Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.8% → 54.9%

Time: 42.3s

Alternatives: 11

Speedup: 21.6×

• could not determine a ground truth

  1. c0 = 4.587571261368582e+281
  2. w = 6.095553888519922e-181
  3. h = -1.0005555491383368e-247
  4. D = -5.208485923913487e-295
  5. d = 1.7360093376896353e+196
  6. M = 4.2332416074852623e+141

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 24.8% 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: 54.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\\ \mathbf{if}\;t_0 \cdot \left(t_1 + \sqrt{t_1 \cdot t_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right)}{d \cdot d}\\ \end{array} \end{array} \]

FPCore

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

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((D * D) * (w * h));
	double tmp;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_0 * (2.0 * (pow((d / D), 2.0) * (c0 / (w * h))));
	} else {
		tmp = 0.25 * ((h * ((D * D) * (M * M))) / (d * d));
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((D * D) * (w * h));
	double tmp;
	if ((t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * (2.0 * (Math.pow((d / D), 2.0) * (c0 / (w * h))));
	} else {
		tmp = 0.25 * ((h * ((D * D) * (M * M))) / (d * d));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	t_1 = (c0 * (d * d)) / ((D * D) * (w * h))
	tmp = 0
	if (t_0 * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))) <= math.inf:
		tmp = t_0 * (2.0 * (math.pow((d / D), 2.0) * (c0 / (w * h))))
	else:
		tmp = 0.25 * ((h * ((D * D) * (M * M))) / (d * d))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(D * D) * Float64(w * h)))
	tmp = 0.0
	if (Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_0 * Float64(2.0 * Float64((Float64(d / D) ^ 2.0) * Float64(c0 / Float64(w * h)))));
	else
		tmp = Float64(0.25 * Float64(Float64(h * Float64(Float64(D * D) * Float64(M * M))) / Float64(d * d)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	t_1 = (c0 * (d * d)) / ((D * D) * (w * h));
	tmp = 0.0;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= Inf)
		tmp = t_0 * (2.0 * (((d / D) ^ 2.0) * (c0 / (w * h))));
	else
		tmp = 0.25 * ((h * ((D * D) * (M * M))) / (d * d));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(2.0 * N[(N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision] * N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(h * N[(N[(D * D), $MachinePrecision] * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 75.9%

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

   3. Simplified 73.8%

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

   4. Taylor expanded in c0 around inf 80.1%

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

      1. associate-/l* 79.3%

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

      2. unpow2 79.3%

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

      3. unpow2 79.3%

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

   6. Simplified 79.3%

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

      1. *-commutative 79.3%

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

      2. *-commutative 79.3%

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

      3. associate-/l* 80.1%

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

      4. times-frac 78.1%

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

      5. frac-times 84.9%

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

      6. *-commutative 84.9%

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

      7. pow2 84.9%

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

      8. *-commutative 84.9%

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

   8. Applied egg-rr 84.9%

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

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

   1. Initial program 0.0%

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

      1. frac-times 0.0%

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

      2. frac-times 0.1%

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

      3. *-commutative 0.1%

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

      4. pow2 0.1%

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

   3. Applied egg-rr 0.1%

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

   4. Taylor expanded in c0 around -inf 1.8%

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

   6. Simplified 26.1%

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

   7. Taylor expanded in c0 around 0 43.7%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   8. Step-by-step derivation

      1. associate-*r* 44.8%

         \[\leadsto 0.25 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2}} \]

      2. unpow2 44.8%

         \[\leadsto 0.25 \cdot \frac{\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot h}{{d}^{2}} \]

      3. unpow2 44.8%

         \[\leadsto 0.25 \cdot \frac{\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot h}{{d}^{2}} \]

      4. *-commutative 44.8%

         \[\leadsto 0.25 \cdot \frac{\color{blue}{h \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right)}}{{d}^{2}} \]

      5. unpow2 44.8%

         \[\leadsto 0.25 \cdot \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right)}{\color{blue}{d \cdot d}} \]

   9. Simplified 44.8%

      \[\leadsto \color{blue}{0.25 \cdot \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right)}{d \cdot d}\\ \end{array} \]

Alternative 2: 43.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c0 \leq -1.6 \cdot 10^{-141} \lor \neg \left(c0 \leq 3.7 \cdot 10^{-179}\right):\\ \;\;\;\;\left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{\frac{c0}{h}}{w}\right) \cdot \frac{2 \cdot \frac{c0}{2}}{w}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (or (<= c0 -1.6e-141) (not (<= c0 3.7e-179)))
   (* (* (pow (/ d D) 2.0) (/ (/ c0 h) w)) (/ (* 2.0 (/ c0 2.0)) w))
   (* 0.25 (/ (* D D) (/ (* d d) (* h (* M M)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((c0 <= -1.6e-141) || !(c0 <= 3.7e-179)) {
		tmp = (pow((d / D), 2.0) * ((c0 / h) / w)) * ((2.0 * (c0 / 2.0)) / w);
	} else {
		tmp = 0.25 * ((D * D) / ((d * d) / (h * (M * M))));
	}
	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 <= (-1.6d-141)) .or. (.not. (c0 <= 3.7d-179))) then
        tmp = (((d_1 / d) ** 2.0d0) * ((c0 / h) / w)) * ((2.0d0 * (c0 / 2.0d0)) / w)
    else
        tmp = 0.25d0 * ((d * d) / ((d_1 * d_1) / (h * (m * m))))
    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 <= -1.6e-141) || !(c0 <= 3.7e-179)) {
		tmp = (Math.pow((d / D), 2.0) * ((c0 / h) / w)) * ((2.0 * (c0 / 2.0)) / w);
	} else {
		tmp = 0.25 * ((D * D) / ((d * d) / (h * (M * M))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (c0 <= -1.6e-141) or not (c0 <= 3.7e-179):
		tmp = (math.pow((d / D), 2.0) * ((c0 / h) / w)) * ((2.0 * (c0 / 2.0)) / w)
	else:
		tmp = 0.25 * ((D * D) / ((d * d) / (h * (M * M))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if ((c0 <= -1.6e-141) || !(c0 <= 3.7e-179))
		tmp = Float64(Float64((Float64(d / D) ^ 2.0) * Float64(Float64(c0 / h) / w)) * Float64(Float64(2.0 * Float64(c0 / 2.0)) / w));
	else
		tmp = Float64(0.25 * Float64(Float64(D * D) / Float64(Float64(d * d) / Float64(h * Float64(M * M)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((c0 <= -1.6e-141) || ~((c0 <= 3.7e-179)))
		tmp = (((d / D) ^ 2.0) * ((c0 / h) / w)) * ((2.0 * (c0 / 2.0)) / w);
	else
		tmp = 0.25 * ((D * D) / ((d * d) / (h * (M * M))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[Or[LessEqual[c0, -1.6e-141], N[Not[LessEqual[c0, 3.7e-179]], $MachinePrecision]], N[(N[(N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(c0 / h), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[(c0 / 2.0), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(D * D), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] / N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c0 < -1.6000000000000001e-141 or 3.6999999999999999e-179 < c0

   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 26.2%

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

   4. Taylor expanded in c0 around inf 36.2%

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

      1. associate-/l* 37.7%

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

      2. unpow2 37.7%

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

      3. unpow2 37.7%

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

   6. Simplified 37.7%

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

      1. *-commutative 37.7%

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

      2. *-commutative 37.7%

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

      3. associate-/l* 36.2%

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

      4. add-cube-cbrt 36.2%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} \cdot \sqrt[3]{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}}\right) \cdot \sqrt[3]{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}}\right)}\right) \]

   8. Applied egg-rr 47.8%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\left(\sqrt[3]{{\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{h \cdot w}} \cdot \sqrt[3]{{\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{h \cdot w}}\right) \cdot \sqrt[3]{{\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{h \cdot w}}\right)}\right) \]
   9. Step-by-step derivation

      1. pow1 47.8%

         \[\leadsto \color{blue}{{\left(\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\left(\sqrt[3]{{\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{h \cdot w}} \cdot \sqrt[3]{{\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{h \cdot w}}\right) \cdot \sqrt[3]{{\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{h \cdot w}}\right)\right)\right)}^{1}} \]

      2. add-cube-cbrt 47.9%

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

      3. associate-*r* 47.9%

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

      4. associate-/r* 47.9%

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

   10. Applied egg-rr 47.9%

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

       1. unpow1 47.9%

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

       2. *-commutative 47.9%

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

       3. associate-/r* 49.7%

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

       4. associate-*l/ 49.7%

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

   12. Simplified 49.7%

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

   if -1.6000000000000001e-141 < c0 < 3.6999999999999999e-179

   1. Initial program 17.8%

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

      1. frac-times 16.0%

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

      2. frac-times 16.1%

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

      3. *-commutative 16.1%

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

      4. pow2 16.1%

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

   3. Applied egg-rr 16.1%

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

   4. Taylor expanded in c0 around -inf 2.6%

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

   6. Simplified 49.9%

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

   7. Taylor expanded in c0 around 0 60.6%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   8. Step-by-step derivation

