4%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 2 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 2.2%

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

   5. Taylor expanded in c0 around -inf 3.0%

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

      1. associate-*r* 3.0%

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

      2. neg-mul-1 3.0%

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

      3. distribute-lft1-in 3.0%

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

      4. metadata-eval 3.0%

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

      5. mul0-lft 35.7%

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

      6. distribute-lft-neg-in 35.7%

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

      7. distribute-rgt-neg-in 35.7%

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

      8. metadata-eval 35.7%

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

   7. Simplified 35.7%

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

   8. Taylor expanded in c0 around 0 45.9%

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

4. Final simplification 54.7%

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

Alternative 3: 38.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 7.2 \cdot 10^{-247}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 2.25 \cdot 10^{-160} \lor \neg \left(M \leq 5000000000\right):\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0}{h \cdot \left(w \cdot {\left(\frac{d}{D}\right)}^{-2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= M 7.2e-247)
   0.0
   (if (or (<= M 2.25e-160) (not (<= M 5000000000.0)))
     (* (/ c0 (* 2.0 w)) (* 2.0 (/ c0 (* h (* w (pow (/ d D) -2.0))))))
     0.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 7.2e-247) {
		tmp = 0.0;
	} else if ((M <= 2.25e-160) || !(M <= 5000000000.0)) {
		tmp = (c0 / (2.0 * w)) * (2.0 * (c0 / (h * (w * pow((d / D), -2.0)))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 7.2d-247) then
        tmp = 0.0d0
    else if ((m <= 2.25d-160) .or. (.not. (m <= 5000000000.0d0))) then
        tmp = (c0 / (2.0d0 * w)) * (2.0d0 * (c0 / (h * (w * ((d_1 / d) ** (-2.0d0))))))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 7.2e-247) {
		tmp = 0.0;
	} else if ((M <= 2.25e-160) || !(M <= 5000000000.0)) {
		tmp = (c0 / (2.0 * w)) * (2.0 * (c0 / (h * (w * Math.pow((d / D), -2.0)))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if M <= 7.2e-247:
		tmp = 0.0
	elif (M <= 2.25e-160) or not (M <= 5000000000.0):
		tmp = (c0 / (2.0 * w)) * (2.0 * (c0 / (h * (w * math.pow((d / D), -2.0)))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (M <= 7.2e-247)
		tmp = 0.0;
	elseif ((M <= 2.25e-160) || !(M <= 5000000000.0))
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(2.0 * Float64(c0 / Float64(h * Float64(w * (Float64(d / D) ^ -2.0))))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (M <= 7.2e-247)
		tmp = 0.0;
	elseif ((M <= 2.25e-160) || ~((M <= 5000000000.0)))
		tmp = (c0 / (2.0 * w)) * (2.0 * (c0 / (h * (w * ((d / D) ^ -2.0)))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[M, 7.2e-247], 0.0, If[Or[LessEqual[M, 2.25e-160], N[Not[LessEqual[M, 5000000000.0]], $MachinePrecision]], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(c0 / N[(h * N[(w * N[Power[N[(d / D), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if M < 7.1999999999999994e-247 or 2.25000000000000013e-160 < M < 5e9

