Henrywood and Agarwal, Equation (13)

Percentage Accurate: 25.8% → 58.1%

Time: 32.0s

Alternatives: 9

Speedup: 151.0×

• could not determine a ground truth

  1. c0 = -9.760100635541865e+280
  2. w = -3.2612175012283304e-301
  3. h = -7.309211467174581e-118
  4. D = 6.515768966133767e-277
  5. d = -1.3313809068355972e+261
  6. M = -8.740880801855367e-302

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: 25.8% 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: 58.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d}{D} \cdot \sqrt{\frac{c0}{w \cdot h}}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_2 := \frac{c0}{2 \cdot w} \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right)\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-105}:\\ \;\;\;\;{t\_0}^{2} \cdot \frac{c0}{w}\\ \mathbf{elif}\;t\_2 \leq 10^{-187}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;{\left(t\_0 \cdot \sqrt{\frac{c0}{w}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ d D) (sqrt (/ c0 (* w h)))))
        (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_2 (* (/ c0 (* 2.0 w)) (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))))
   (if (<= t_2 -2e-105)
     (* (pow t_0 2.0) (/ c0 w))
     (if (<= t_2 1e-187)
       t_2
       (if (<= t_2 INFINITY) (pow (* t_0 (sqrt (/ c0 w))) 2.0) 0.0)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (d / D) * sqrt((c0 / (w * h)));
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_2 = (c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= -2e-105) {
		tmp = pow(t_0, 2.0) * (c0 / w);
	} else if (t_2 <= 1e-187) {
		tmp = t_2;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = pow((t_0 * sqrt((c0 / w))), 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 = (d / D) * Math.sqrt((c0 / (w * h)));
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_2 = (c0 / (2.0 * w)) * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= -2e-105) {
		tmp = Math.pow(t_0, 2.0) * (c0 / w);
	} else if (t_2 <= 1e-187) {
		tmp = t_2;
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = Math.pow((t_0 * Math.sqrt((c0 / w))), 2.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (d / D) * math.sqrt((c0 / (w * h)))
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	t_2 = (c0 / (2.0 * w)) * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))
	tmp = 0
	if t_2 <= -2e-105:
		tmp = math.pow(t_0, 2.0) * (c0 / w)
	elif t_2 <= 1e-187:
		tmp = t_2
	elif t_2 <= math.inf:
		tmp = math.pow((t_0 * math.sqrt((c0 / w))), 2.0)
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(d / D) * sqrt(Float64(c0 / Float64(w * h))))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	t_2 = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M)))))
	tmp = 0.0
	if (t_2 <= -2e-105)
		tmp = Float64((t_0 ^ 2.0) * Float64(c0 / w));
	elseif (t_2 <= 1e-187)
		tmp = t_2;
	elseif (t_2 <= Inf)
		tmp = Float64(t_0 * sqrt(Float64(c0 / w))) ^ 2.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (d / D) * sqrt((c0 / (w * h)));
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	t_2 = (c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	tmp = 0.0;
	if (t_2 <= -2e-105)
		tmp = (t_0 ^ 2.0) * (c0 / w);
	elseif (t_2 <= 1e-187)
		tmp = t_2;
	elseif (t_2 <= Inf)
		tmp = (t_0 * sqrt((c0 / w))) ^ 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(d / D), $MachinePrecision] * N[Sqrt[N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-105], N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(c0 / w), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e-187], t$95$2, If[LessEqual[t$95$2, Infinity], N[Power[N[(t$95$0 * N[Sqrt[N[(c0 / w), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $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))))) < -1.99999999999999993e-105

   1. Initial program 79.8%

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

   3. Simplified 81.9%

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

   5. Taylor expanded in c0 around inf 81.5%

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

      1. *-commutative 81.5%

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

      2. *-commutative 81.5%

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

      3. associate-*r* 81.5%

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

      4. associate-/r* 81.4%

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

      5. *-commutative 81.4%

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

   7. Simplified 81.4%

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

      1. pow1 81.4%

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

      2. associate-*r* 81.4%

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

      3. associate-/l/ 81.4%

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

      4. div-inv 81.4%

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

      5. metadata-eval 81.4%

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

      6. associate-/l/ 81.5%

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

      7. *-commutative 81.5%

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

   9. Applied egg-rr 81.5%

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

       1. unpow1 81.5%

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

       2. *-commutative 81.5%

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

       3. *-commutative 81.5%

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

       4. associate-*r* 81.5%

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

       5. *-commutative 81.5%

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

       6. times-frac 81.9%

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

       7. unpow2 81.9%

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

       8. unpow2 81.9%

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

       9. times-frac 84.4%

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

       10. unpow2 84.4%

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

       11. associate-*l* 84.4%

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

       12. metadata-eval 84.4%

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

       13. *-rgt-identity 84.4%

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

   11. Simplified 84.4%

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

       1. add-sqr-sqrt 84.4%

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

       2. pow2 84.4%

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

       3. div-inv 84.4%

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

       4. associate-*l/ 84.5%

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

       5. clear-num 84.4%

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

       6. sqrt-prod 84.4%

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

       7. unpow2 84.4%

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

       8. sqrt-prod 53.7%

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

       9. add-sqr-sqrt 92.8%

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

   13. Applied egg-rr 92.8%

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

   if -1.99999999999999993e-105 < (*.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))))) < 1e-187