      1. associate-/l* 60.6%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{{M}^{2} \cdot h}}} \]

      2. unpow2 60.6%

         \[\leadsto 0.25 \cdot \frac{\color{blue}{D \cdot D}}{\frac{{d}^{2}}{{M}^{2} \cdot h}} \]

      3. unpow2 60.6%

         \[\leadsto 0.25 \cdot \frac{D \cdot D}{\frac{\color{blue}{d \cdot d}}{{M}^{2} \cdot h}} \]

      4. *-commutative 60.6%

         \[\leadsto 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{\color{blue}{h \cdot {M}^{2}}}} \]

      5. unpow2 60.6%

         \[\leadsto 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \color{blue}{\left(M \cdot M\right)}}} \]

   9. Simplified 60.6%

      \[\leadsto \color{blue}{0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -1.6 \cdot 10^{-141} \lor \neg \left(c0 \leq 3.7 \cdot 10^{-179}\right):\\ \;\;\;\;\left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{\frac{c0}{h}}{w}\right) \cdot \frac{2 \cdot \frac{c0}{2}}{w}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\\ \end{array} \]

Alternative 3: 38.6% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ \mathbf{if}\;D \cdot D \leq 5 \cdot 10^{-205}:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \left(\left(d \cdot d\right) \cdot \frac{c0}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}\right)\right)\\ \mathbf{elif}\;D \cdot D \leq 4 \cdot 10^{-59}:\\ \;\;\;\;0.25 \cdot \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right)}{d \cdot d}\\ \mathbf{elif}\;D \cdot D \leq 4 \cdot 10^{+304}:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \frac{c0}{\frac{D \cdot D}{d} \cdot \frac{w \cdot h}{d}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{D}{c0}} \cdot \left(\frac{d}{D} \cdot \frac{\frac{d}{h}}{w \cdot w}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w))))
   (if (<= (* D D) 5e-205)
     (* t_0 (* 2.0 (* (* d d) (/ c0 (* D (* D (* w h)))))))
     (if (<= (* D D) 4e-59)
       (* 0.25 (/ (* h (* (* D D) (* M M))) (* d d)))
       (if (<= (* D D) 4e+304)
         (* t_0 (* 2.0 (/ c0 (* (/ (* D D) d) (/ (* w h) d)))))
         (* (/ c0 (/ D c0)) (* (/ d D) (/ (/ d h) (* w w)))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double tmp;
	if ((D * D) <= 5e-205) {
		tmp = t_0 * (2.0 * ((d * d) * (c0 / (D * (D * (w * h))))));
	} else if ((D * D) <= 4e-59) {
		tmp = 0.25 * ((h * ((D * D) * (M * M))) / (d * d));
	} else if ((D * D) <= 4e+304) {
		tmp = t_0 * (2.0 * (c0 / (((D * D) / d) * ((w * h) / d))));
	} else {
		tmp = (c0 / (D / c0)) * ((d / D) * ((d / h) / (w * w)));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 / (2.0d0 * w)
    if ((d * d) <= 5d-205) then
        tmp = t_0 * (2.0d0 * ((d_1 * d_1) * (c0 / (d * (d * (w * h))))))
    else if ((d * d) <= 4d-59) then
        tmp = 0.25d0 * ((h * ((d * d) * (m * m))) / (d_1 * d_1))
    else if ((d * d) <= 4d+304) then
        tmp = t_0 * (2.0d0 * (c0 / (((d * d) / d_1) * ((w * h) / d_1))))
    else
        tmp = (c0 / (d / c0)) * ((d_1 / d) * ((d_1 / h) / (w * w)))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double tmp;
	if ((D * D) <= 5e-205) {
		tmp = t_0 * (2.0 * ((d * d) * (c0 / (D * (D * (w * h))))));
	} else if ((D * D) <= 4e-59) {
		tmp = 0.25 * ((h * ((D * D) * (M * M))) / (d * d));
	} else if ((D * D) <= 4e+304) {
		tmp = t_0 * (2.0 * (c0 / (((D * D) / d) * ((w * h) / d))));
	} else {
		tmp = (c0 / (D / c0)) * ((d / D) * ((d / h) / (w * w)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	tmp = 0
	if (D * D) <= 5e-205:
		tmp = t_0 * (2.0 * ((d * d) * (c0 / (D * (D * (w * h))))))
	elif (D * D) <= 4e-59:
		tmp = 0.25 * ((h * ((D * D) * (M * M))) / (d * d))
	elif (D * D) <= 4e+304:
		tmp = t_0 * (2.0 * (c0 / (((D * D) / d) * ((w * h) / d))))
	else:
		tmp = (c0 / (D / c0)) * ((d / D) * ((d / h) / (w * w)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	tmp = 0.0
	if (Float64(D * D) <= 5e-205)
		tmp = Float64(t_0 * Float64(2.0 * Float64(Float64(d * d) * Float64(c0 / Float64(D * Float64(D * Float64(w * h)))))));
	elseif (Float64(D * D) <= 4e-59)
		tmp = Float64(0.25 * Float64(Float64(h * Float64(Float64(D * D) * Float64(M * M))) / Float64(d * d)));
	elseif (Float64(D * D) <= 4e+304)
		tmp = Float64(t_0 * Float64(2.0 * Float64(c0 / Float64(Float64(Float64(D * D) / d) * Float64(Float64(w * h) / d)))));
	else
		tmp = Float64(Float64(c0 / Float64(D / c0)) * Float64(Float64(d / D) * Float64(Float64(d / h) / Float64(w * w))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	tmp = 0.0;
	if ((D * D) <= 5e-205)
		tmp = t_0 * (2.0 * ((d * d) * (c0 / (D * (D * (w * h))))));
	elseif ((D * D) <= 4e-59)
		tmp = 0.25 * ((h * ((D * D) * (M * M))) / (d * d));
	elseif ((D * D) <= 4e+304)
		tmp = t_0 * (2.0 * (c0 / (((D * D) / d) * ((w * h) / d))));
	else
		tmp = (c0 / (D / c0)) * ((d / D) * ((d / h) / (w * w)));
	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]}, If[LessEqual[N[(D * D), $MachinePrecision], 5e-205], N[(t$95$0 * N[(2.0 * N[(N[(d * d), $MachinePrecision] * N[(c0 / N[(D * N[(D * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(D * D), $MachinePrecision], 4e-59], N[(0.25 * N[(N[(h * N[(N[(D * D), $MachinePrecision] * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(D * D), $MachinePrecision], 4e+304], N[(t$95$0 * N[(2.0 * N[(c0 / N[(N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision] * N[(N[(w * h), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0 / N[(D / c0), $MachinePrecision]), $MachinePrecision] * N[(N[(d / D), $MachinePrecision] * N[(N[(d / h), $MachinePrecision] / N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 D D) < 5.00000000000000001e-205

   1. Initial program 27.9%

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

   3. Simplified 28.7%

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

   4. Taylor expanded in c0 around inf 40.3%

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

      1. associate-/l* 42.5%

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

      2. unpow2 42.5%

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

      3. unpow2 42.5%

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

   6. Simplified 42.5%

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

      1. associate-/r/ 42.5%

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

      2. associate-*l* 45.5%

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

   8. Applied egg-rr 45.5%

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

   if 5.00000000000000001e-205 < (*.f64 D D) < 4.0000000000000001e-59

   1. Initial program 24.5%

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

      1. frac-times 22.9%

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

      2. frac-times 22.9%

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

      3. *-commutative 22.9%

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

      4. pow2 22.9%

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

   3. Applied egg-rr 22.9%

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

   4. Taylor expanded in c0 around -inf 2.6%

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

   6. Simplified 39.8%

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

   7. Taylor expanded in c0 around 0 54.9%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   8. Step-by-step derivation

      1. associate-*r* 56.6%

         \[\leadsto 0.25 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2}} \]

      2. unpow2 56.6%

         \[\leadsto 0.25 \cdot \frac{\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot h}{{d}^{2}} \]

      3. unpow2 56.6%

         \[\leadsto 0.25 \cdot \frac{\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot h}{{d}^{2}} \]

      4. *-commutative 56.6%

         \[\leadsto 0.25 \cdot \frac{\color{blue}{h \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right)}}{{d}^{2}} \]

      5. unpow2 56.6%

         \[\leadsto 0.25 \cdot \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right)}{\color{blue}{d \cdot d}} \]