   1. Initial program 19.9%

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

   3. Simplified 20.5%

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

   5. Taylor expanded in c0 around -inf 5.1%

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

      1. associate-*r* 5.1%

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

      2. neg-mul-1 5.1%

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

      3. distribute-lft1-in 5.1%

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

      4. metadata-eval 5.1%

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

      5. mul0-lft 32.5%

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

      6. distribute-lft-neg-in 32.5%

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

      7. distribute-rgt-neg-in 32.5%

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

      8. metadata-eval 32.5%

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

   7. Simplified 32.5%

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

   8. Taylor expanded in c0 around 0 41.9%

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

   if 7.1999999999999994e-247 < M < 2.25000000000000013e-160 or 5e9 < M

   1. Initial program 27.9%

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

   3. Simplified 27.9%

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

   5. Taylor expanded in c0 around inf 47.2%

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

      1. *-commutative 47.2%

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

      2. associate-/l/ 47.1%

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

      3. associate-*l/ 47.1%

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

      4. associate-/l* 47.1%

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

      5. associate-/r* 48.6%

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

      6. associate-/r* 50.0%

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

      7. *-commutative 50.0%

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

      8. associate-/l/ 48.7%

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

      9. *-commutative 48.7%

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

   7. Simplified 48.7%

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

      1. *-un-lft-identity 48.7%

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

      2. clear-num 48.7%

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

      3. unpow2 48.7%

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

      4. unpow2 48.7%

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

      5. frac-times 59.2%

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

      6. pow2 59.2%

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

      7. pow-flip 59.2%

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

      8. metadata-eval 59.2%

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

   9. Applied egg-rr 59.2%

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

       1. *-lft-identity 59.2%

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

   11. Simplified 59.2%

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

       1. pow1 59.2%

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

       2. associate-*l* 60.7%

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

   13. Applied egg-rr 60.7%

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

       1. unpow1 60.7%

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

       2. *-commutative 60.7%

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

   15. Simplified 60.7%

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

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 7.2 \cdot 10^{-247}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 2.25 \cdot 10^{-160} \lor \neg \left(M \leq 5000000000\right):\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0}{h \cdot \left(w \cdot {\left(\frac{d}{D}\right)}^{-2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 38.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{d}{D}\right)}^{-2}\\ \mathbf{if}\;M \leq 3.2 \cdot 10^{-246}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 3.2 \cdot 10^{-160}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0}{h \cdot \left(w \cdot t\_0\right)}\right)\\ \mathbf{elif}\;M \leq 98000000:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot t\_0}}{w}\right)}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (pow (/ d D) -2.0)))
   (if (<= M 3.2e-246)
     0.0
     (if (<= M 3.2e-160)
       (* (/ c0 (* 2.0 w)) (* 2.0 (/ c0 (* h (* w t_0)))))
       (if (<= M 98000000.0)
         0.0
         (/ (* c0 (* 2.0 (/ (/ c0 (* h t_0)) w))) (* 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 tmp;
	if (M <= 3.2e-246) {
		tmp = 0.0;
	} else if (M <= 3.2e-160) {
		tmp = (c0 / (2.0 * w)) * (2.0 * (c0 / (h * (w * t_0))));
	} else if (M <= 98000000.0) {
		tmp = 0.0;
	} else {
		tmp = (c0 * (2.0 * ((c0 / (h * t_0)) / w))) / (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) :: t_0
    real(8) :: tmp
    t_0 = (d_1 / d) ** (-2.0d0)
    if (m <= 3.2d-246) then
        tmp = 0.0d0
    else if (m <= 3.2d-160) then
        tmp = (c0 / (2.0d0 * w)) * (2.0d0 * (c0 / (h * (w * t_0))))
    else if (m <= 98000000.0d0) then
        tmp = 0.0d0
    else
        tmp = (c0 * (2.0d0 * ((c0 / (h * t_0)) / w))) / (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 t_0 = Math.pow((d / D), -2.0);
	double tmp;
	if (M <= 3.2e-246) {
		tmp = 0.0;
	} else if (M <= 3.2e-160) {
		tmp = (c0 / (2.0 * w)) * (2.0 * (c0 / (h * (w * t_0))));
	} else if (M <= 98000000.0) {
		tmp = 0.0;
	} else {
		tmp = (c0 * (2.0 * ((c0 / (h * t_0)) / w))) / (2.0 * w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = math.pow((d / D), -2.0)
	tmp = 0
	if M <= 3.2e-246:
		tmp = 0.0
	elif M <= 3.2e-160:
		tmp = (c0 / (2.0 * w)) * (2.0 * (c0 / (h * (w * t_0))))
	elif M <= 98000000.0:
		tmp = 0.0
	else:
		tmp = (c0 * (2.0 * ((c0 / (h * t_0)) / w))) / (2.0 * w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(d / D) ^ -2.0
	tmp = 0.0
	if (M <= 3.2e-246)
		tmp = 0.0;
	elseif (M <= 3.2e-160)
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(2.0 * Float64(c0 / Float64(h * Float64(w * t_0)))));
	elseif (M <= 98000000.0)
		tmp = 0.0;
	else
		tmp = Float64(Float64(c0 * Float64(2.0 * Float64(Float64(c0 / Float64(h * t_0)) / w))) / 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;
	tmp = 0.0;
	if (M <= 3.2e-246)
		tmp = 0.0;
	elseif (M <= 3.2e-160)
		tmp = (c0 / (2.0 * w)) * (2.0 * (c0 / (h * (w * t_0))));
	elseif (M <= 98000000.0)
		tmp = 0.0;
	else
		tmp = (c0 * (2.0 * ((c0 / (h * t_0)) / w))) / (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]}, If[LessEqual[M, 3.2e-246], 0.0, If[LessEqual[M, 3.2e-160], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(c0 / N[(h * N[(w * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 98000000.0], 0.0, N[(N[(c0 * N[(2.0 * N[(N[(c0 / N[(h * t$95$0), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if M < 3.2000000000000001e-246 or 3.20000000000000009e-160 < M < 9.8e7