   1. Initial program 69.4%

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

   if 1e-187 < (*.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 69.2%

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

   3. Simplified 69.2%

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

   5. Taylor expanded in c0 around inf 69.3%

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

      1. *-commutative 69.3%

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

      2. *-commutative 69.3%

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

      3. associate-*r* 66.5%

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

      4. associate-/r* 66.2%

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

      5. *-commutative 66.2%

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

   7. Simplified 66.2%

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

      1. pow1 66.2%

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

      2. associate-*r* 66.2%

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

      3. associate-/l/ 66.2%

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

      4. div-inv 66.2%

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

      5. metadata-eval 66.2%

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

      6. associate-/l/ 66.5%

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

      7. *-commutative 66.5%

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

   9. Applied egg-rr 66.5%

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

       1. unpow1 66.5%

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

       2. *-commutative 66.5%

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

       3. *-commutative 66.5%

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

       4. associate-*r* 69.3%

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

       5. *-commutative 69.3%

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

       6. times-frac 74.4%

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

       7. unpow2 74.4%

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

       8. unpow2 74.4%

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

       9. times-frac 81.8%

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

       10. unpow2 81.8%

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

       11. associate-*l* 81.8%

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

       12. metadata-eval 81.8%

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

       13. *-rgt-identity 81.8%

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

   11. Simplified 71.2%

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

       1. add-sqr-sqrt 71.0%

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

       2. pow2 71.0%

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

       3. sqrt-prod 71.1%

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

       4. div-inv 71.1%

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

       5. associate-*l/ 81.7%

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

       6. clear-num 81.8%

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

       7. sqrt-prod 84.3%

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

       8. unpow2 84.3%

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

       9. sqrt-prod 43.3%

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

       10. add-sqr-sqrt 94.4%

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

   13. Applied egg-rr 94.4%

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

   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 3.1%

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

   5. Taylor expanded in c0 around -inf 3.1%

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

      1. associate-*r* 3.1%

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

      2. neg-mul-1 3.1%

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

      3. distribute-lft1-in 3.1%

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

      4. metadata-eval 3.1%

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

      5. mul0-lft 38.6%

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

      6. distribute-lft-neg-in 38.6%

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

      7. distribute-rgt-neg-in 38.6%

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

      8. metadata-eval 38.6%

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

   7. Simplified 38.6%

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

   8. Taylor expanded in c0 around 0 45.7%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq -2 \cdot 10^{-105}:\\ \;\;\;\;{\left(\frac{d}{D} \cdot \sqrt{\frac{c0}{w \cdot h}}\right)}^{2} \cdot \frac{c0}{w}\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 10^{-187}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;{\left(\left(\frac{d}{D} \cdot \sqrt{\frac{c0}{w \cdot h}}\right) \cdot \sqrt{\frac{c0}{w}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 57.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_2 := t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right)\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-105}:\\ \;\;\;\;{\left(\frac{d}{D} \cdot \sqrt{\frac{c0}{w \cdot h}}\right)}^{2} \cdot \frac{c0}{w}\\ \mathbf{elif}\;t\_2 \leq 10^{-187}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{2 \cdot t\_0}{\frac{h}{\frac{c0}{w} \cdot {\left(\frac{d}{D}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w)))
        (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_2 (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))))
   (if (<= t_2 -2e-105)
     (* (pow (* (/ d D) (sqrt (/ c0 (* w h)))) 2.0) (/ c0 w))
     (if (<= t_2 1e-187)
       t_2
       (if (<= t_2 INFINITY)
         (/ (* 2.0 t_0) (/ h (* (/ c0 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)) / ((w * h) * (D * D));
	double t_2 = t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= -2e-105) {
		tmp = pow(((d / D) * sqrt((c0 / (w * h)))), 2.0) * (c0 / w);
	} else if (t_2 <= 1e-187) {
		tmp = t_2;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (2.0 * t_0) / (h / ((c0 / 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)) / ((w * h) * (D * D));
	double t_2 = t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= -2e-105) {
		tmp = Math.pow(((d / D) * Math.sqrt((c0 / (w * h)))), 2.0) * (c0 / w);
	} else if (t_2 <= 1e-187) {
		tmp = t_2;
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = (2.0 * t_0) / (h / ((c0 / 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)) / ((w * h) * (D * D))
	t_2 = t_0 * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))
	tmp = 0
	if t_2 <= -2e-105:
		tmp = math.pow(((d / D) * math.sqrt((c0 / (w * h)))), 2.0) * (c0 / w)
	elif t_2 <= 1e-187:
		tmp = t_2
	elif t_2 <= math.inf:
		tmp = (2.0 * t_0) / (h / ((c0 / 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(w * h) * Float64(D * D)))
	t_2 = Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M)))))
	tmp = 0.0
	if (t_2 <= -2e-105)
		tmp = Float64((Float64(Float64(d / D) * sqrt(Float64(c0 / Float64(w * h)))) ^ 2.0) * Float64(c0 / w));
	elseif (t_2 <= 1e-187)
		tmp = t_2;
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(2.0 * t_0) / Float64(h / Float64(Float64(c0 / 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)) / ((w * h) * (D * D));
	t_2 = t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	tmp = 0.0;
	if (t_2 <= -2e-105)
		tmp = (((d / D) * sqrt((c0 / (w * h)))) ^ 2.0) * (c0 / w);
	elseif (t_2 <= 1e-187)
		tmp = t_2;
	elseif (t_2 <= Inf)
		tmp = (2.0 * t_0) / (h / ((c0 / 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[(w * h), $MachinePrecision] * N[(D * D), $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, -2e-105], N[(N[Power[N[(N[(d / D), $MachinePrecision] * N[Sqrt[N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(c0 / w), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e-187], t$95$2, If[LessEqual[t$95$2, Infinity], N[(N[(2.0 * t$95$0), $MachinePrecision] / N[(h / N[(N[(c0 / w), $MachinePrecision] * N[Power[N[(d / D), $MachinePrecision], 2.0], $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))))) < -1.99999999999999993e-105

   1. Initial program 79.8%

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

   3. Simplified 81.9%

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

   5. Taylor expanded in c0 around inf 81.5%

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

      1. *-commutative 81.5%

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

      2. *-commutative 81.5%

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

      3. associate-*r* 81.5%

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

      4. associate-/r* 81.4%

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

      5. *-commutative 81.4%

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

   7. Simplified 81.4%

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

      1. pow1 81.4%

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

      2. associate-*r* 81.4%

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

      3. associate-/l/ 81.4%

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

      4. div-inv 81.4%

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

      5. metadata-eval 81.4%

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

      6. associate-/l/ 81.5%

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

      7. *-commutative 81.5%

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

   9. Applied egg-rr 81.5%

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

       1. unpow1 81.5%

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

       2. *-commutative 81.5%

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

       3. *-commutative 81.5%

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

       4. associate-*r* 81.5%

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

       5. *-commutative 81.5%

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

       6. times-frac 81.9%

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

       7. unpow2 81.9%

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

       8. unpow2 81.9%

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

       9. times-frac 84.4%

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

       10. unpow2 84.4%

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

       11. associate-*l* 84.4%

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

       12. metadata-eval 84.4%

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

       13. *-rgt-identity 84.4%

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

   11. Simplified 84.4%

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

       1. add-sqr-sqrt 84.4%

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

       2. pow2 84.4%

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

       3. div-inv 84.4%

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

       4. associate-*l/ 84.5%

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

       5. clear-num 84.4%

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

       6. sqrt-prod 84.4%

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

       7. unpow2 84.4%

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

       8. sqrt-prod 53.7%

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

       9. add-sqr-sqrt 92.8%

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

   13. Applied egg-rr 92.8%

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

   if -1.99999999999999993e-105 < (*.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))))) < 1e-187

   1. Initial program 69.4%

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

   if 1e-187 < (*.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 69.2%

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

   3. Simplified 69.2%

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

   5. Taylor expanded in c0 around inf 69.3%

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

      1. *-commutative 69.3%

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

      2. *-commutative 69.3%

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

      3. associate-*r* 66.5%

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

      4. associate-/r* 66.2%

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

      5. *-commutative 66.2%

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

   7. Simplified 66.2%

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

      1. clear-num 66.2%

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

      2. inv-pow 66.2%

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

      3. *-commutative 66.2%

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

      4. associate-/l* 63.2%

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

   9. Applied egg-rr 63.2%

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

       1. unpow-1 63.2%

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

       2. associate-/l* 68.6%

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

       3. associate-/r/ 74.0%

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

   11. Simplified 74.0%

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

       1. associate-*l/ 71.5%

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

       2. un-div-inv 71.5%

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

       3. associate-/r/ 69.1%

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

       4. associate-*l/ 69.1%

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

   13. Applied egg-rr 69.1%

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

       1. times-frac 66.5%

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

       2. associate-*r/ 69.1%

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

       3. associate-*l/ 69.1%

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

       4. associate-*r/ 69.1%

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

       5. associate-/r* 69.1%

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

       6. associate-*l/ 66.2%

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

       7. associate-/l* 71.6%

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

       8. associate-/r* 69.8%

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

       9. times-frac 76.9%

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

       10. unpow2 76.9%

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

       11. unpow2 76.9%

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

       12. times-frac 84.3%

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

       13. unpow2 84.3%

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

       14. *-commutative 84.3%

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

   15. Simplified 84.3%

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

   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 3.1%

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

   5. Taylor expanded in c0 around -inf 3.1%

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

      1. associate-*r* 3.1%

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

      2. neg-mul-1 3.1%

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

      3. distribute-lft1-in 3.1%

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

      4. metadata-eval 3.1%

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

      5. mul0-lft 38.6%

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

      6. distribute-lft-neg-in 38.6%

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

      7. distribute-rgt-neg-in 38.6%

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

      8. metadata-eval 38.6%

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

   7. Simplified 38.6%

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

   8. Taylor expanded in c0 around 0 45.7%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq -2 \cdot 10^{-105}:\\ \;\;\;\;{\left(\frac{d}{D} \cdot \sqrt{\frac{c0}{w \cdot h}}\right)}^{2} \cdot \frac{c0}{w}\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 10^{-187}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{2 \cdot \frac{c0}{2 \cdot w}}{\frac{h}{\frac{c0}{w} \cdot {\left(\frac{d}{D}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_2 := t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right)\\ \mathbf{if}\;t\_2 \leq 10^{-187}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{2 \cdot t\_0}{\frac{h}{\frac{c0}{w} \cdot {\left(\frac{d}{D}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w)))
        (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_2 (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))))
   (if (<= t_2 1e-187)
     t_2
     (if (<= t_2 INFINITY)
       (/ (* 2.0 t_0) (/ h (* (/ c0 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)) / ((w * h) * (D * D));
	double t_2 = t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= 1e-187) {
		tmp = t_2;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (2.0 * t_0) / (h / ((c0 / 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)) / ((w * h) * (D * D));
	double t_2 = t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= 1e-187) {
		tmp = t_2;
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = (2.0 * t_0) / (h / ((c0 / 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)) / ((w * h) * (D * D))
	t_2 = t_0 * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))
	tmp = 0
	if t_2 <= 1e-187:
		tmp = t_2
	elif t_2 <= math.inf:
		tmp = (2.0 * t_0) / (h / ((c0 / 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(w * h) * Float64(D * D)))
	t_2 = Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M)))))
	tmp = 0.0
	if (t_2 <= 1e-187)
		tmp = t_2;
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(2.0 * t_0) / Float64(h / Float64(Float64(c0 / 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)) / ((w * h) * (D * D));
	t_2 = t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	tmp = 0.0;
	if (t_2 <= 1e-187)
		tmp = t_2;
	elseif (t_2 <= Inf)
		tmp = (2.0 * t_0) / (h / ((c0 / 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[(w * h), $MachinePrecision] * N[(D * D), $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, 1e-187], t$95$2, If[LessEqual[t$95$2, Infinity], N[(N[(2.0 * t$95$0), $MachinePrecision] / N[(h / N[(N[(c0 / w), $MachinePrecision] * N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]]]