   9. Simplified 56.6%

      \[\leadsto \color{blue}{0.25 \cdot \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \]

   if 4.0000000000000001e-59 < (*.f64 D D) < 3.9999999999999998e304

   1. Initial program 32.0%

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

   3. Simplified 30.2%

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

   4. Taylor expanded in c0 around inf 40.1%

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

      1. associate-/l* 42.2%

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

      2. unpow2 42.2%

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

      3. unpow2 42.2%

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

   6. Simplified 42.2%

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

   7. Taylor expanded in D around 0 42.2%

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

      1. unpow2 42.2%

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

      2. associate-*r* 42.1%

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

      3. unpow2 42.1%

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

      4. associate-/r* 52.5%

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

      5. associate-*r* 52.6%

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

      6. unpow2 52.6%

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

      7. associate-*r/ 52.8%

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

      8. associate-*l/ 52.8%

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

      9. unpow2 52.8%

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

   9. Simplified 52.8%

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

   if 3.9999999999999998e304 < (*.f64 D D)

   1. Initial program 0.1%

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

   3. Simplified 0.1%

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

   4. Taylor expanded in c0 around inf 0.2%

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

      1. pow1 0.2%

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

      2. pow2 0.2%

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

      3. unpow2 0.2%

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

   6. Applied egg-rr 0.2%

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

      1. unpow1 0.2%

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

      2. associate-*l* 28.0%

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

   8. Simplified 28.0%

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

      1. pow2 28.0%

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

      2. pow2 28.0%

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

      3. times-frac 31.6%

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

   10. Applied egg-rr 31.6%

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

       1. associate-/l* 34.8%

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

       2. times-frac 38.4%

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

       3. unpow2 38.4%

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

       4. associate-/r* 38.1%

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

       5. unpow2 38.1%

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

   12. Simplified 38.1%

       \[\leadsto \color{blue}{\frac{c0}{\frac{D}{c0}} \cdot \left(\frac{d}{D} \cdot \frac{\frac{d}{h}}{w \cdot w}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \cdot D \leq 5 \cdot 10^{-205}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\left(d \cdot d\right) \cdot \frac{c0}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}\right)\right)\\ \mathbf{elif}\;D \cdot D \leq 4 \cdot 10^{-59}:\\ \;\;\;\;0.25 \cdot \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right)}{d \cdot d}\\ \mathbf{elif}\;D \cdot D \leq 4 \cdot 10^{+304}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0}{\frac{D \cdot D}{d} \cdot \frac{w \cdot h}{d}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{D}{c0}} \cdot \left(\frac{d}{D} \cdot \frac{\frac{d}{h}}{w \cdot w}\right)\\ \end{array} \]

Alternative 4: 39.0% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{\frac{D}{c0}} \cdot \left(\frac{d}{D} \cdot \frac{\frac{d}{h}}{w \cdot w}\right)\\ \mathbf{if}\;D \cdot D \leq 2 \cdot 10^{-209}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;D \cdot D \leq 4 \cdot 10^{-59}:\\ \;\;\;\;0.25 \cdot \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right)}{d \cdot d}\\ \mathbf{elif}\;D \cdot D \leq 2 \cdot 10^{+286}:\\ \;\;\;\;c0 \cdot \left(\left(c0 \cdot \frac{d}{D \cdot D}\right) \cdot \frac{d}{h \cdot \left(w \cdot w\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ c0 (/ D c0)) (* (/ d D) (/ (/ d h) (* w w))))))
   (if (<= (* D D) 2e-209)
     t_0
     (if (<= (* D D) 4e-59)
       (* 0.25 (/ (* h (* (* D D) (* M M))) (* d d)))
       (if (<= (* D D) 2e+286)
         (* c0 (* (* c0 (/ d (* D D))) (/ d (* h (* w w)))))
         t_0)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 / (D / c0)) * ((d / D) * ((d / h) / (w * w)));
	double tmp;
	if ((D * D) <= 2e-209) {
		tmp = t_0;
	} else if ((D * D) <= 4e-59) {
		tmp = 0.25 * ((h * ((D * D) * (M * M))) / (d * d));
	} else if ((D * D) <= 2e+286) {
		tmp = c0 * ((c0 * (d / (D * D))) * (d / (h * (w * w))));
	} else {
		tmp = 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) :: tmp
    t_0 = (c0 / (d / c0)) * ((d_1 / d) * ((d_1 / h) / (w * w)))
    if ((d * d) <= 2d-209) then
        tmp = t_0
    else if ((d * d) <= 4d-59) then
        tmp = 0.25d0 * ((h * ((d * d) * (m * m))) / (d_1 * d_1))
    else if ((d * d) <= 2d+286) then
        tmp = c0 * ((c0 * (d_1 / (d * d))) * (d_1 / (h * (w * w))))
    else
        tmp = 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 = (c0 / (D / c0)) * ((d / D) * ((d / h) / (w * w)));
	double tmp;
	if ((D * D) <= 2e-209) {
		tmp = t_0;
	} else if ((D * D) <= 4e-59) {
		tmp = 0.25 * ((h * ((D * D) * (M * M))) / (d * d));
	} else if ((D * D) <= 2e+286) {
		tmp = c0 * ((c0 * (d / (D * D))) * (d / (h * (w * w))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 / (D / c0)) * ((d / D) * ((d / h) / (w * w)))
	tmp = 0
	if (D * D) <= 2e-209:
		tmp = t_0
	elif (D * D) <= 4e-59:
		tmp = 0.25 * ((h * ((D * D) * (M * M))) / (d * d))
	elif (D * D) <= 2e+286:
		tmp = c0 * ((c0 * (d / (D * D))) * (d / (h * (w * w))))
	else:
		tmp = t_0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 / Float64(D / c0)) * Float64(Float64(d / D) * Float64(Float64(d / h) / Float64(w * w))))
	tmp = 0.0
	if (Float64(D * D) <= 2e-209)
		tmp = t_0;
	elseif (Float64(D * D) <= 4e-59)
		tmp = Float64(0.25 * Float64(Float64(h * Float64(Float64(D * D) * Float64(M * M))) / Float64(d * d)));
	elseif (Float64(D * D) <= 2e+286)
		tmp = Float64(c0 * Float64(Float64(c0 * Float64(d / Float64(D * D))) * Float64(d / Float64(h * Float64(w * w)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 / (D / c0)) * ((d / D) * ((d / h) / (w * w)));
	tmp = 0.0;
	if ((D * D) <= 2e-209)
		tmp = t_0;
	elseif ((D * D) <= 4e-59)
		tmp = 0.25 * ((h * ((D * D) * (M * M))) / (d * d));
	elseif ((D * D) <= 2e+286)
		tmp = c0 * ((c0 * (d / (D * D))) * (d / (h * (w * w))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 / N[(D / c0), $MachinePrecision]), $MachinePrecision] * N[(N[(d / D), $MachinePrecision] * N[(N[(d / h), $MachinePrecision] / N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(D * D), $MachinePrecision], 2e-209], t$95$0, If[LessEqual[N[(D * D), $MachinePrecision], 4e-59], N[(0.25 * N[(N[(h * N[(N[(D * D), $MachinePrecision] * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(D * D), $MachinePrecision], 2e+286], N[(c0 * N[(N[(c0 * N[(d / N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[(h * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 D D) < 2.0000000000000001e-209 or 2.00000000000000007e286 < (*.f64 D D)

   1. Initial program 24.1%

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

   3. Simplified 24.2%

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

   4. Taylor expanded in c0 around inf 25.3%

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

      1. pow1 25.3%

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

      2. pow2 25.3%

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

      3. unpow2 25.3%

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

   6. Applied egg-rr 25.3%

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

      1. unpow1 25.3%

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

      2. associate-*l* 33.7%

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

   8. Simplified 33.7%

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

      1. pow2 33.7%

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

      2. pow2 33.7%

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

      3. times-frac 36.9%

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

   10. Applied egg-rr 36.9%

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

       1. associate-/l* 40.6%

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

       2. times-frac 42.8%

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

       3. unpow2 42.8%

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

       4. associate-/r* 42.8%

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

       5. unpow2 42.8%

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

   12. Simplified 42.8%

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

   if 2.0000000000000001e-209 < (*.f64 D D) < 4.0000000000000001e-59

   1. Initial program 25.1%

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

      1. frac-times 23.6%

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

      2. frac-times 23.6%

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

      3. *-commutative 23.6%

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

      4. pow2 23.6%

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

   3. Applied egg-rr 23.6%

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

   4. Taylor expanded in c0 around -inf 2.4%

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

   6. Simplified 39.4%

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

   7. Taylor expanded in c0 around 0 55.7%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   8. Step-by-step derivation

      1. associate-*r* 57.4%

         \[\leadsto 0.25 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2}} \]