   1. Initial program 19.9%

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

   3. Simplified 20.5%

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

   5. Taylor expanded in c0 around -inf 5.1%

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

      1. associate-*r* 5.1%

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

      2. neg-mul-1 5.1%

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

      3. distribute-lft1-in 5.1%

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

      4. metadata-eval 5.1%

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

      5. mul0-lft 32.5%

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

      6. distribute-lft-neg-in 32.5%

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

      7. distribute-rgt-neg-in 32.5%

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

      8. metadata-eval 32.5%

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

   7. Simplified 32.5%

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

   8. Taylor expanded in c0 around 0 41.9%

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

   if 3.2000000000000001e-246 < M < 3.20000000000000009e-160

   1. Initial program 67.1%

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

   3. Simplified 67.1%

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

   5. Taylor expanded in c0 around inf 56.4%

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

      1. *-commutative 56.4%

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

      2. associate-/l/ 56.4%

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

      3. associate-*l/ 56.4%

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

      4. associate-/l* 56.4%

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

      5. associate-/r* 56.4%

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

      6. associate-/r* 56.4%

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

      7. *-commutative 56.4%

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

      8. associate-/l/ 56.4%

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

      9. *-commutative 56.4%

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

   7. Simplified 56.4%

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

      1. *-un-lft-identity 56.4%

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

      2. clear-num 56.4%

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

      3. unpow2 56.4%

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

      4. unpow2 56.4%

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

      5. frac-times 61.9%

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

      6. pow2 61.9%

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

      7. pow-flip 61.9%

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

      8. metadata-eval 61.9%

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

   9. Applied egg-rr 61.9%

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

       1. *-lft-identity 61.9%

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

   11. Simplified 61.9%

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

       1. pow1 61.9%

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

       2. associate-*l* 67.4%

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

   13. Applied egg-rr 67.4%

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

       1. unpow1 67.4%

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

       2. *-commutative 67.4%

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

   15. Simplified 67.4%

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

   if 9.8e7 < M

   1. Initial program 14.1%

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

   3. Simplified 14.1%

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

   5. Taylor expanded in c0 around inf 43.9%

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

      1. *-commutative 43.9%

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

      2. associate-/l/ 43.9%

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

      3. associate-*l/ 43.9%

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

      4. associate-/l* 43.9%

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

      5. associate-/r* 45.9%

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

      6. associate-/r* 47.8%

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

      7. *-commutative 47.8%

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

      8. associate-/l/ 46.0%