Derivation

1. Split input into 3 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))))) < 1e-187

   1. Initial program 76.5%

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

   if 1e-187 < (*.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 69.2%

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

   3. Simplified 69.2%

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

   5. Taylor expanded in c0 around inf 69.3%

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

      1. *-commutative 69.3%

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

      2. *-commutative 69.3%

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

      3. associate-*r* 66.5%

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

      4. associate-/r* 66.2%

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

      5. *-commutative 66.2%

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

   7. Simplified 66.2%

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

      1. clear-num 66.2%

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

      2. inv-pow 66.2%

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

      3. *-commutative 66.2%

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

      4. associate-/l* 63.2%

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

   9. Applied egg-rr 63.2%

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

       1. unpow-1 63.2%

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

       2. associate-/l* 68.6%

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

       3. associate-/r/ 74.0%

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

   11. Simplified 74.0%

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

       1. associate-*l/ 71.5%

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

       2. un-div-inv 71.5%

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

       3. associate-/r/ 69.1%

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

       4. associate-*l/ 69.1%

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

   13. Applied egg-rr 69.1%

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

       1. times-frac 66.5%

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

       2. associate-*r/ 69.1%

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

       3. associate-*l/ 69.1%

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

       4. associate-*r/ 69.1%

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

       5. associate-/r* 69.1%

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

       6. associate-*l/ 66.2%

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

       7. associate-/l* 71.6%

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

       8. associate-/r* 69.8%

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

       9. times-frac 76.9%

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

       10. unpow2 76.9%

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

       11. unpow2 76.9%

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

       12. times-frac 84.3%

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

       13. unpow2 84.3%

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

       14. *-commutative 84.3%

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

   15. Simplified 84.3%

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

   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 3.1%

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

   5. Taylor expanded in c0 around -inf 3.1%

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

      1. associate-*r* 3.1%

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

      2. neg-mul-1 3.1%

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

      3. distribute-lft1-in 3.1%

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

      4. metadata-eval 3.1%

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

      5. mul0-lft 38.6%

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

      6. distribute-lft-neg-in 38.6%

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

      7. distribute-rgt-neg-in 38.6%

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

      8. metadata-eval 38.6%

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

   7. Simplified 38.6%

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

   8. Taylor expanded in c0 around 0 45.7%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 10^{-187}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{2 \cdot \frac{c0}{2 \cdot w}}{\frac{h}{\frac{c0}{w} \cdot {\left(\frac{d}{D}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 46.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c0 \leq -1.5 \cdot 10^{+275}:\\ \;\;\;\;\frac{c0}{w} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w \cdot \frac{h}{c0}}\\ \mathbf{elif}\;c0 \leq -2.45 \cdot 10^{+247}:\\ \;\;\;\;0\\ \mathbf{elif}\;c0 \leq -4 \cdot 10^{-25} \lor \neg \left(c0 \leq 1.15 \cdot 10^{-55}\right):\\ \;\;\;\;\frac{2 \cdot \frac{c0}{2 \cdot w}}{\frac{h}{\frac{c0}{w} \cdot {\left(\frac{d}{D}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{{c0}^{2}}{\frac{w}{0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= c0 -1.5e+275)
   (* (/ c0 w) (/ (* (/ d D) (/ d D)) (* w (/ h c0))))
   (if (<= c0 -2.45e+247)
     0.0
     (if (or (<= c0 -4e-25) (not (<= c0 1.15e-55)))
       (/ (* 2.0 (/ c0 (* 2.0 w))) (/ h (* (/ c0 w) (pow (/ d D) 2.0))))
       (* -0.5 (/ (pow c0 2.0) (/ w 0.0)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (c0 <= -1.5e+275) {
		tmp = (c0 / w) * (((d / D) * (d / D)) / (w * (h / c0)));
	} else if (c0 <= -2.45e+247) {
		tmp = 0.0;
	} else if ((c0 <= -4e-25) || !(c0 <= 1.15e-55)) {
		tmp = (2.0 * (c0 / (2.0 * w))) / (h / ((c0 / w) * pow((d / D), 2.0)));
	} else {
		tmp = -0.5 * (pow(c0, 2.0) / (w / 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 (c0 <= (-1.5d+275)) then
        tmp = (c0 / w) * (((d_1 / d) * (d_1 / d)) / (w * (h / c0)))
    else if (c0 <= (-2.45d+247)) then
        tmp = 0.0d0
    else if ((c0 <= (-4d-25)) .or. (.not. (c0 <= 1.15d-55))) then
        tmp = (2.0d0 * (c0 / (2.0d0 * w))) / (h / ((c0 / w) * ((d_1 / d) ** 2.0d0)))
    else
        tmp = (-0.5d0) * ((c0 ** 2.0d0) / (w / 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 (c0 <= -1.5e+275) {
		tmp = (c0 / w) * (((d / D) * (d / D)) / (w * (h / c0)));
	} else if (c0 <= -2.45e+247) {
		tmp = 0.0;
	} else if ((c0 <= -4e-25) || !(c0 <= 1.15e-55)) {
		tmp = (2.0 * (c0 / (2.0 * w))) / (h / ((c0 / w) * Math.pow((d / D), 2.0)));
	} else {
		tmp = -0.5 * (Math.pow(c0, 2.0) / (w / 0.0));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if c0 <= -1.5e+275:
		tmp = (c0 / w) * (((d / D) * (d / D)) / (w * (h / c0)))
	elif c0 <= -2.45e+247:
		tmp = 0.0
	elif (c0 <= -4e-25) or not (c0 <= 1.15e-55):
		tmp = (2.0 * (c0 / (2.0 * w))) / (h / ((c0 / w) * math.pow((d / D), 2.0)))
	else:
		tmp = -0.5 * (math.pow(c0, 2.0) / (w / 0.0))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (c0 <= -1.5e+275)
		tmp = Float64(Float64(c0 / w) * Float64(Float64(Float64(d / D) * Float64(d / D)) / Float64(w * Float64(h / c0))));
	elseif (c0 <= -2.45e+247)
		tmp = 0.0;
	elseif ((c0 <= -4e-25) || !(c0 <= 1.15e-55))
		tmp = Float64(Float64(2.0 * Float64(c0 / Float64(2.0 * w))) / Float64(h / Float64(Float64(c0 / w) * (Float64(d / D) ^ 2.0))));
	else
		tmp = Float64(-0.5 * Float64((c0 ^ 2.0) / Float64(w / 0.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (c0 <= -1.5e+275)
		tmp = (c0 / w) * (((d / D) * (d / D)) / (w * (h / c0)));
	elseif (c0 <= -2.45e+247)
		tmp = 0.0;
	elseif ((c0 <= -4e-25) || ~((c0 <= 1.15e-55)))
		tmp = (2.0 * (c0 / (2.0 * w))) / (h / ((c0 / w) * ((d / D) ^ 2.0)));
	else
		tmp = -0.5 * ((c0 ^ 2.0) / (w / 0.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[c0, -1.5e+275], N[(N[(c0 / w), $MachinePrecision] * N[(N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] / N[(w * N[(h / c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c0, -2.45e+247], 0.0, If[Or[LessEqual[c0, -4e-25], N[Not[LessEqual[c0, 1.15e-55]], $MachinePrecision]], N[(N[(2.0 * N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(h / N[(N[(c0 / w), $MachinePrecision] * N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(N[Power[c0, 2.0], $MachinePrecision] / N[(w / 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c0 < -1.50000000000000002e275

   1. Initial program 31.1%

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

   3. Simplified 31.1%

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

   5. Taylor expanded in c0 around inf 31.1%

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

      1. *-commutative 31.1%

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

      2. *-commutative 31.1%

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

      3. associate-*r* 31.1%

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

      4. associate-/r* 31.1%

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

      5. *-commutative 31.1%

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

   7. Simplified 31.1%

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

      1. pow1 31.1%

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

      2. associate-*r* 31.1%

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

      3. associate-/l/ 31.1%

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

      4. div-inv 31.1%

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

      5. metadata-eval 31.1%

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

      6. associate-/l/ 31.1%

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

      7. *-commutative 31.1%

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

   9. Applied egg-rr 31.1%

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

       1. unpow1 31.1%

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

       2. *-commutative 31.1%

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

       3. *-commutative 31.1%

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

       4. associate-*r* 31.1%

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

       5. *-commutative 31.1%

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

       6. times-frac 31.3%

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

       7. unpow2 31.3%

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

       8. unpow2 31.3%

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

       9. times-frac 60.5%

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

       10. unpow2 60.5%

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

       11. associate-*l* 60.5%

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

       12. metadata-eval 60.5%

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

       13. *-rgt-identity 60.5%

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

   11. Simplified 70.3%

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

       1. unpow2 70.3%

          \[\leadsto \frac{\color{blue}{\frac{d}{D} \cdot \frac{d}{D}}}{\frac{h}{c0} \cdot w} \cdot \frac{c0}{w} \]