      2. unpow2 57.4%

         \[\leadsto 0.25 \cdot \frac{\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot h}{{d}^{2}} \]

      3. unpow2 57.4%

         \[\leadsto 0.25 \cdot \frac{\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot h}{{d}^{2}} \]

      4. *-commutative 57.4%

         \[\leadsto 0.25 \cdot \frac{\color{blue}{h \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right)}}{{d}^{2}} \]

      5. unpow2 57.4%

         \[\leadsto 0.25 \cdot \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right)}{\color{blue}{d \cdot d}} \]

   9. Simplified 57.4%

      \[\leadsto \color{blue}{0.25 \cdot \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \]

   if 4.0000000000000001e-59 < (*.f64 D D) < 2.00000000000000007e286

   1. Initial program 27.5%

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

   3. Simplified 27.8%

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

   4. Taylor expanded in c0 around inf 24.5%

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

      1. pow1 24.5%

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

      2. pow2 24.5%

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

      3. unpow2 24.5%

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

   6. Applied egg-rr 24.5%

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

      1. unpow1 24.5%

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

      2. associate-*l* 24.5%

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

   8. Simplified 24.5%

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

      1. pow2 24.5%

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

      2. pow2 24.5%

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

      3. associate-*r/ 20.9%

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

      4. associate-*r* 20.9%

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

      5. associate-*l* 27.7%

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

      6. times-frac 36.6%

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

   10. Applied egg-rr 36.6%

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

       1. associate-*r* 40.9%

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

   12. Simplified 40.9%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \cdot D \leq 2 \cdot 10^{-209}:\\ \;\;\;\;\frac{c0}{\frac{D}{c0}} \cdot \left(\frac{d}{D} \cdot \frac{\frac{d}{h}}{w \cdot w}\right)\\ \mathbf{elif}\;D \cdot D \leq 4 \cdot 10^{-59}:\\ \;\;\;\;0.25 \cdot \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right)}{d \cdot d}\\ \mathbf{elif}\;D \cdot D \leq 2 \cdot 10^{+286}:\\ \;\;\;\;c0 \cdot \left(\left(c0 \cdot \frac{d}{D \cdot D}\right) \cdot \frac{d}{h \cdot \left(w \cdot w\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{D}{c0}} \cdot \left(\frac{d}{D} \cdot \frac{\frac{d}{h}}{w \cdot w}\right)\\ \end{array} \]

Alternative 5: 39.7% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ \mathbf{if}\;d \cdot d \leq 5 \cdot 10^{-284}:\\ \;\;\;\;\frac{c0}{2} \cdot \frac{0.5 \cdot \frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\frac{h}{d} \cdot \frac{w}{d}\right)}{c0}}{w}\\ \mathbf{elif}\;d \cdot d \leq 2 \cdot 10^{+304}:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \left(\left(d \cdot d\right) \cdot \frac{c0}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \frac{c0}{\frac{D \cdot D}{d} \cdot \frac{w \cdot h}{d}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w))))
   (if (<= (* d d) 5e-284)
     (*
      (/ c0 2.0)
      (/ (* 0.5 (/ (* (* (* D D) (* M M)) (* (/ h d) (/ w d))) c0)) w))
     (if (<= (* d d) 2e+304)
       (* t_0 (* 2.0 (* (* d d) (/ c0 (* D (* D (* w h)))))))
       (* t_0 (* 2.0 (/ c0 (* (/ (* D D) d) (/ (* w h) d)))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double tmp;
	if ((d * d) <= 5e-284) {
		tmp = (c0 / 2.0) * ((0.5 * ((((D * D) * (M * M)) * ((h / d) * (w / d))) / c0)) / w);
	} else if ((d * d) <= 2e+304) {
		tmp = t_0 * (2.0 * ((d * d) * (c0 / (D * (D * (w * h))))));
	} else {
		tmp = t_0 * (2.0 * (c0 / (((D * D) / d) * ((w * h) / d))));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 / (2.0d0 * w)
    if ((d_1 * d_1) <= 5d-284) then
        tmp = (c0 / 2.0d0) * ((0.5d0 * ((((d * d) * (m * m)) * ((h / d_1) * (w / d_1))) / c0)) / w)
    else if ((d_1 * d_1) <= 2d+304) then
        tmp = t_0 * (2.0d0 * ((d_1 * d_1) * (c0 / (d * (d * (w * h))))))
    else
        tmp = t_0 * (2.0d0 * (c0 / (((d * d) / d_1) * ((w * h) / d_1))))
    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 / (2.0 * w);
	double tmp;
	if ((d * d) <= 5e-284) {
		tmp = (c0 / 2.0) * ((0.5 * ((((D * D) * (M * M)) * ((h / d) * (w / d))) / c0)) / w);
	} else if ((d * d) <= 2e+304) {
		tmp = t_0 * (2.0 * ((d * d) * (c0 / (D * (D * (w * h))))));
	} else {
		tmp = t_0 * (2.0 * (c0 / (((D * D) / d) * ((w * h) / d))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	tmp = 0
	if (d * d) <= 5e-284:
		tmp = (c0 / 2.0) * ((0.5 * ((((D * D) * (M * M)) * ((h / d) * (w / d))) / c0)) / w)
	elif (d * d) <= 2e+304:
		tmp = t_0 * (2.0 * ((d * d) * (c0 / (D * (D * (w * h))))))
	else:
		tmp = t_0 * (2.0 * (c0 / (((D * D) / d) * ((w * h) / d))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	tmp = 0.0
	if (Float64(d * d) <= 5e-284)
		tmp = Float64(Float64(c0 / 2.0) * Float64(Float64(0.5 * Float64(Float64(Float64(Float64(D * D) * Float64(M * M)) * Float64(Float64(h / d) * Float64(w / d))) / c0)) / w));
	elseif (Float64(d * d) <= 2e+304)
		tmp = Float64(t_0 * Float64(2.0 * Float64(Float64(d * d) * Float64(c0 / Float64(D * Float64(D * Float64(w * h)))))));
	else
		tmp = Float64(t_0 * Float64(2.0 * Float64(c0 / Float64(Float64(Float64(D * D) / d) * Float64(Float64(w * h) / d)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	tmp = 0.0;
	if ((d * d) <= 5e-284)
		tmp = (c0 / 2.0) * ((0.5 * ((((D * D) * (M * M)) * ((h / d) * (w / d))) / c0)) / w);
	elseif ((d * d) <= 2e+304)
		tmp = t_0 * (2.0 * ((d * d) * (c0 / (D * (D * (w * h))))));
	else
		tmp = t_0 * (2.0 * (c0 / (((D * D) / d) * ((w * h) / d))));
	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]}, If[LessEqual[N[(d * d), $MachinePrecision], 5e-284], N[(N[(c0 / 2.0), $MachinePrecision] * N[(N[(0.5 * N[(N[(N[(N[(D * D), $MachinePrecision] * N[(M * M), $MachinePrecision]), $MachinePrecision] * N[(N[(h / d), $MachinePrecision] * N[(w / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(d * d), $MachinePrecision], 2e+304], N[(t$95$0 * N[(2.0 * N[(N[(d * d), $MachinePrecision] * N[(c0 / N[(D * N[(D * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(2.0 * N[(c0 / N[(N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision] * N[(N[(w * h), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 d d) < 4.99999999999999973e-284

   1. Initial program 4.0%

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

      1. frac-times 4.0%

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

      2. frac-times 4.1%

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

      3. *-commutative 4.1%

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

      4. pow2 4.1%

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

   3. Applied egg-rr 4.1%

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

   4. Taylor expanded in c0 around -inf 0.0%

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

   6. Simplified 0.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(0 + 0.5 \cdot \frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \left(h \cdot w\right)}{c0 \cdot \left(d \cdot d\right)}\right)} \]
   7. Step-by-step derivation

      1. +-lft-identity 0.1%

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

      2. associate-*l/ 4.0%

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

      3. *-commutative 4.0%

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

      4. times-frac 7.9%

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

   8. Applied egg-rr 7.9%

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

      1. times-frac 7.9%

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

      2. *-commutative 7.9%

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

      3. associate-*l/ 11.9%

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

      4. times-frac 32.8%

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

   10. Simplified 32.8%

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

   if 4.99999999999999973e-284 < (*.f64 d d) < 1.9999999999999999e304

   1. Initial program 29.6%

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

   3. Simplified 30.0%

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

   4. Taylor expanded in c0 around inf 43.2%

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

      1. associate-/l* 45.1%

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

      2. unpow2 45.1%

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

      3. unpow2 45.1%

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

   6. Simplified 45.1%

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

      1. associate-/r/ 46.4%

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

      2. associate-*l* 54.2%

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

   8. Applied egg-rr 54.2%

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

   if 1.9999999999999999e304 < (*.f64 d d)