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

      9. *-commutative 46.0%

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

   7. Simplified 46.0%

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

      1. associate-*l/ 46.1%

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

      2. associate-/r* 47.8%

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

      3. clear-num 47.8%

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

      4. unpow2 47.8%

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

      5. unpow2 47.8%

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

      6. frac-times 59.9%

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

      7. pow2 59.9%

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

      8. pow-flip 59.9%

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

      9. metadata-eval 59.9%

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

      10. *-commutative 59.9%

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

   9. Applied egg-rr 59.9%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3.2 \cdot 10^{-246}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 3.2 \cdot 10^{-160}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0}{h \cdot \left(w \cdot {\left(\frac{d}{D}\right)}^{-2}\right)}\right)\\ \mathbf{elif}\;M \leq 98000000:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot {\left(\frac{d}{D}\right)}^{-2}}}{w}\right)}{2 \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 38.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{d}{D}\right)}^{-2}\\ \mathbf{if}\;M \leq 4 \cdot 10^{-248}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 2.8 \cdot 10^{-160}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0}{h \cdot \left(w \cdot t\_0\right)}\right)\\ \mathbf{elif}\;M \leq 2450000000:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(c0 \cdot 2\right) \cdot \frac{c0}{h \cdot t\_0}}{w}}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (pow (/ d D) -2.0)))
   (if (<= M 4e-248)
     0.0
     (if (<= M 2.8e-160)
       (* (/ c0 (* 2.0 w)) (* 2.0 (/ c0 (* h (* w t_0)))))
       (if (<= M 2450000000.0)
         0.0
         (/ (/ (* (* c0 2.0) (/ c0 (* h t_0))) w) (* 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 tmp;
	if (M <= 4e-248) {
		tmp = 0.0;
	} else if (M <= 2.8e-160) {
		tmp = (c0 / (2.0 * w)) * (2.0 * (c0 / (h * (w * t_0))));
	} else if (M <= 2450000000.0) {
		tmp = 0.0;
	} else {
		tmp = (((c0 * 2.0) * (c0 / (h * t_0))) / w) / (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) :: t_0
    real(8) :: tmp
    t_0 = (d_1 / d) ** (-2.0d0)
    if (m <= 4d-248) then
        tmp = 0.0d0
    else if (m <= 2.8d-160) then
        tmp = (c0 / (2.0d0 * w)) * (2.0d0 * (c0 / (h * (w * t_0))))
    else if (m <= 2450000000.0d0) then
        tmp = 0.0d0
    else
        tmp = (((c0 * 2.0d0) * (c0 / (h * t_0))) / w) / (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 t_0 = Math.pow((d / D), -2.0);
	double tmp;
	if (M <= 4e-248) {
		tmp = 0.0;
	} else if (M <= 2.8e-160) {
		tmp = (c0 / (2.0 * w)) * (2.0 * (c0 / (h * (w * t_0))));
	} else if (M <= 2450000000.0) {
		tmp = 0.0;
	} else {
		tmp = (((c0 * 2.0) * (c0 / (h * t_0))) / w) / (2.0 * w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = math.pow((d / D), -2.0)
	tmp = 0
	if M <= 4e-248:
		tmp = 0.0
	elif M <= 2.8e-160:
		tmp = (c0 / (2.0 * w)) * (2.0 * (c0 / (h * (w * t_0))))
	elif M <= 2450000000.0:
		tmp = 0.0
	else:
		tmp = (((c0 * 2.0) * (c0 / (h * t_0))) / w) / (2.0 * w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(d / D) ^ -2.0
	tmp = 0.0
	if (M <= 4e-248)
		tmp = 0.0;
	elseif (M <= 2.8e-160)
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(2.0 * Float64(c0 / Float64(h * Float64(w * t_0)))));
	elseif (M <= 2450000000.0)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(Float64(c0 * 2.0) * Float64(c0 / Float64(h * t_0))) / w) / 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;
	tmp = 0.0;
	if (M <= 4e-248)
		tmp = 0.0;
	elseif (M <= 2.8e-160)
		tmp = (c0 / (2.0 * w)) * (2.0 * (c0 / (h * (w * t_0))));
	elseif (M <= 2450000000.0)
		tmp = 0.0;
	else
		tmp = (((c0 * 2.0) * (c0 / (h * t_0))) / w) / (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]}, If[LessEqual[M, 4e-248], 0.0, If[LessEqual[M, 2.8e-160], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(c0 / N[(h * N[(w * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 2450000000.0], 0.0, N[(N[(N[(N[(c0 * 2.0), $MachinePrecision] * N[(c0 / N[(h * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if M < 3.99999999999999992e-248 or 2.80000000000000016e-160 < M < 2.45e9