   13. Applied egg-rr 70.3%

       \[\leadsto \frac{\color{blue}{\frac{d}{D} \cdot \frac{d}{D}}}{\frac{h}{c0} \cdot w} \cdot \frac{c0}{w} \]

   if -1.50000000000000002e275 < c0 < -2.4499999999999999e247

   1. Initial program 12.5%

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

   3. Simplified 12.5%

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

   5. Taylor expanded in c0 around -inf 0.0%

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

      1. associate-*r* 0.0%

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

      2. neg-mul-1 0.0%

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

      3. distribute-lft1-in 0.0%

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

      4. metadata-eval 0.0%

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

      5. mul0-lft 50.2%

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

      6. distribute-lft-neg-in 50.2%

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

      7. distribute-rgt-neg-in 50.2%

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

      8. metadata-eval 50.2%

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

   7. Simplified 50.2%

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

   8. Taylor expanded in c0 around 0 76.3%

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

   if -2.4499999999999999e247 < c0 < -4.00000000000000015e-25 or 1.15000000000000006e-55 < c0

   1. Initial program 28.4%

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

   3. Simplified 30.3%

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

   5. Taylor expanded in c0 around inf 38.7%

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

      1. *-commutative 38.7%

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

      2. *-commutative 38.7%

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

      3. associate-*r* 38.6%

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

      4. associate-/r* 39.3%

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

      5. *-commutative 39.3%

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

   7. Simplified 39.3%

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

      1. clear-num 39.3%

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

      2. inv-pow 39.3%

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

      3. *-commutative 39.3%

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

      4. associate-/l* 39.9%

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

   9. Applied egg-rr 39.9%

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

       1. unpow-1 39.9%

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

       2. associate-/l* 41.6%

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

       3. associate-/r/ 42.3%

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

   11. Simplified 42.3%

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

       1. associate-*l/ 41.0%

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

       2. un-div-inv 41.0%

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

       3. associate-/r/ 41.5%

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

       4. associate-*l/ 41.6%

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

   13. Applied egg-rr 41.6%

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

       1. times-frac 42.3%

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

       2. associate-*r/ 41.6%

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

       3. associate-*l/ 41.4%

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

       4. associate-*r/ 41.4%

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

       5. associate-/r* 41.4%

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

       6. associate-*l/ 39.3%

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

       7. associate-/l* 41.0%

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

       8. associate-/r* 40.9%

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

       9. times-frac 42.9%

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

       10. unpow2 42.9%

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

       11. unpow2 42.9%

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

       12. times-frac 54.0%

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

       13. unpow2 54.0%

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

       14. *-commutative 54.0%

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

   15. Simplified 54.0%

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

   if -4.00000000000000015e-25 < c0 < 1.15000000000000006e-55

   1. Initial program 23.2%

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

   3. Taylor expanded in c0 around -inf 6.6%

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

      1. associate-/l* 6.6%

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

      2. distribute-lft1-in 6.6%

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

      3. metadata-eval 6.6%

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

      4. mul0-lft 53.4%

         \[\leadsto -0.5 \cdot \frac{{c0}^{2}}{\frac{w}{\color{blue}{0}}} \]

   5. Simplified 53.4%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{{c0}^{2}}{\frac{w}{0}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -1.5 \cdot 10^{+275}:\\ \;\;\;\;\frac{c0}{w} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w \cdot \frac{h}{c0}}\\ \mathbf{elif}\;c0 \leq -2.45 \cdot 10^{+247}:\\ \;\;\;\;0\\ \mathbf{elif}\;c0 \leq -4 \cdot 10^{-25} \lor \neg \left(c0 \leq 1.15 \cdot 10^{-55}\right):\\ \;\;\;\;\frac{2 \cdot \frac{c0}{2 \cdot w}}{\frac{h}{\frac{c0}{w} \cdot {\left(\frac{d}{D}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{{c0}^{2}}{\frac{w}{0}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 45.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := w \cdot \frac{h}{c0}\\ \mathbf{if}\;c0 \leq -6 \cdot 10^{+274}:\\ \;\;\;\;\frac{c0}{w} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{t\_0}\\ \mathbf{elif}\;c0 \leq -6.2 \cdot 10^{+246}:\\ \;\;\;\;0\\ \mathbf{elif}\;c0 \leq -2.9 \cdot 10^{-29} \lor \neg \left(c0 \leq 6.6 \cdot 10^{-55}\right):\\ \;\;\;\;\frac{\frac{c0}{w} \cdot {\left(\frac{d}{D}\right)}^{2}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{{c0}^{2}}{\frac{w}{0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* w (/ h c0))))
   (if (<= c0 -6e+274)
     (* (/ c0 w) (/ (* (/ d D) (/ d D)) t_0))
     (if (<= c0 -6.2e+246)
       0.0
       (if (or (<= c0 -2.9e-29) (not (<= c0 6.6e-55)))
         (/ (* (/ c0 w) (pow (/ d D) 2.0)) t_0)
         (* -0.5 (/ (pow c0 2.0) (/ w 0.0))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = w * (h / c0);
	double tmp;
	if (c0 <= -6e+274) {
		tmp = (c0 / w) * (((d / D) * (d / D)) / t_0);
	} else if (c0 <= -6.2e+246) {
		tmp = 0.0;
	} else if ((c0 <= -2.9e-29) || !(c0 <= 6.6e-55)) {
		tmp = ((c0 / w) * pow((d / D), 2.0)) / t_0;
	} else {
		tmp = -0.5 * (pow(c0, 2.0) / (w / 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) :: t_0
    real(8) :: tmp
    t_0 = w * (h / c0)
    if (c0 <= (-6d+274)) then
        tmp = (c0 / w) * (((d_1 / d) * (d_1 / d)) / t_0)
    else if (c0 <= (-6.2d+246)) then
        tmp = 0.0d0
    else if ((c0 <= (-2.9d-29)) .or. (.not. (c0 <= 6.6d-55))) then
        tmp = ((c0 / w) * ((d_1 / d) ** 2.0d0)) / t_0
    else
        tmp = (-0.5d0) * ((c0 ** 2.0d0) / (w / 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 t_0 = w * (h / c0);
	double tmp;
	if (c0 <= -6e+274) {
		tmp = (c0 / w) * (((d / D) * (d / D)) / t_0);
	} else if (c0 <= -6.2e+246) {
		tmp = 0.0;
	} else if ((c0 <= -2.9e-29) || !(c0 <= 6.6e-55)) {
		tmp = ((c0 / w) * Math.pow((d / D), 2.0)) / t_0;
	} else {
		tmp = -0.5 * (Math.pow(c0, 2.0) / (w / 0.0));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = w * (h / c0)
	tmp = 0
	if c0 <= -6e+274:
		tmp = (c0 / w) * (((d / D) * (d / D)) / t_0)
	elif c0 <= -6.2e+246:
		tmp = 0.0
	elif (c0 <= -2.9e-29) or not (c0 <= 6.6e-55):
		tmp = ((c0 / w) * math.pow((d / D), 2.0)) / t_0
	else:
		tmp = -0.5 * (math.pow(c0, 2.0) / (w / 0.0))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(w * Float64(h / c0))
	tmp = 0.0
	if (c0 <= -6e+274)
		tmp = Float64(Float64(c0 / w) * Float64(Float64(Float64(d / D) * Float64(d / D)) / t_0));
	elseif (c0 <= -6.2e+246)
		tmp = 0.0;
	elseif ((c0 <= -2.9e-29) || !(c0 <= 6.6e-55))
		tmp = Float64(Float64(Float64(c0 / w) * (Float64(d / D) ^ 2.0)) / t_0);
	else
		tmp = Float64(-0.5 * Float64((c0 ^ 2.0) / Float64(w / 0.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = w * (h / c0);
	tmp = 0.0;
	if (c0 <= -6e+274)
		tmp = (c0 / w) * (((d / D) * (d / D)) / t_0);
	elseif (c0 <= -6.2e+246)
		tmp = 0.0;
	elseif ((c0 <= -2.9e-29) || ~((c0 <= 6.6e-55)))
		tmp = ((c0 / w) * ((d / D) ^ 2.0)) / t_0;
	else
		tmp = -0.5 * ((c0 ^ 2.0) / (w / 0.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(w * N[(h / c0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c0, -6e+274], N[(N[(c0 / w), $MachinePrecision] * N[(N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[c0, -6.2e+246], 0.0, If[Or[LessEqual[c0, -2.9e-29], N[Not[LessEqual[c0, 6.6e-55]], $MachinePrecision]], N[(N[(N[(c0 / w), $MachinePrecision] * N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(-0.5 * N[(N[Power[c0, 2.0], $MachinePrecision] / N[(w / 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if c0 < -5.99999999999999991e274