   1. Initial program 24.2%

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

   3. Simplified 24.2%

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

   4. Taylor expanded in c0 around inf 31.8%

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

      1. associate-/l* 31.8%

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

      2. unpow2 31.8%

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

      3. unpow2 31.8%

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

   6. Simplified 31.8%

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

   7. Taylor expanded in D around 0 31.8%

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

      1. unpow2 31.8%

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

      2. associate-*r* 34.9%

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

      3. unpow2 34.9%

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

      4. associate-/r* 46.1%

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

      5. associate-*r* 43.1%

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

      6. unpow2 43.1%

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

      7. associate-*r/ 41.2%

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

      8. associate-*l/ 40.2%

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

      9. unpow2 40.2%

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

   9. Simplified 40.2%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \cdot d \leq 5 \cdot 10^{-284}:\\ \;\;\;\;\frac{c0}{2} \cdot \frac{0.5 \cdot \frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\frac{h}{d} \cdot \frac{w}{d}\right)}{c0}}{w}\\ \mathbf{elif}\;d \cdot d \leq 2 \cdot 10^{+304}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\left(d \cdot d\right) \cdot \frac{c0}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0}{\frac{D \cdot D}{d} \cdot \frac{w \cdot h}{d}}\right)\\ \end{array} \]

Alternative 6: 39.4% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ \mathbf{if}\;w \leq -4.1 \cdot 10^{+93}:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d \cdot d}{h \cdot \left(D \cdot D\right)}\right)\right)\\ \mathbf{elif}\;w \leq 1.3 \cdot 10^{-126}:\\ \;\;\;\;\frac{c0}{\frac{D}{c0}} \cdot \left(\frac{d}{D} \cdot \frac{\frac{d}{h}}{w \cdot w}\right)\\ \mathbf{elif}\;w \leq 2.2 \cdot 10^{-86} \lor \neg \left(w \leq 5.9 \cdot 10^{+153}\right):\\ \;\;\;\;t_0 \cdot 0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\left(c0 \cdot \frac{d}{D \cdot D}\right) \cdot \frac{d}{h \cdot \left(w \cdot w\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w))))
   (if (<= w -4.1e+93)
     (* t_0 (* 2.0 (* (/ c0 w) (/ (* d d) (* h (* D D))))))
     (if (<= w 1.3e-126)
       (* (/ c0 (/ D c0)) (* (/ d D) (/ (/ d h) (* w w))))
       (if (or (<= w 2.2e-86) (not (<= w 5.9e+153)))
         (* t_0 0.0)
         (* c0 (* (* c0 (/ d (* D D))) (/ d (* h (* w w))))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double tmp;
	if (w <= -4.1e+93) {
		tmp = t_0 * (2.0 * ((c0 / w) * ((d * d) / (h * (D * D)))));
	} else if (w <= 1.3e-126) {
		tmp = (c0 / (D / c0)) * ((d / D) * ((d / h) / (w * w)));
	} else if ((w <= 2.2e-86) || !(w <= 5.9e+153)) {
		tmp = t_0 * 0.0;
	} else {
		tmp = c0 * ((c0 * (d / (D * D))) * (d / (h * (w * w))));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 / (2.0d0 * w)
    if (w <= (-4.1d+93)) then
        tmp = t_0 * (2.0d0 * ((c0 / w) * ((d_1 * d_1) / (h * (d * d)))))
    else if (w <= 1.3d-126) then
        tmp = (c0 / (d / c0)) * ((d_1 / d) * ((d_1 / h) / (w * w)))
    else if ((w <= 2.2d-86) .or. (.not. (w <= 5.9d+153))) then
        tmp = t_0 * 0.0d0
    else
        tmp = c0 * ((c0 * (d_1 / (d * d))) * (d_1 / (h * (w * w))))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double tmp;
	if (w <= -4.1e+93) {
		tmp = t_0 * (2.0 * ((c0 / w) * ((d * d) / (h * (D * D)))));
	} else if (w <= 1.3e-126) {
		tmp = (c0 / (D / c0)) * ((d / D) * ((d / h) / (w * w)));
	} else if ((w <= 2.2e-86) || !(w <= 5.9e+153)) {
		tmp = t_0 * 0.0;
	} else {
		tmp = c0 * ((c0 * (d / (D * D))) * (d / (h * (w * w))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	tmp = 0
	if w <= -4.1e+93:
		tmp = t_0 * (2.0 * ((c0 / w) * ((d * d) / (h * (D * D)))))
	elif w <= 1.3e-126:
		tmp = (c0 / (D / c0)) * ((d / D) * ((d / h) / (w * w)))
	elif (w <= 2.2e-86) or not (w <= 5.9e+153):
		tmp = t_0 * 0.0
	else:
		tmp = c0 * ((c0 * (d / (D * D))) * (d / (h * (w * w))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	tmp = 0.0
	if (w <= -4.1e+93)
		tmp = Float64(t_0 * Float64(2.0 * Float64(Float64(c0 / w) * Float64(Float64(d * d) / Float64(h * Float64(D * D))))));
	elseif (w <= 1.3e-126)
		tmp = Float64(Float64(c0 / Float64(D / c0)) * Float64(Float64(d / D) * Float64(Float64(d / h) / Float64(w * w))));
	elseif ((w <= 2.2e-86) || !(w <= 5.9e+153))
		tmp = Float64(t_0 * 0.0);
	else
		tmp = Float64(c0 * Float64(Float64(c0 * Float64(d / Float64(D * D))) * Float64(d / Float64(h * Float64(w * w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	tmp = 0.0;
	if (w <= -4.1e+93)
		tmp = t_0 * (2.0 * ((c0 / w) * ((d * d) / (h * (D * D)))));
	elseif (w <= 1.3e-126)
		tmp = (c0 / (D / c0)) * ((d / D) * ((d / h) / (w * w)));
	elseif ((w <= 2.2e-86) || ~((w <= 5.9e+153)))
		tmp = t_0 * 0.0;
	else
		tmp = c0 * ((c0 * (d / (D * D))) * (d / (h * (w * w))));
	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]}, If[LessEqual[w, -4.1e+93], N[(t$95$0 * N[(2.0 * N[(N[(c0 / w), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / N[(h * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 1.3e-126], N[(N[(c0 / N[(D / c0), $MachinePrecision]), $MachinePrecision] * N[(N[(d / D), $MachinePrecision] * N[(N[(d / h), $MachinePrecision] / N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[w, 2.2e-86], N[Not[LessEqual[w, 5.9e+153]], $MachinePrecision]], N[(t$95$0 * 0.0), $MachinePrecision], N[(c0 * N[(N[(c0 * N[(d / N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[(h * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if w < -4.1000000000000001e93