   1. Initial program 19.9%

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

   3. Simplified 20.5%

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

   5. Taylor expanded in c0 around -inf 5.1%

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

      1. associate-*r* 5.1%

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

      2. neg-mul-1 5.1%

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

      3. distribute-lft1-in 5.1%

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

      4. metadata-eval 5.1%

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

      5. mul0-lft 32.5%

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

      6. distribute-lft-neg-in 32.5%

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

      7. distribute-rgt-neg-in 32.5%

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

      8. metadata-eval 32.5%

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

   7. Simplified 32.5%

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

   8. Taylor expanded in c0 around 0 41.9%

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

   if 3.99999999999999992e-248 < M < 2.80000000000000016e-160

   1. Initial program 67.1%

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

   3. Simplified 67.1%

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

   5. Taylor expanded in c0 around inf 56.4%

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

      1. *-commutative 56.4%

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

      2. associate-/l/ 56.4%

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

      3. associate-*l/ 56.4%

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

      4. associate-/l* 56.4%

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

      5. associate-/r* 56.4%

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

      6. associate-/r* 56.4%

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

      7. *-commutative 56.4%

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

      8. associate-/l/ 56.4%

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

      9. *-commutative 56.4%

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

   7. Simplified 56.4%

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

      1. *-un-lft-identity 56.4%

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

      2. clear-num 56.4%

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

      3. unpow2 56.4%

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

      4. unpow2 56.4%

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

      5. frac-times 61.9%

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

      6. pow2 61.9%

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

      7. pow-flip 61.9%

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

      8. metadata-eval 61.9%

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

   9. Applied egg-rr 61.9%

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

       1. *-lft-identity 61.9%

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

   11. Simplified 61.9%

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

       1. pow1 61.9%

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

       2. associate-*l* 67.4%

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

   13. Applied egg-rr 67.4%

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

       1. unpow1 67.4%

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

       2. *-commutative 67.4%

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

   15. Simplified 67.4%

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

   if 2.45e9 < M

   1. Initial program 14.1%

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

   3. Simplified 14.1%

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

   5. Taylor expanded in c0 around inf 43.9%

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

      1. *-commutative 43.9%

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

      2. associate-/l/ 43.9%

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

      3. associate-*l/ 43.9%

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

      4. associate-/l* 43.9%

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

      5. associate-/r* 45.9%

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

      6. associate-/r* 47.8%

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