   1. Initial program 31.1%

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

   3. Simplified 31.1%

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

   5. Taylor expanded in c0 around inf 31.1%

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

      1. *-commutative 31.1%

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

      2. *-commutative 31.1%

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

      3. associate-*r* 31.1%

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

      4. associate-/r* 31.1%

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

      5. *-commutative 31.1%

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

   7. Simplified 31.1%

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

      1. pow1 31.1%

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

      2. associate-*r* 31.1%

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

      3. associate-/l/ 31.1%

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

      4. div-inv 31.1%

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

      5. metadata-eval 31.1%

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

      6. associate-/l/ 31.1%

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

      7. *-commutative 31.1%

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

   9. Applied egg-rr 31.1%

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

       1. unpow1 31.1%

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

       2. *-commutative 31.1%

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

       3. *-commutative 31.1%

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

       4. associate-*r* 31.1%

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

       5. *-commutative 31.1%

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

       6. times-frac 31.3%

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

       7. unpow2 31.3%

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

       8. unpow2 31.3%

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

       9. times-frac 60.5%

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

       10. unpow2 60.5%

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

       11. associate-*l* 60.5%

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

       12. metadata-eval 60.5%

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

       13. *-rgt-identity 60.5%

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

   11. Simplified 70.3%

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

       1. unpow2 70.3%

          \[\leadsto \frac{\color{blue}{\frac{d}{D} \cdot \frac{d}{D}}}{\frac{h}{c0} \cdot w} \cdot \frac{c0}{w} \]

   13. Applied egg-rr 70.3%

       \[\leadsto \frac{\color{blue}{\frac{d}{D} \cdot \frac{d}{D}}}{\frac{h}{c0} \cdot w} \cdot \frac{c0}{w} \]

   if -5.99999999999999991e274 < c0 < -6.19999999999999977e246

   1. Initial program 12.5%

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

   3. Simplified 12.5%

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

   5. Taylor expanded in c0 around -inf 0.0%

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

      1. associate-*r* 0.0%

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

      2. neg-mul-1 0.0%

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

      3. distribute-lft1-in 0.0%

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

      4. metadata-eval 0.0%

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

      5. mul0-lft 50.2%

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

      6. distribute-lft-neg-in 50.2%

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

      7. distribute-rgt-neg-in 50.2%

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

      8. metadata-eval 50.2%

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

   7. Simplified 50.2%

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

   8. Taylor expanded in c0 around 0 76.3%

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

   if -6.19999999999999977e246 < c0 < -2.90000000000000024e-29 or 6.5999999999999999e-55 < c0

   1. Initial program 28.4%

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

   3. Simplified 30.3%

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

   5. Taylor expanded in c0 around inf 38.7%

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

      1. *-commutative 38.7%

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

      2. *-commutative 38.7%

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

      3. associate-*r* 38.6%

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

      4. associate-/r* 39.3%

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

      5. *-commutative 39.3%

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

   7. Simplified 39.3%

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

      1. pow1 39.3%

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

      2. associate-*r* 39.3%

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

      3. associate-/l/ 39.3%

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

      4. div-inv 39.3%

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

      5. metadata-eval 39.3%

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

      6. associate-/l/ 38.6%

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

      7. *-commutative 38.6%

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

   9. Applied egg-rr 38.6%

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

       1. unpow1 38.6%

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

       2. *-commutative 38.6%

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

       3. *-commutative 38.6%

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

       4. associate-*r* 38.7%

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

       5. *-commutative 38.7%

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

       6. times-frac 40.2%

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

       7. unpow2 40.2%

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

       8. unpow2 40.2%

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

       9. times-frac 49.9%

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

       10. unpow2 49.9%

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

       11. associate-*l* 49.9%

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

       12. metadata-eval 49.9%

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

       13. *-rgt-identity 49.9%

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

   11. Simplified 52.0%

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

       1. associate-*l/ 52.0%

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

   13. Applied egg-rr 52.0%

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

   if -2.90000000000000024e-29 < c0 < 6.5999999999999999e-55

   1. Initial program 23.2%

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

   3. Taylor expanded in c0 around -inf 6.6%

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

      1. associate-/l* 6.6%

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

      2. distribute-lft1-in 6.6%

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

      3. metadata-eval 6.6%

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

      4. mul0-lft 53.4%

         \[\leadsto -0.5 \cdot \frac{{c0}^{2}}{\frac{w}{\color{blue}{0}}} \]

   5. Simplified 53.4%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{{c0}^{2}}{\frac{w}{0}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -6 \cdot 10^{+274}:\\ \;\;\;\;\frac{c0}{w} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w \cdot \frac{h}{c0}}\\ \mathbf{elif}\;c0 \leq -6.2 \cdot 10^{+246}:\\ \;\;\;\;0\\ \mathbf{elif}\;c0 \leq -2.9 \cdot 10^{-29} \lor \neg \left(c0 \leq 6.6 \cdot 10^{-55}\right):\\ \;\;\;\;\frac{\frac{c0}{w} \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot \frac{h}{c0}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{{c0}^{2}}{\frac{w}{0}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 45.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := w \cdot \frac{h}{c0}\\ t_1 := {\left(\frac{d}{D}\right)}^{2}\\ \mathbf{if}\;c0 \leq -1.65 \cdot 10^{+274}:\\ \;\;\;\;\frac{c0}{w} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{t\_0}\\ \mathbf{elif}\;c0 \leq -6.5 \cdot 10^{+247}:\\ \;\;\;\;0\\ \mathbf{elif}\;c0 \leq -3.9 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{c0}{w} \cdot t\_1}{t\_0}\\ \mathbf{elif}\;c0 \leq 6.8 \cdot 10^{-56}:\\ \;\;\;\;-0.5 \cdot \frac{{c0}^{2}}{\frac{w}{0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{w}{\frac{t\_1}{t\_0}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* w (/ h c0))) (t_1 (pow (/ d D) 2.0)))
   (if (<= c0 -1.65e+274)
     (* (/ c0 w) (/ (* (/ d D) (/ d D)) t_0))
     (if (<= c0 -6.5e+247)
       0.0
       (if (<= c0 -3.9e-29)
         (/ (* (/ c0 w) t_1) t_0)
         (if (<= c0 6.8e-56)
           (* -0.5 (/ (pow c0 2.0) (/ w 0.0)))
           (/ c0 (/ w (/ t_1 t_0)))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = w * (h / c0);
	double t_1 = pow((d / D), 2.0);
	double tmp;
	if (c0 <= -1.65e+274) {
		tmp = (c0 / w) * (((d / D) * (d / D)) / t_0);
	} else if (c0 <= -6.5e+247) {
		tmp = 0.0;
	} else if (c0 <= -3.9e-29) {
		tmp = ((c0 / w) * t_1) / t_0;
	} else if (c0 <= 6.8e-56) {
		tmp = -0.5 * (pow(c0, 2.0) / (w / 0.0));
	} else {
		tmp = c0 / (w / (t_1 / t_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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = w * (h / c0)
    t_1 = (d_1 / d) ** 2.0d0
    if (c0 <= (-1.65d+274)) then
        tmp = (c0 / w) * (((d_1 / d) * (d_1 / d)) / t_0)
    else if (c0 <= (-6.5d+247)) then
        tmp = 0.0d0
    else if (c0 <= (-3.9d-29)) then
        tmp = ((c0 / w) * t_1) / t_0
    else if (c0 <= 6.8d-56) then
        tmp = (-0.5d0) * ((c0 ** 2.0d0) / (w / 0.0d0))
    else
        tmp = c0 / (w / (t_1 / t_0))
    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 = w * (h / c0);
	double t_1 = Math.pow((d / D), 2.0);
	double tmp;
	if (c0 <= -1.65e+274) {
		tmp = (c0 / w) * (((d / D) * (d / D)) / t_0);
	} else if (c0 <= -6.5e+247) {
		tmp = 0.0;
	} else if (c0 <= -3.9e-29) {
		tmp = ((c0 / w) * t_1) / t_0;
	} else if (c0 <= 6.8e-56) {
		tmp = -0.5 * (Math.pow(c0, 2.0) / (w / 0.0));
	} else {
		tmp = c0 / (w / (t_1 / t_0));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = w * (h / c0)
	t_1 = math.pow((d / D), 2.0)
	tmp = 0
	if c0 <= -1.65e+274:
		tmp = (c0 / w) * (((d / D) * (d / D)) / t_0)
	elif c0 <= -6.5e+247:
		tmp = 0.0
	elif c0 <= -3.9e-29:
		tmp = ((c0 / w) * t_1) / t_0
	elif c0 <= 6.8e-56:
		tmp = -0.5 * (math.pow(c0, 2.0) / (w / 0.0))
	else:
		tmp = c0 / (w / (t_1 / t_0))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(w * Float64(h / c0))
	t_1 = Float64(d / D) ^ 2.0
	tmp = 0.0
	if (c0 <= -1.65e+274)
		tmp = Float64(Float64(c0 / w) * Float64(Float64(Float64(d / D) * Float64(d / D)) / t_0));
	elseif (c0 <= -6.5e+247)
		tmp = 0.0;
	elseif (c0 <= -3.9e-29)
		tmp = Float64(Float64(Float64(c0 / w) * t_1) / t_0);
	elseif (c0 <= 6.8e-56)
		tmp = Float64(-0.5 * Float64((c0 ^ 2.0) / Float64(w / 0.0)));
	else
		tmp = Float64(c0 / Float64(w / Float64(t_1 / t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = w * (h / c0);
	t_1 = (d / D) ^ 2.0;
	tmp = 0.0;
	if (c0 <= -1.65e+274)
		tmp = (c0 / w) * (((d / D) * (d / D)) / t_0);
	elseif (c0 <= -6.5e+247)
		tmp = 0.0;
	elseif (c0 <= -3.9e-29)
		tmp = ((c0 / w) * t_1) / t_0;
	elseif (c0 <= 6.8e-56)
		tmp = -0.5 * ((c0 ^ 2.0) / (w / 0.0));
	else
		tmp = c0 / (w / (t_1 / t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(w * N[(h / c0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[c0, -1.65e+274], N[(N[(c0 / w), $MachinePrecision] * N[(N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[c0, -6.5e+247], 0.0, If[LessEqual[c0, -3.9e-29], N[(N[(N[(c0 / w), $MachinePrecision] * t$95$1), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[c0, 6.8e-56], N[(-0.5 * N[(N[Power[c0, 2.0], $MachinePrecision] / N[(w / 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[(w / N[(t$95$1 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if c0 < -1.65000000000000007e274