   1. Initial program 24.6%

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

   3. Simplified 24.6%

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

   4. Taylor expanded in c0 around inf 28.7%

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

      1. associate-*r/ 28.7%

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

      2. associate-*r* 32.5%

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

      3. *-commutative 32.5%

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

      4. unpow2 32.5%

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

      5. *-commutative 32.5%

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

      6. associate-*r/ 32.5%

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

      7. times-frac 39.3%

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

      8. unpow2 39.3%

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

      9. unpow2 39.3%

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

      10. *-commutative 39.3%

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

      11. unpow2 39.3%

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

   6. Simplified 39.3%

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

   if -4.1000000000000001e93 < w < 1.3e-126

   1. Initial program 29.6%

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

   3. Simplified 30.4%

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

   4. Taylor expanded in c0 around inf 29.7%

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

      1. pow1 29.7%

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

      2. pow2 29.7%

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

      3. unpow2 29.7%

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

   6. Applied egg-rr 29.7%

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

      1. unpow1 29.7%

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

      2. associate-*l* 35.4%

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

   8. Simplified 35.4%

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

      1. pow2 35.4%

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

      2. pow2 35.4%

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

      3. times-frac 39.4%

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

   10. Applied egg-rr 39.4%

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

       1. associate-/l* 42.2%

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

       2. times-frac 48.5%

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

       3. unpow2 48.5%

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

       4. associate-/r* 49.1%

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

       5. unpow2 49.1%

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

   12. Simplified 49.1%

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

   if 1.3e-126 < w < 2.2000000000000002e-86 or 5.9000000000000002e153 < w

   1. Initial program 14.2%

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

   3. Simplified 12.2%

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

   4. Taylor expanded in c0 around -inf 6.4%

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

      1. mul-1-neg 6.4%

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

      2. distribute-lft-in 6.4%

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

   6. Simplified 47.7%

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

   if 2.2000000000000002e-86 < w < 5.9000000000000002e153

   1. Initial program 18.7%

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

   3. Simplified 18.9%

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

   4. Taylor expanded in c0 around inf 36.3%

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

      1. pow1 36.3%

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

      2. pow2 36.3%

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

      3. unpow2 36.3%

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

   6. Applied egg-rr 36.3%

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

      1. unpow1 36.3%

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

      2. associate-*l* 38.5%

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

   8. Simplified 38.5%

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

      1. pow2 38.5%

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

      2. pow2 38.5%

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

      3. associate-*r/ 34.8%

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

      4. associate-*r* 32.6%

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

      5. associate-*l* 35.0%

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

      6. times-frac 40.3%

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

   10. Applied egg-rr 40.3%

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

       1. associate-*r* 46.1%

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

   12. Simplified 46.1%

       \[\leadsto \color{blue}{c0 \cdot \left(\left(c0 \cdot \frac{d}{D \cdot D}\right) \cdot \frac{d}{h \cdot \left(w \cdot w\right)}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -4.1 \cdot 10^{+93}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d \cdot d}{h \cdot \left(D \cdot D\right)}\right)\right)\\ \mathbf{elif}\;w \leq 1.3 \cdot 10^{-126}:\\ \;\;\;\;\frac{c0}{\frac{D}{c0}} \cdot \left(\frac{d}{D} \cdot \frac{\frac{d}{h}}{w \cdot w}\right)\\ \mathbf{elif}\;w \leq 2.2 \cdot 10^{-86} \lor \neg \left(w \leq 5.9 \cdot 10^{+153}\right):\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot 0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\left(c0 \cdot \frac{d}{D \cdot D}\right) \cdot \frac{d}{h \cdot \left(w \cdot w\right)}\right)\\ \end{array} \]

Alternative 7: 39.0% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c0 \leq -1.2 \cdot 10^{-141}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\left(d \cdot d\right) \cdot \frac{c0}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}\right)\right)\\ \mathbf{elif}\;c0 \leq 1.08 \cdot 10^{-163}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\\ \mathbf{elif}\;c0 \leq 3 \cdot 10^{+50} \lor \neg \left(c0 \leq 6.5 \cdot 10^{+129}\right):\\ \;\;\;\;\frac{c0}{\frac{D}{c0}} \cdot \left(\frac{d}{D} \cdot \frac{\frac{d}{h}}{w \cdot w}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c0 \cdot c0}{\frac{w}{0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= c0 -1.2e-141)
   (* (/ c0 (* 2.0 w)) (* 2.0 (* (* d d) (/ c0 (* D (* D (* w h)))))))
   (if (<= c0 1.08e-163)
     (* 0.25 (/ (* D D) (/ (* d d) (* h (* M M)))))
     (if (or (<= c0 3e+50) (not (<= c0 6.5e+129)))
       (* (/ c0 (/ D c0)) (* (/ d D) (/ (/ d h) (* w w))))
       (* -0.5 (/ (* c0 c0) (/ w 0.0)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (c0 <= -1.2e-141) {
		tmp = (c0 / (2.0 * w)) * (2.0 * ((d * d) * (c0 / (D * (D * (w * h))))));
	} else if (c0 <= 1.08e-163) {
		tmp = 0.25 * ((D * D) / ((d * d) / (h * (M * M))));
	} else if ((c0 <= 3e+50) || !(c0 <= 6.5e+129)) {
		tmp = (c0 / (D / c0)) * ((d / D) * ((d / h) / (w * w)));
	} else {
		tmp = -0.5 * ((c0 * c0) / (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 <= (-1.2d-141)) then
        tmp = (c0 / (2.0d0 * w)) * (2.0d0 * ((d_1 * d_1) * (c0 / (d * (d * (w * h))))))
    else if (c0 <= 1.08d-163) then
        tmp = 0.25d0 * ((d * d) / ((d_1 * d_1) / (h * (m * m))))
    else if ((c0 <= 3d+50) .or. (.not. (c0 <= 6.5d+129))) then
        tmp = (c0 / (d / c0)) * ((d_1 / d) * ((d_1 / h) / (w * w)))
    else
        tmp = (-0.5d0) * ((c0 * c0) / (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 <= -1.2e-141) {
		tmp = (c0 / (2.0 * w)) * (2.0 * ((d * d) * (c0 / (D * (D * (w * h))))));
	} else if (c0 <= 1.08e-163) {
		tmp = 0.25 * ((D * D) / ((d * d) / (h * (M * M))));
	} else if ((c0 <= 3e+50) || !(c0 <= 6.5e+129)) {
		tmp = (c0 / (D / c0)) * ((d / D) * ((d / h) / (w * w)));
	} else {
		tmp = -0.5 * ((c0 * c0) / (w / 0.0));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if c0 <= -1.2e-141:
		tmp = (c0 / (2.0 * w)) * (2.0 * ((d * d) * (c0 / (D * (D * (w * h))))))
	elif c0 <= 1.08e-163:
		tmp = 0.25 * ((D * D) / ((d * d) / (h * (M * M))))
	elif (c0 <= 3e+50) or not (c0 <= 6.5e+129):
		tmp = (c0 / (D / c0)) * ((d / D) * ((d / h) / (w * w)))
	else:
		tmp = -0.5 * ((c0 * c0) / (w / 0.0))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (c0 <= -1.2e-141)
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(2.0 * Float64(Float64(d * d) * Float64(c0 / Float64(D * Float64(D * Float64(w * h)))))));
	elseif (c0 <= 1.08e-163)
		tmp = Float64(0.25 * Float64(Float64(D * D) / Float64(Float64(d * d) / Float64(h * Float64(M * M)))));
	elseif ((c0 <= 3e+50) || !(c0 <= 6.5e+129))
		tmp = Float64(Float64(c0 / Float64(D / c0)) * Float64(Float64(d / D) * Float64(Float64(d / h) / Float64(w * w))));
	else
		tmp = Float64(-0.5 * Float64(Float64(c0 * c0) / Float64(w / 0.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (c0 <= -1.2e-141)
		tmp = (c0 / (2.0 * w)) * (2.0 * ((d * d) * (c0 / (D * (D * (w * h))))));
	elseif (c0 <= 1.08e-163)
		tmp = 0.25 * ((D * D) / ((d * d) / (h * (M * M))));
	elseif ((c0 <= 3e+50) || ~((c0 <= 6.5e+129)))
		tmp = (c0 / (D / c0)) * ((d / D) * ((d / h) / (w * w)));
	else
		tmp = -0.5 * ((c0 * c0) / (w / 0.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[c0, -1.2e-141], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[(d * d), $MachinePrecision] * N[(c0 / N[(D * N[(D * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c0, 1.08e-163], N[(0.25 * N[(N[(D * D), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] / N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[c0, 3e+50], N[Not[LessEqual[c0, 6.5e+129]], $MachinePrecision]], N[(N[(c0 / N[(D / c0), $MachinePrecision]), $MachinePrecision] * N[(N[(d / D), $MachinePrecision] * N[(N[(d / h), $MachinePrecision] / N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(N[(c0 * c0), $MachinePrecision] / N[(w / 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c0 < -1.2e-141

   1. Initial program 24.6%

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

   3. Simplified 23.7%

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

   4. Taylor expanded in c0 around inf 36.8%

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

      1. associate-/l* 37.8%

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

      2. unpow2 37.8%

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

      3. unpow2 37.8%

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

   6. Simplified 37.8%

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

      1. associate-/r/ 37.7%

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

      2. associate-*l* 44.6%

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

   8. Applied egg-rr 44.6%

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

   if -1.2e-141 < c0 < 1.0800000000000001e-163

   1. Initial program 19.4%

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

      1. frac-times 17.7%

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

      2. frac-times 17.7%

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

      3. *-commutative 17.7%

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

      4. pow2 17.7%

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

   3. Applied egg-rr 17.7%

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

   4. Taylor expanded in c0 around -inf 2.5%

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

   6. Simplified 50.0%

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

   7. Taylor expanded in c0 around 0 60.1%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   8. Step-by-step derivation

      1. associate-/l* 60.1%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{{M}^{2} \cdot h}}} \]

      2. unpow2 60.1%

         \[\leadsto 0.25 \cdot \frac{\color{blue}{D \cdot D}}{\frac{{d}^{2}}{{M}^{2} \cdot h}} \]

      3. unpow2 60.1%

         \[\leadsto 0.25 \cdot \frac{D \cdot D}{\frac{\color{blue}{d \cdot d}}{{M}^{2} \cdot h}} \]

      4. *-commutative 60.1%

         \[\leadsto 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{\color{blue}{h \cdot {M}^{2}}}} \]

      5. unpow2 60.1%

         \[\leadsto 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \color{blue}{\left(M \cdot M\right)}}} \]