      7. *-commutative 47.8%

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

      8. associate-/l/ 46.0%

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

      9. *-commutative 46.0%

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

   7. Simplified 46.0%

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

      1. associate-*l/ 46.1%

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

      2. associate-/r* 47.8%

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

      3. clear-num 47.8%

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

      4. unpow2 47.8%

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

      5. unpow2 47.8%

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

      6. frac-times 59.9%

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

      7. pow2 59.9%

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

      8. pow-flip 59.9%

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

      9. metadata-eval 59.9%

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

      10. *-commutative 59.9%

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

   9. Applied egg-rr 59.9%

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

       1. pow1 59.9%

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

       2. associate-*r* 59.9%

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

       3. associate-/r* 60.0%

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

   11. Applied egg-rr 60.0%

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

       1. unpow1 60.0%

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

       2. associate-*r/ 60.1%

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

       3. associate-/l/ 60.0%

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

       4. *-commutative 60.0%

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

   13. Simplified 60.0%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4 \cdot 10^{-248}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 2.8 \cdot 10^{-160}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0}{h \cdot \left(w \cdot {\left(\frac{d}{D}\right)}^{-2}\right)}\right)\\ \mathbf{elif}\;M \leq 2450000000:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(c0 \cdot 2\right) \cdot \frac{c0}{h \cdot {\left(\frac{d}{D}\right)}^{-2}}}{w}}{2 \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 38.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 7.5 \cdot 10^{-249}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 2.2 \cdot 10^{-160} \lor \neg \left(M \leq 1500000000\right):\\ \;\;\;\;\frac{c0}{w} \cdot \frac{\frac{\frac{c0}{h}}{{\left(\frac{d}{D}\right)}^{-2}}}{w}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= M 7.5e-249)
   0.0
   (if (or (<= M 2.2e-160) (not (<= M 1500000000.0)))
     (* (/ c0 w) (/ (/ (/ c0 h) (pow (/ d D) -2.0)) w))
     0.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 7.5e-249) {
		tmp = 0.0;
	} else if ((M <= 2.2e-160) || !(M <= 1500000000.0)) {
		tmp = (c0 / w) * (((c0 / h) / pow((d / D), -2.0)) / w);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 7.5d-249) then
        tmp = 0.0d0
    else if ((m <= 2.2d-160) .or. (.not. (m <= 1500000000.0d0))) then
        tmp = (c0 / w) * (((c0 / h) / ((d_1 / d) ** (-2.0d0))) / w)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 7.5e-249) {
		tmp = 0.0;
	} else if ((M <= 2.2e-160) || !(M <= 1500000000.0)) {
		tmp = (c0 / w) * (((c0 / h) / Math.pow((d / D), -2.0)) / w);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if M <= 7.5e-249:
		tmp = 0.0
	elif (M <= 2.2e-160) or not (M <= 1500000000.0):
		tmp = (c0 / w) * (((c0 / h) / math.pow((d / D), -2.0)) / w)
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (M <= 7.5e-249)
		tmp = 0.0;
	elseif ((M <= 2.2e-160) || !(M <= 1500000000.0))
		tmp = Float64(Float64(c0 / w) * Float64(Float64(Float64(c0 / h) / (Float64(d / D) ^ -2.0)) / w));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (M <= 7.5e-249)
		tmp = 0.0;
	elseif ((M <= 2.2e-160) || ~((M <= 1500000000.0)))
		tmp = (c0 / w) * (((c0 / h) / ((d / D) ^ -2.0)) / w);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[M, 7.5e-249], 0.0, If[Or[LessEqual[M, 2.2e-160], N[Not[LessEqual[M, 1500000000.0]], $MachinePrecision]], N[(N[(c0 / w), $MachinePrecision] * N[(N[(N[(c0 / h), $MachinePrecision] / N[Power[N[(d / D), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if M < 7.50000000000000034e-249 or 2.2e-160 < M < 1.5e9