   1. Initial program 31.1%

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

   3. Simplified 31.1%

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

   5. Taylor expanded in c0 around inf 31.1%

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

      1. *-commutative 31.1%

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

      2. *-commutative 31.1%

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

      3. associate-*r* 31.1%

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

      4. associate-/r* 31.1%

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

      5. *-commutative 31.1%

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

   7. Simplified 31.1%

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

      1. pow1 31.1%

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

      2. associate-*r* 31.1%

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

      3. associate-/l/ 31.1%

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

      4. div-inv 31.1%

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

      5. metadata-eval 31.1%

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

      6. associate-/l/ 31.1%

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

      7. *-commutative 31.1%

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

   9. Applied egg-rr 31.1%

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

       1. unpow1 31.1%

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

       2. *-commutative 31.1%

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

       3. *-commutative 31.1%

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

       4. associate-*r* 31.1%

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

       5. *-commutative 31.1%

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

       6. times-frac 31.3%

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

       7. unpow2 31.3%

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

       8. unpow2 31.3%

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

       9. times-frac 60.5%

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

       10. unpow2 60.5%

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

       11. associate-*l* 60.5%

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

       12. metadata-eval 60.5%

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

       13. *-rgt-identity 60.5%

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

   11. Simplified 70.3%

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

       1. unpow2 70.3%

          \[\leadsto \frac{\color{blue}{\frac{d}{D} \cdot \frac{d}{D}}}{\frac{h}{c0} \cdot w} \cdot \frac{c0}{w} \]

   13. Applied egg-rr 70.3%

       \[\leadsto \frac{\color{blue}{\frac{d}{D} \cdot \frac{d}{D}}}{\frac{h}{c0} \cdot w} \cdot \frac{c0}{w} \]

   if -1.65000000000000007e274 < c0 < -6.50000000000000023e247

   1. Initial program 12.5%

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

   3. Simplified 12.5%

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

   5. Taylor expanded in c0 around -inf 0.0%

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

      1. associate-*r* 0.0%

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

      2. neg-mul-1 0.0%

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

      3. distribute-lft1-in 0.0%

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

      4. metadata-eval 0.0%

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

      5. mul0-lft 50.2%

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

      6. distribute-lft-neg-in 50.2%

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

      7. distribute-rgt-neg-in 50.2%

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

      8. metadata-eval 50.2%

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

   7. Simplified 50.2%

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

   8. Taylor expanded in c0 around 0 76.3%

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

   if -6.50000000000000023e247 < c0 < -3.8999999999999998e-29

   1. Initial program 30.2%

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

   3. Simplified 31.7%

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

   5. Taylor expanded in c0 around inf 42.2%

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

      1. *-commutative 42.2%

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

      2. *-commutative 42.2%

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

      3. associate-*r* 43.7%

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

      4. associate-/r* 45.0%

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

      5. *-commutative 45.0%

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

   7. Simplified 45.0%

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

      1. pow1 45.0%

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

      2. associate-*r* 45.0%

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

      3. associate-/l/ 45.0%

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

      4. div-inv 45.0%

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

      5. metadata-eval 45.0%

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

      6. associate-/l/ 43.7%

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

      7. *-commutative 43.7%

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

   9. Applied egg-rr 43.7%

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

       1. unpow1 43.7%

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

       2. *-commutative 43.7%

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

       3. *-commutative 43.7%

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

       4. associate-*r* 42.2%

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

       5. *-commutative 42.2%

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

       6. times-frac 43.0%

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

       7. unpow2 43.0%

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

       8. unpow2 43.0%

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

       9. times-frac 50.8%

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

       10. unpow2 50.8%

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

       11. associate-*l* 50.8%

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

       12. metadata-eval 50.8%

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

       13. *-rgt-identity 50.8%

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

   11. Simplified 55.0%

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

       1. associate-*l/ 55.0%

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

   13. Applied egg-rr 55.0%

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

   if -3.8999999999999998e-29 < c0 < 6.79999999999999964e-56

   1. Initial program 23.2%

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

   3. Taylor expanded in c0 around -inf 6.6%

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

      1. associate-/l* 6.6%

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

      2. distribute-lft1-in 6.6%

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

      3. metadata-eval 6.6%

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

      4. mul0-lft 53.4%

         \[\leadsto -0.5 \cdot \frac{{c0}^{2}}{\frac{w}{\color{blue}{0}}} \]

   5. Simplified 53.4%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{{c0}^{2}}{\frac{w}{0}}} \]

   if 6.79999999999999964e-56 < c0

   1. Initial program 26.9%

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

   3. Simplified 29.2%

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

   5. Taylor expanded in c0 around inf 35.8%

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

      1. *-commutative 35.8%

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

      2. *-commutative 35.8%

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

      3. associate-*r* 34.4%

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

      4. associate-/r* 34.4%

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

      5. *-commutative 34.4%

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

   7. Simplified 34.4%

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

      1. associate-*l/ 34.6%

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

      2. associate-/l/ 34.6%

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

      3. *-commutative 34.6%

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

      4. *-commutative 34.6%

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

   9. Applied egg-rr 34.6%

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

       1. associate-/l* 35.8%

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

       2. *-commutative 35.8%

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

       3. *-commutative 35.8%

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

       4. associate-*r* 37.2%

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

       5. times-frac 37.2%

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

       6. metadata-eval 37.2%

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

       7. *-commutative 37.2%

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

       8. times-frac 39.0%

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

       9. unpow2 39.0%

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

       10. unpow2 39.0%

           \[\leadsto \frac{c0}{1 \cdot \frac{w}{\frac{d \cdot d}{\color{blue}{D \cdot D}} \cdot \frac{c0}{h \cdot w}}} \]