   9. Simplified 60.1%

      \[\leadsto \color{blue}{0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}} \]

   if 1.0800000000000001e-163 < c0 < 2.9999999999999998e50 or 6.4999999999999995e129 < c0

   1. Initial program 29.2%

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

   3. Simplified 30.3%

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

   4. Taylor expanded in c0 around inf 32.4%

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

      1. pow1 32.4%

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

      2. pow2 32.4%

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

      3. unpow2 32.4%

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

   6. Applied egg-rr 32.4%

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

      1. unpow1 32.4%

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

      2. associate-*l* 38.0%

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

   8. Simplified 38.0%

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

      1. pow2 38.0%

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

      2. pow2 38.0%

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

      3. times-frac 40.8%

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

   10. Applied egg-rr 40.8%

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

       1. associate-/l* 41.8%

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

       2. times-frac 48.5%

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

       3. unpow2 48.5%

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

       4. associate-/r* 48.4%

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

       5. unpow2 48.4%

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

   12. Simplified 48.4%

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

   if 2.9999999999999998e50 < c0 < 6.4999999999999995e129

   1. Initial program 18.5%

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

   3. Simplified 18.8%

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

   4. Taylor expanded in c0 around -inf 5.0%

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

      1. associate-/l* 5.0%

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

      2. unpow2 5.0%

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

      3. distribute-lft1-in 5.0%

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

      4. metadata-eval 5.0%

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

      5. mul0-lft 42.6%

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

   6. Simplified 42.6%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -1.2 \cdot 10^{-141}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\left(d \cdot d\right) \cdot \frac{c0}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}\right)\right)\\ \mathbf{elif}\;c0 \leq 1.08 \cdot 10^{-163}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\\ \mathbf{elif}\;c0 \leq 3 \cdot 10^{+50} \lor \neg \left(c0 \leq 6.5 \cdot 10^{+129}\right):\\ \;\;\;\;\frac{c0}{\frac{D}{c0}} \cdot \left(\frac{d}{D} \cdot \frac{\frac{d}{h}}{w \cdot w}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c0 \cdot c0}{\frac{w}{0}}\\ \end{array} \]

Alternative 8: 38.5% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c0 \cdot \left(\left(c0 \cdot \frac{d}{D \cdot D}\right) \cdot \frac{d}{h \cdot \left(w \cdot w\right)}\right)\\ \mathbf{if}\;w \leq -4.3 \cdot 10^{-99}:\\ \;\;\;\;0.25 \cdot \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right)}{d \cdot d}\\ \mathbf{elif}\;w \leq 1.5 \cdot 10^{-217}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;w \leq 2.5 \cdot 10^{-86}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\\ \mathbf{elif}\;w \leq 2.1 \cdot 10^{+139}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot 0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* c0 (* (* c0 (/ d (* D D))) (/ d (* h (* w w)))))))
   (if (<= w -4.3e-99)
     (* 0.25 (/ (* h (* (* D D) (* M M))) (* d d)))
     (if (<= w 1.5e-217)
       t_0
       (if (<= w 2.5e-86)
         (* 0.25 (/ (* D D) (/ (* d d) (* h (* M M)))))
         (if (<= w 2.1e+139) t_0 (* (/ 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 * ((c0 * (d / (D * D))) * (d / (h * (w * w))));
	double tmp;
	if (w <= -4.3e-99) {
		tmp = 0.25 * ((h * ((D * D) * (M * M))) / (d * d));
	} else if (w <= 1.5e-217) {
		tmp = t_0;
	} else if (w <= 2.5e-86) {
		tmp = 0.25 * ((D * D) / ((d * d) / (h * (M * M))));
	} else if (w <= 2.1e+139) {
		tmp = t_0;
	} else {
		tmp = (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 * ((c0 * (d_1 / (d * d))) * (d_1 / (h * (w * w))))
    if (w <= (-4.3d-99)) then
        tmp = 0.25d0 * ((h * ((d * d) * (m * m))) / (d_1 * d_1))
    else if (w <= 1.5d-217) then
        tmp = t_0
    else if (w <= 2.5d-86) then
        tmp = 0.25d0 * ((d * d) / ((d_1 * d_1) / (h * (m * m))))
    else if (w <= 2.1d+139) then
        tmp = t_0
    else
        tmp = (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 * ((c0 * (d / (D * D))) * (d / (h * (w * w))));
	double tmp;
	if (w <= -4.3e-99) {
		tmp = 0.25 * ((h * ((D * D) * (M * M))) / (d * d));
	} else if (w <= 1.5e-217) {
		tmp = t_0;
	} else if (w <= 2.5e-86) {
		tmp = 0.25 * ((D * D) / ((d * d) / (h * (M * M))));
	} else if (w <= 2.1e+139) {
		tmp = t_0;
	} else {
		tmp = (c0 / (2.0 * w)) * 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 * ((c0 * (d / (D * D))) * (d / (h * (w * w))))
	tmp = 0
	if w <= -4.3e-99:
		tmp = 0.25 * ((h * ((D * D) * (M * M))) / (d * d))
	elif w <= 1.5e-217:
		tmp = t_0
	elif w <= 2.5e-86:
		tmp = 0.25 * ((D * D) / ((d * d) / (h * (M * M))))
	elif w <= 2.1e+139:
		tmp = t_0
	else:
		tmp = (c0 / (2.0 * w)) * 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 * Float64(Float64(c0 * Float64(d / Float64(D * D))) * Float64(d / Float64(h * Float64(w * w)))))
	tmp = 0.0
	if (w <= -4.3e-99)
		tmp = Float64(0.25 * Float64(Float64(h * Float64(Float64(D * D) * Float64(M * M))) / Float64(d * d)));
	elseif (w <= 1.5e-217)
		tmp = t_0;
	elseif (w <= 2.5e-86)
		tmp = Float64(0.25 * Float64(Float64(D * D) / Float64(Float64(d * d) / Float64(h * Float64(M * M)))));
	elseif (w <= 2.1e+139)
		tmp = t_0;
	else
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * 0.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 * ((c0 * (d / (D * D))) * (d / (h * (w * w))));
	tmp = 0.0;
	if (w <= -4.3e-99)
		tmp = 0.25 * ((h * ((D * D) * (M * M))) / (d * d));
	elseif (w <= 1.5e-217)
		tmp = t_0;
	elseif (w <= 2.5e-86)
		tmp = 0.25 * ((D * D) / ((d * d) / (h * (M * M))));
	elseif (w <= 2.1e+139)
		tmp = t_0;
	else
		tmp = (c0 / (2.0 * w)) * 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 * N[(N[(c0 * N[(d / N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[(h * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, -4.3e-99], N[(0.25 * N[(N[(h * N[(N[(D * D), $MachinePrecision] * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 1.5e-217], t$95$0, If[LessEqual[w, 2.5e-86], N[(0.25 * N[(N[(D * D), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] / N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 2.1e+139], t$95$0, N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * 0.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if w < -4.2999999999999999e-99

   1. Initial program 18.5%

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

      1. frac-times 18.5%

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

      2. frac-times 18.5%

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

      3. *-commutative 18.5%

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

      4. pow2 18.5%

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

   3. Applied egg-rr 18.5%

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

   4. Taylor expanded in c0 around -inf 6.1%

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

   6. Simplified 23.3%

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

   7. Taylor expanded in c0 around 0 34.2%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   8. Step-by-step derivation

      1. associate-*r* 36.0%

         \[\leadsto 0.25 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2}} \]

      2. unpow2 36.0%

         \[\leadsto 0.25 \cdot \frac{\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot h}{{d}^{2}} \]

      3. unpow2 36.0%

         \[\leadsto 0.25 \cdot \frac{\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot h}{{d}^{2}} \]

      4. *-commutative 36.0%

         \[\leadsto 0.25 \cdot \frac{\color{blue}{h \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right)}}{{d}^{2}} \]

      5. unpow2 36.0%

         \[\leadsto 0.25 \cdot \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right)}{\color{blue}{d \cdot d}} \]

   9. Simplified 36.0%

      \[\leadsto \color{blue}{0.25 \cdot \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \]

   if -4.2999999999999999e-99 < w < 1.50000000000000002e-217 or 2.4999999999999999e-86 < w < 2.0999999999999999e139