   1. Initial program 19.9%

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

   3. Simplified 20.5%

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

   5. Taylor expanded in c0 around -inf 5.1%

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

      1. associate-*r* 5.1%

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

      2. neg-mul-1 5.1%

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

      3. distribute-lft1-in 5.1%

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

      4. metadata-eval 5.1%

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

      5. mul0-lft 32.5%

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

      6. distribute-lft-neg-in 32.5%

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

      7. distribute-rgt-neg-in 32.5%

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

      8. metadata-eval 32.5%

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

   7. Simplified 32.5%

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

   8. Taylor expanded in c0 around 0 41.9%

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

   if 7.50000000000000034e-249 < M < 2.2e-160 or 1.5e9 < M

   1. Initial program 27.9%

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

   3. Simplified 27.9%

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

   5. Taylor expanded in c0 around inf 47.2%

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

      1. *-commutative 47.2%

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

      2. associate-/l/ 47.1%

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

      3. associate-*l/ 47.1%

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

      4. associate-/l* 47.1%

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

      5. associate-/r* 48.6%

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

      6. associate-/r* 50.0%

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

      7. *-commutative 50.0%

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

      8. associate-/l/ 48.7%

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

      9. *-commutative 48.7%

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

   7. Simplified 48.7%

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

      1. *-un-lft-identity 48.7%

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

      2. clear-num 48.7%

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

      3. unpow2 48.7%

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

      4. unpow2 48.7%

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

      5. frac-times 59.2%

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

      6. pow2 59.2%

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

      7. pow-flip 59.2%

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

      8. metadata-eval 59.2%

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

   9. Applied egg-rr 59.2%

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

       1. *-lft-identity 59.2%

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

   11. Simplified 59.2%

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

       1. pow1 59.2%

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

       2. associate-*r* 59.2%

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

       3. associate-/l/ 59.2%

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

       4. div-inv 59.2%

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

       5. metadata-eval 59.2%

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

       6. associate-/r* 60.4%

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

       7. associate-/r* 60.5%

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

   13. Applied egg-rr 60.5%

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

       1. unpow1 60.5%

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

       2. *-commutative 60.5%

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

       3. associate-*l* 60.5%

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

       4. metadata-eval 60.5%

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

       5. *-rgt-identity 60.5%

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

   15. Simplified 60.5%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 7.5 \cdot 10^{-249}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 2.2 \cdot 10^{-160} \lor \neg \left(M \leq 1500000000\right):\\ \;\;\;\;\frac{c0}{w} \cdot \frac{\frac{\frac{c0}{h}}{{\left(\frac{d}{D}\right)}^{-2}}}{w}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 38.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 5 \cdot 10^{-248}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 1.9 \cdot 10^{-160} \lor \neg \left(M \leq 21500000\right):\\ \;\;\;\;\frac{c0}{w} \cdot \frac{\frac{c0}{h}}{w \cdot {\left(\frac{d}{D}\right)}^{-2}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= M 5e-248)
   0.0
   (if (or (<= M 1.9e-160) (not (<= M 21500000.0)))
     (* (/ c0 w) (/ (/ c0 h) (* w (pow (/ d D) -2.0))))
     0.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 5e-248) {
		tmp = 0.0;
	} else if ((M <= 1.9e-160) || !(M <= 21500000.0)) {
		tmp = (c0 / w) * ((c0 / h) / (w * pow((d / D), -2.0)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 5d-248) then
        tmp = 0.0d0
    else if ((m <= 1.9d-160) .or. (.not. (m <= 21500000.0d0))) then
        tmp = (c0 / w) * ((c0 / h) / (w * ((d_1 / d) ** (-2.0d0))))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 5e-248) {
		tmp = 0.0;
	} else if ((M <= 1.9e-160) || !(M <= 21500000.0)) {
		tmp = (c0 / w) * ((c0 / h) / (w * Math.pow((d / D), -2.0)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if M <= 5e-248:
		tmp = 0.0
	elif (M <= 1.9e-160) or not (M <= 21500000.0):
		tmp = (c0 / w) * ((c0 / h) / (w * math.pow((d / D), -2.0)))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (M <= 5e-248)
		tmp = 0.0;
	elseif ((M <= 1.9e-160) || !(M <= 21500000.0))
		tmp = Float64(Float64(c0 / w) * Float64(Float64(c0 / h) / Float64(w * (Float64(d / D) ^ -2.0))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (M <= 5e-248)
		tmp = 0.0;
	elseif ((M <= 1.9e-160) || ~((M <= 21500000.0)))
		tmp = (c0 / w) * ((c0 / h) / (w * ((d / D) ^ -2.0)));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[M, 5e-248], 0.0, If[Or[LessEqual[M, 1.9e-160], N[Not[LessEqual[M, 21500000.0]], $MachinePrecision]], N[(N[(c0 / w), $MachinePrecision] * N[(N[(c0 / h), $MachinePrecision] / N[(w * N[Power[N[(d / D), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if M < 5.0000000000000001e-248 or 1.8999999999999999e-160 < M < 2.15e7