       11. times-frac 50.4%

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

       12. unpow2 50.4%

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

       13. associate-*r/ 48.2%

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

       14. times-frac 51.8%

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

       15. associate-/r/ 53.0%

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

       16. associate-/r/ 50.6%

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

   11. Simplified 50.6%

       \[\leadsto \color{blue}{\frac{c0}{1 \cdot \frac{w}{\frac{{\left(\frac{d}{D}\right)}^{2}}{\frac{h}{c0} \cdot w}}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -1.65 \cdot 10^{+274}:\\ \;\;\;\;\frac{c0}{w} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w \cdot \frac{h}{c0}}\\ \mathbf{elif}\;c0 \leq -6.5 \cdot 10^{+247}:\\ \;\;\;\;0\\ \mathbf{elif}\;c0 \leq -3.9 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{c0}{w} \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot \frac{h}{c0}}\\ \mathbf{elif}\;c0 \leq 6.8 \cdot 10^{-56}:\\ \;\;\;\;-0.5 \cdot \frac{{c0}^{2}}{\frac{w}{0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{w}{\frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot \frac{h}{c0}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 45.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{w} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w \cdot \frac{h}{c0}}\\ \mathbf{if}\;c0 \leq -5.1 \cdot 10^{+277}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c0 \leq -5.5 \cdot 10^{+246}:\\ \;\;\;\;0\\ \mathbf{elif}\;c0 \leq -8.8 \cdot 10^{-29} \lor \neg \left(c0 \leq 2.7 \cdot 10^{-55}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{{c0}^{2}}{\frac{w}{0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ c0 w) (/ (* (/ d D) (/ d D)) (* w (/ h c0))))))
   (if (<= c0 -5.1e+277)
     t_0
     (if (<= c0 -5.5e+246)
       0.0
       (if (or (<= c0 -8.8e-29) (not (<= c0 2.7e-55)))
         t_0
         (* -0.5 (/ (pow c0 2.0) (/ w 0.0))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 / w) * (((d / D) * (d / D)) / (w * (h / c0)));
	double tmp;
	if (c0 <= -5.1e+277) {
		tmp = t_0;
	} else if (c0 <= -5.5e+246) {
		tmp = 0.0;
	} else if ((c0 <= -8.8e-29) || !(c0 <= 2.7e-55)) {
		tmp = t_0;
	} else {
		tmp = -0.5 * (pow(c0, 2.0) / (w / 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) :: t_0
    real(8) :: tmp
    t_0 = (c0 / w) * (((d_1 / d) * (d_1 / d)) / (w * (h / c0)))
    if (c0 <= (-5.1d+277)) then
        tmp = t_0
    else if (c0 <= (-5.5d+246)) then
        tmp = 0.0d0
    else if ((c0 <= (-8.8d-29)) .or. (.not. (c0 <= 2.7d-55))) then
        tmp = t_0
    else
        tmp = (-0.5d0) * ((c0 ** 2.0d0) / (w / 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 t_0 = (c0 / w) * (((d / D) * (d / D)) / (w * (h / c0)));
	double tmp;
	if (c0 <= -5.1e+277) {
		tmp = t_0;
	} else if (c0 <= -5.5e+246) {
		tmp = 0.0;
	} else if ((c0 <= -8.8e-29) || !(c0 <= 2.7e-55)) {
		tmp = t_0;
	} else {
		tmp = -0.5 * (Math.pow(c0, 2.0) / (w / 0.0));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 / w) * (((d / D) * (d / D)) / (w * (h / c0)))
	tmp = 0
	if c0 <= -5.1e+277:
		tmp = t_0
	elif c0 <= -5.5e+246:
		tmp = 0.0
	elif (c0 <= -8.8e-29) or not (c0 <= 2.7e-55):
		tmp = t_0
	else:
		tmp = -0.5 * (math.pow(c0, 2.0) / (w / 0.0))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 / w) * Float64(Float64(Float64(d / D) * Float64(d / D)) / Float64(w * Float64(h / c0))))
	tmp = 0.0
	if (c0 <= -5.1e+277)
		tmp = t_0;
	elseif (c0 <= -5.5e+246)
		tmp = 0.0;
	elseif ((c0 <= -8.8e-29) || !(c0 <= 2.7e-55))
		tmp = t_0;
	else
		tmp = Float64(-0.5 * Float64((c0 ^ 2.0) / Float64(w / 0.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 / w) * (((d / D) * (d / D)) / (w * (h / c0)));
	tmp = 0.0;
	if (c0 <= -5.1e+277)
		tmp = t_0;
	elseif (c0 <= -5.5e+246)
		tmp = 0.0;
	elseif ((c0 <= -8.8e-29) || ~((c0 <= 2.7e-55)))
		tmp = t_0;
	else
		tmp = -0.5 * ((c0 ^ 2.0) / (w / 0.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 / w), $MachinePrecision] * N[(N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] / N[(w * N[(h / c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c0, -5.1e+277], t$95$0, If[LessEqual[c0, -5.5e+246], 0.0, If[Or[LessEqual[c0, -8.8e-29], N[Not[LessEqual[c0, 2.7e-55]], $MachinePrecision]], t$95$0, N[(-0.5 * N[(N[Power[c0, 2.0], $MachinePrecision] / N[(w / 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if c0 < -5.0999999999999998e277 or -5.50000000000000028e246 < c0 < -8.79999999999999961e-29 or 2.70000000000000004e-55 < c0

   1. Initial program 28.6%

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

   3. Simplified 30.4%

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

   5. Taylor expanded in c0 around inf 38.2%

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

      1. *-commutative 38.2%

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

      2. *-commutative 38.2%

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

      3. associate-*r* 38.2%

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

      4. associate-/r* 38.7%

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

      5. *-commutative 38.7%

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

   7. Simplified 38.7%

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

      1. pow1 38.7%

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

      2. associate-*r* 38.7%

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

      3. associate-/l/ 38.7%

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

      4. div-inv 38.7%

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

      5. metadata-eval 38.7%

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

      6. associate-/l/ 38.2%

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

      7. *-commutative 38.2%

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

   9. Applied egg-rr 38.2%

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

       1. unpow1 38.2%

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

       2. *-commutative 38.2%

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

       3. *-commutative 38.2%

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

       4. associate-*r* 38.2%

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

       5. *-commutative 38.2%

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

       6. times-frac 39.6%

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

       7. unpow2 39.6%

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

       8. unpow2 39.6%

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

       9. times-frac 50.6%

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

       10. unpow2 50.6%

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

       11. associate-*l* 50.6%

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

       12. metadata-eval 50.6%

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

       13. *-rgt-identity 50.6%

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

   11. Simplified 53.2%

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

       1. unpow2 53.2%

          \[\leadsto \frac{\color{blue}{\frac{d}{D} \cdot \frac{d}{D}}}{\frac{h}{c0} \cdot w} \cdot \frac{c0}{w} \]

   13. Applied egg-rr 53.2%

       \[\leadsto \frac{\color{blue}{\frac{d}{D} \cdot \frac{d}{D}}}{\frac{h}{c0} \cdot w} \cdot \frac{c0}{w} \]

   if -5.0999999999999998e277 < c0 < -5.50000000000000028e246

   1. Initial program 12.5%

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

   3. Simplified 12.5%

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

   5. Taylor expanded in c0 around -inf 0.0%

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

      1. associate-*r* 0.0%

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

      2. neg-mul-1 0.0%

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

      3. distribute-lft1-in 0.0%

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

      4. metadata-eval 0.0%

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

      5. mul0-lft 50.2%

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

      6. distribute-lft-neg-in 50.2%

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

      7. distribute-rgt-neg-in 50.2%

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

      8. metadata-eval 50.2%

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

   7. Simplified 50.2%

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

   8. Taylor expanded in c0 around 0 76.3%

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

   if -8.79999999999999961e-29 < c0 < 2.70000000000000004e-55

   1. Initial program 23.2%

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

   3. Taylor expanded in c0 around -inf 6.6%

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

      1. associate-/l* 6.6%

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

      2. distribute-lft1-in 6.6%

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

      3. metadata-eval 6.6%

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

      4. mul0-lft 53.4%

         \[\leadsto -0.5 \cdot \frac{{c0}^{2}}{\frac{w}{\color{blue}{0}}} \]