   1. Initial program 32.4%

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

   3. Simplified 33.2%

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

   4. Taylor expanded in c0 around inf 37.6%

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

      1. pow1 37.6%

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

      2. pow2 37.6%

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

      3. unpow2 37.6%

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

   6. Applied egg-rr 37.6%

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

      1. unpow1 37.6%

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

      2. associate-*l* 41.6%

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

   8. Simplified 41.6%

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

      1. pow2 41.6%

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

      2. pow2 41.6%

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

      3. associate-*r/ 43.4%

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

      4. associate-*r* 39.4%

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

      5. associate-*l* 44.1%

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

      6. times-frac 47.7%

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

   10. Applied egg-rr 47.7%

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

       1. associate-*r* 50.6%

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

   12. Simplified 50.6%

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

   if 1.50000000000000002e-217 < w < 2.4999999999999999e-86

   1. Initial program 15.1%

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

      1. frac-times 15.2%

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

      2. frac-times 15.7%

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

      3. *-commutative 15.7%

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

      4. pow2 15.7%

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

   3. Applied egg-rr 15.7%

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

   4. Taylor expanded in c0 around -inf 0.1%

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

   6. Simplified 23.4%

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

   7. Taylor expanded in c0 around 0 43.7%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   8. Step-by-step derivation

      1. associate-/l* 43.7%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{{M}^{2} \cdot h}}} \]

      2. unpow2 43.7%

         \[\leadsto 0.25 \cdot \frac{\color{blue}{D \cdot D}}{\frac{{d}^{2}}{{M}^{2} \cdot h}} \]

      3. unpow2 43.7%

         \[\leadsto 0.25 \cdot \frac{D \cdot D}{\frac{\color{blue}{d \cdot d}}{{M}^{2} \cdot h}} \]

      4. *-commutative 43.7%

         \[\leadsto 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{\color{blue}{h \cdot {M}^{2}}}} \]

      5. unpow2 43.7%

         \[\leadsto 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \color{blue}{\left(M \cdot M\right)}}} \]

   9. Simplified 43.7%

      \[\leadsto \color{blue}{0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}} \]

   if 2.0999999999999999e139 < w

   1. Initial program 15.8%

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

   3. Simplified 13.0%

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

   4. Taylor expanded in c0 around -inf 8.9%

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

      1. mul-1-neg 8.9%

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

      2. distribute-lft-in 8.9%

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

   6. Simplified 42.3%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -4.3 \cdot 10^{-99}:\\ \;\;\;\;0.25 \cdot \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right)}{d \cdot d}\\ \mathbf{elif}\;w \leq 1.5 \cdot 10^{-217}:\\ \;\;\;\;c0 \cdot \left(\left(c0 \cdot \frac{d}{D \cdot D}\right) \cdot \frac{d}{h \cdot \left(w \cdot w\right)}\right)\\ \mathbf{elif}\;w \leq 2.5 \cdot 10^{-86}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\\ \mathbf{elif}\;w \leq 2.1 \cdot 10^{+139}:\\ \;\;\;\;c0 \cdot \left(\left(c0 \cdot \frac{d}{D \cdot D}\right) \cdot \frac{d}{h \cdot \left(w \cdot w\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot 0\\ \end{array} \]

Alternative 9: 35.9% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \cdot d \leq 7.8 \cdot 10^{+301}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot 0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= (* d d) 7.8e+301)
   (* 0.25 (/ (* D D) (/ (* d d) (* h (* M M)))))
   (* (/ c0 (* 2.0 w)) 0.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((d * d) <= 7.8e+301) {
		tmp = 0.25 * ((D * D) / ((d * d) / (h * (M * M))));
	} else {
		tmp = (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 ((d_1 * d_1) <= 7.8d+301) then
        tmp = 0.25d0 * ((d * d) / ((d_1 * d_1) / (h * (m * m))))
    else
        tmp = (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 ((d * d) <= 7.8e+301) {
		tmp = 0.25 * ((D * D) / ((d * d) / (h * (M * M))));
	} else {
		tmp = (c0 / (2.0 * w)) * 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (d * d) <= 7.8e+301:
		tmp = 0.25 * ((D * D) / ((d * d) / (h * (M * M))))
	else:
		tmp = (c0 / (2.0 * w)) * 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (Float64(d * d) <= 7.8e+301)
		tmp = Float64(0.25 * Float64(Float64(D * D) / Float64(Float64(d * d) / Float64(h * Float64(M * M)))));
	else
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * 0.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((d * d) <= 7.8e+301)
		tmp = 0.25 * ((D * D) / ((d * d) / (h * (M * M))));
	else
		tmp = (c0 / (2.0 * w)) * 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[N[(d * d), $MachinePrecision], 7.8e+301], N[(0.25 * N[(N[(D * D), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] / N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * 0.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 d d) < 7.8000000000000003e301

   1. Initial program 25.7%

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

      1. frac-times 24.6%

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

      2. frac-times 24.6%

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

      3. *-commutative 24.6%

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

      4. pow2 24.6%

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

   3. Applied egg-rr 24.6%

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

   4. Taylor expanded in c0 around -inf 5.4%

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

   6. Simplified 17.4%

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

   7. Taylor expanded in c0 around 0 37.0%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   8. Step-by-step derivation

      1. associate-/l* 36.0%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{{M}^{2} \cdot h}}} \]

      2. unpow2 36.0%

         \[\leadsto 0.25 \cdot \frac{\color{blue}{D \cdot D}}{\frac{{d}^{2}}{{M}^{2} \cdot h}} \]

      3. unpow2 36.0%

         \[\leadsto 0.25 \cdot \frac{D \cdot D}{\frac{\color{blue}{d \cdot d}}{{M}^{2} \cdot h}} \]

      4. *-commutative 36.0%

         \[\leadsto 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{\color{blue}{h \cdot {M}^{2}}}} \]

      5. unpow2 36.0%

         \[\leadsto 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \color{blue}{\left(M \cdot M\right)}}} \]

   9. Simplified 36.0%

      \[\leadsto \color{blue}{0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}} \]

   if 7.8000000000000003e301 < (*.f64 d d)

   1. Initial program 23.7%

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

   3. Simplified 23.8%

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

   4. Taylor expanded in c0 around -inf 0.0%

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

      1. mul-1-neg 0.0%

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

      2. distribute-lft-in 0.0%

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

   6. Simplified 33.1%

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

4. Final simplification 34.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \cdot d \leq 7.8 \cdot 10^{+301}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot 0\\ \end{array} \]

Alternative 10: 34.8% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ 0.25 \cdot \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right)}{d \cdot d} \end{array} \]

FPCore

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

C

double code(double c0, double w, double h, double D, double d, double M) {
	return 0.25 * ((h * ((D * D) * (M * M))) / (d * d));
}

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 = 0.25d0 * ((h * ((d * d) * (m * m))) / (d_1 * d_1))
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return 0.25 * ((h * ((D * D) * (M * M))) / (d * d));
}

Python

def code(c0, w, h, D, d, M):
	return 0.25 * ((h * ((D * D) * (M * M))) / (d * d))

Julia

function code(c0, w, h, D, d, M)
	return Float64(0.25 * Float64(Float64(h * Float64(Float64(D * D) * Float64(M * M))) / Float64(d * d)))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = 0.25 * ((h * ((D * D) * (M * M))) / (d * d));
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(0.25 * N[(N[(h * N[(N[(D * D), $MachinePrecision] * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 24.9%

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

   1. frac-times 24.2%

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

   2. frac-times 24.3%

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

   3. *-commutative 24.3%

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

   4. pow2 24.3%

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

3. Applied egg-rr 24.3%

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

4. Taylor expanded in c0 around -inf 3.3%

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

6. Simplified 20.2%

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

7. Taylor expanded in c0 around 0 33.7%

   \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
8. Step-by-step derivation

   1. associate-*r* 33.7%

      \[\leadsto 0.25 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2}} \]

   2. unpow2 33.7%

      \[\leadsto 0.25 \cdot \frac{\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot h}{{d}^{2}} \]

   3. unpow2 33.7%

      \[\leadsto 0.25 \cdot \frac{\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot h}{{d}^{2}} \]

   4. *-commutative 33.7%

      \[\leadsto 0.25 \cdot \frac{\color{blue}{h \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right)}}{{d}^{2}} \]

   5. unpow2 33.7%

      \[\leadsto 0.25 \cdot \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right)}{\color{blue}{d \cdot d}} \]

9. Simplified 33.7%

   \[\leadsto \color{blue}{0.25 \cdot \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \]

10. Final simplification 33.7%

    \[\leadsto 0.25 \cdot \frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right)}{d \cdot d} \]

Alternative 11: 30.0% accurate, 21.6× speedup

Math

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

FPCore

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

C

double code(double c0, double w, double h, double D, double d, double M) {
	return (c0 / (2.0 * w)) * 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 = (c0 / (2.0d0 * w)) * 0.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.9%

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

3. Simplified 25.1%

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

4. Taylor expanded in c0 around -inf 2.6%

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

   1. mul-1-neg 2.6%

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

   2. distribute-lft-in 2.2%

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

6. Simplified 25.0%

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

7. Final simplification 25.0%

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

Reproduce

herbie shell --seed 2023280 
(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))))))