   1. Initial program 19.9%

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

   3. Simplified 20.5%

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

   5. Taylor expanded in c0 around -inf 5.1%

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

      1. associate-*r* 5.1%

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

      2. neg-mul-1 5.1%

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

      3. distribute-lft1-in 5.1%

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

      4. metadata-eval 5.1%

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

      5. mul0-lft 32.5%

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

      6. distribute-lft-neg-in 32.5%

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

      7. distribute-rgt-neg-in 32.5%

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

      8. metadata-eval 32.5%

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

   7. Simplified 32.5%

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

   8. Taylor expanded in c0 around 0 41.9%

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

   if 5.0000000000000001e-248 < M < 1.8999999999999999e-160 or 2.15e7 < M

   1. Initial program 27.9%

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

   3. Simplified 27.9%

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

   5. Taylor expanded in c0 around inf 47.2%

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

      1. *-commutative 47.2%

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

      2. associate-/l/ 47.1%

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

      3. associate-*l/ 47.1%

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

      4. associate-/l* 47.1%

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

      5. associate-/r* 48.6%

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

      6. associate-/r* 50.0%

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

      7. *-commutative 50.0%

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

      8. associate-/l/ 48.7%

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

      9. *-commutative 48.7%

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

   7. Simplified 48.7%

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

      1. *-un-lft-identity 48.7%

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

      2. clear-num 48.7%

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

      3. unpow2 48.7%

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

      4. unpow2 48.7%

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

      5. frac-times 59.2%

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

      6. pow2 59.2%

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

      7. pow-flip 59.2%

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

      8. metadata-eval 59.2%

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

   9. Applied egg-rr 59.2%

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

       1. *-lft-identity 59.2%

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

   11. Simplified 59.2%

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

       1. pow1 59.2%

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

       2. associate-*r* 59.2%

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

       3. associate-/l/ 59.2%

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

       4. div-inv 59.2%

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

       5. metadata-eval 59.2%

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

       6. associate-/r* 60.4%

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

       7. associate-/r* 60.5%

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

   13. Applied egg-rr 60.5%

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

       1. unpow1 60.5%

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

       2. *-commutative 60.5%

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

       3. associate-/l/ 60.5%

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

       4. associate-*l* 60.5%

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

       5. metadata-eval 60.5%

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

   15. Simplified 60.5%

       \[\leadsto \color{blue}{\frac{\frac{c0}{h}}{w \cdot {\left(\frac{d}{D}\right)}^{-2}} \cdot \left(\frac{c0}{w} \cdot 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 5 \cdot 10^{-248}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 1.9 \cdot 10^{-160} \lor \neg \left(M \leq 21500000\right):\\ \;\;\;\;\frac{c0}{w} \cdot \frac{\frac{c0}{h}}{w \cdot {\left(\frac{d}{D}\right)}^{-2}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 33.5% accurate, 151.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.1%

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

3. Simplified 22.5%

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

5. Taylor expanded in c0 around -inf 4.5%

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

   1. associate-*r* 4.5%

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

   2. neg-mul-1 4.5%

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

   3. distribute-lft1-in 4.5%

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

   4. metadata-eval 4.5%

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

   5. mul0-lft 28.1%

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

   6. distribute-lft-neg-in 28.1%

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

   7. distribute-rgt-neg-in 28.1%

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

   8. metadata-eval 28.1%

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

7. Simplified 28.1%

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

8. Taylor expanded in c0 around 0 35.4%

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

9. Final simplification 35.4%

   \[\leadsto 0 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024030 
(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))))))