   5. Simplified 53.4%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{{c0}^{2}}{\frac{w}{0}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -5.1 \cdot 10^{+277}:\\ \;\;\;\;\frac{c0}{w} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w \cdot \frac{h}{c0}}\\ \mathbf{elif}\;c0 \leq -5.5 \cdot 10^{+246}:\\ \;\;\;\;0\\ \mathbf{elif}\;c0 \leq -8.8 \cdot 10^{-29} \lor \neg \left(c0 \leq 2.7 \cdot 10^{-55}\right):\\ \;\;\;\;\frac{c0}{w} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w \cdot \frac{h}{c0}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{{c0}^{2}}{\frac{w}{0}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 45.6% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{w} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w \cdot \frac{h}{c0}}\\ \mathbf{if}\;c0 \leq -2.6 \cdot 10^{+273}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c0 \leq -4.1 \cdot 10^{+247}:\\ \;\;\;\;0\\ \mathbf{elif}\;c0 \leq -1.4 \cdot 10^{-26} \lor \neg \left(c0 \leq 2.65 \cdot 10^{-55}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(c0 \cdot 0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ c0 w) (/ (* (/ d D) (/ d D)) (* w (/ h c0))))))
   (if (<= c0 -2.6e+273)
     t_0
     (if (<= c0 -4.1e+247)
       0.0
       (if (or (<= c0 -1.4e-26) (not (<= c0 2.65e-55)))
         t_0
         (* (/ c0 (* 2.0 w)) (* c0 0.0)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 / w) * (((d / D) * (d / D)) / (w * (h / c0)));
	double tmp;
	if (c0 <= -2.6e+273) {
		tmp = t_0;
	} else if (c0 <= -4.1e+247) {
		tmp = 0.0;
	} else if ((c0 <= -1.4e-26) || !(c0 <= 2.65e-55)) {
		tmp = t_0;
	} else {
		tmp = (c0 / (2.0 * w)) * (c0 * 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) :: t_0
    real(8) :: tmp
    t_0 = (c0 / w) * (((d_1 / d) * (d_1 / d)) / (w * (h / c0)))
    if (c0 <= (-2.6d+273)) then
        tmp = t_0
    else if (c0 <= (-4.1d+247)) then
        tmp = 0.0d0
    else if ((c0 <= (-1.4d-26)) .or. (.not. (c0 <= 2.65d-55))) then
        tmp = t_0
    else
        tmp = (c0 / (2.0d0 * w)) * (c0 * 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 t_0 = (c0 / w) * (((d / D) * (d / D)) / (w * (h / c0)));
	double tmp;
	if (c0 <= -2.6e+273) {
		tmp = t_0;
	} else if (c0 <= -4.1e+247) {
		tmp = 0.0;
	} else if ((c0 <= -1.4e-26) || !(c0 <= 2.65e-55)) {
		tmp = t_0;
	} else {
		tmp = (c0 / (2.0 * w)) * (c0 * 0.0);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 / w) * (((d / D) * (d / D)) / (w * (h / c0)))
	tmp = 0
	if c0 <= -2.6e+273:
		tmp = t_0
	elif c0 <= -4.1e+247:
		tmp = 0.0
	elif (c0 <= -1.4e-26) or not (c0 <= 2.65e-55):
		tmp = t_0
	else:
		tmp = (c0 / (2.0 * w)) * (c0 * 0.0)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 / w) * Float64(Float64(Float64(d / D) * Float64(d / D)) / Float64(w * Float64(h / c0))))
	tmp = 0.0
	if (c0 <= -2.6e+273)
		tmp = t_0;
	elseif (c0 <= -4.1e+247)
		tmp = 0.0;
	elseif ((c0 <= -1.4e-26) || !(c0 <= 2.65e-55))
		tmp = t_0;
	else
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(c0 * 0.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 / w) * (((d / D) * (d / D)) / (w * (h / c0)));
	tmp = 0.0;
	if (c0 <= -2.6e+273)
		tmp = t_0;
	elseif (c0 <= -4.1e+247)
		tmp = 0.0;
	elseif ((c0 <= -1.4e-26) || ~((c0 <= 2.65e-55)))
		tmp = t_0;
	else
		tmp = (c0 / (2.0 * w)) * (c0 * 0.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 / w), $MachinePrecision] * N[(N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] / N[(w * N[(h / c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c0, -2.6e+273], t$95$0, If[LessEqual[c0, -4.1e+247], 0.0, If[Or[LessEqual[c0, -1.4e-26], N[Not[LessEqual[c0, 2.65e-55]], $MachinePrecision]], t$95$0, N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(c0 * 0.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if c0 < -2.59999999999999993e273 or -4.1000000000000002e247 < c0 < -1.4000000000000001e-26 or 2.6500000000000001e-55 < c0

   1. Initial program 28.6%

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

   3. Simplified 30.4%

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

   5. Taylor expanded in c0 around inf 38.2%

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

      1. *-commutative 38.2%

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

      2. *-commutative 38.2%

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

      3. associate-*r* 38.2%

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

      4. associate-/r* 38.7%

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

      5. *-commutative 38.7%

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

   7. Simplified 38.7%

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

      1. pow1 38.7%

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

      2. associate-*r* 38.7%

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

      3. associate-/l/ 38.7%

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

      4. div-inv 38.7%

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

      5. metadata-eval 38.7%

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

      6. associate-/l/ 38.2%

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

      7. *-commutative 38.2%

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

   9. Applied egg-rr 38.2%

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

       1. unpow1 38.2%

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

       2. *-commutative 38.2%

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

       3. *-commutative 38.2%

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

       4. associate-*r* 38.2%

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

       5. *-commutative 38.2%

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

       6. times-frac 39.6%

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

       7. unpow2 39.6%

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

       8. unpow2 39.6%

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

       9. times-frac 50.6%

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

       10. unpow2 50.6%

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

       11. associate-*l* 50.6%

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

       12. metadata-eval 50.6%

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

       13. *-rgt-identity 50.6%

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

   11. Simplified 53.2%

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

       1. unpow2 53.2%

          \[\leadsto \frac{\color{blue}{\frac{d}{D} \cdot \frac{d}{D}}}{\frac{h}{c0} \cdot w} \cdot \frac{c0}{w} \]

   13. Applied egg-rr 53.2%

       \[\leadsto \frac{\color{blue}{\frac{d}{D} \cdot \frac{d}{D}}}{\frac{h}{c0} \cdot w} \cdot \frac{c0}{w} \]

   if -2.59999999999999993e273 < c0 < -4.1000000000000002e247

   1. Initial program 12.5%

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

   3. Simplified 12.5%

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

   5. Taylor expanded in c0 around -inf 0.0%

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

      1. associate-*r* 0.0%

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

      2. neg-mul-1 0.0%

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

      3. distribute-lft1-in 0.0%

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

      4. metadata-eval 0.0%

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

      5. mul0-lft 50.2%

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

      6. distribute-lft-neg-in 50.2%

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

      7. distribute-rgt-neg-in 50.2%

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

      8. metadata-eval 50.2%

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

   7. Simplified 50.2%

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

   8. Taylor expanded in c0 around 0 76.3%

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

   if -1.4000000000000001e-26 < c0 < 2.6500000000000001e-55

   1. Initial program 23.2%

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

   3. Simplified 23.3%

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

   5. Taylor expanded in c0 around -inf 6.6%

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

      1. associate-*r* 6.6%

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

      2. neg-mul-1 6.6%

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

      3. distribute-lft1-in 6.6%

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

      4. metadata-eval 6.6%

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

      5. mul0-lft 53.4%

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

      6. distribute-lft-neg-in 53.4%

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

      7. distribute-rgt-neg-in 53.4%

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

      8. metadata-eval 53.4%

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

   7. Simplified 53.4%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -2.6 \cdot 10^{+273}:\\ \;\;\;\;\frac{c0}{w} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w \cdot \frac{h}{c0}}\\ \mathbf{elif}\;c0 \leq -4.1 \cdot 10^{+247}:\\ \;\;\;\;0\\ \mathbf{elif}\;c0 \leq -1.4 \cdot 10^{-26} \lor \neg \left(c0 \leq 2.65 \cdot 10^{-55}\right):\\ \;\;\;\;\frac{c0}{w} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w \cdot \frac{h}{c0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(c0 \cdot 0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 33.6% 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 26.1%

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

3. Simplified 27.3%

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

5. Taylor expanded in c0 around -inf 6.0%

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

   1. associate-*r* 6.0%

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

   2. neg-mul-1 6.0%

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

   3. distribute-lft1-in 6.0%

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

   4. metadata-eval 6.0%

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

   5. mul0-lft 30.0%

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

   6. distribute-lft-neg-in 30.0%

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

   7. distribute-rgt-neg-in 30.0%

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

   8. metadata-eval 30.0%

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

7. Simplified 30.0%

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

8. Taylor expanded in c0 around 0 34.6%

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

9. Final simplification 34.6%

   \[\leadsto 0 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024034 
(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))))))