Henrywood and Agarwal, Equation (13)

Percentage Accurate: 25.0% → 64.9%

Time: 36.1s

Alternatives: 9

Speedup: 151.0×

• could not determine a ground truth

  1. c0 = 2.933115961484664e+305
  2. w = 1.3052986161963722e-273
  3. h = -3.7960161496881435e-238
  4. D = 2.376940948469917e-232
  5. d = -1.7644225618836123e+208
  6. M = 7.884419113188641e+65

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 25.0% 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: 64.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(2.0 * N[(N[(N[(c0 * d), $MachinePrecision] / D), $MachinePrecision] * N[(d / N[(h * N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(M * N[(h * M), $MachinePrecision]), $MachinePrecision] / N[Power[N[(d / D), $MachinePrecision], 2.0], $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 73.3%

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

   3. Simplified 74.0%

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

   4. Taylor expanded in c0 around inf 74.4%

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

      1. unpow2 74.4%

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

      2. *-commutative 74.4%

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

      3. associate-/l/ 75.2%

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

      4. associate-*r* 76.0%

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

      5. associate-*r/ 74.7%

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

      6. unpow2 74.7%

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

      7. times-frac 77.4%

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

      8. *-commutative 77.4%

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

   6. Simplified 77.4%

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

      1. add-cbrt-cube 73.9%

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

      2. associate-/l/ 73.9%

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

      3. associate-/l/ 73.8%

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

      4. associate-/l/ 73.9%

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

   8. Applied egg-rr 73.9%

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

      1. associate-*l* 73.9%

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

      2. associate-/r* 74.0%

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

      3. associate-/r* 73.9%

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

      4. associate-/r* 73.9%

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

   10. Simplified 73.9%

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

   11. Taylor expanded in d around 0 79.9%

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

       1. associate-*r* 78.7%

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

   13. Simplified 78.7%

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

   14. Taylor expanded in d around 0 79.9%

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

       1. associate-*r* 78.7%

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

       2. *-commutative 78.7%

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

       3. associate-*l* 81.1%

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

   16. Simplified 81.1%

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

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

   1. Initial program 0.0%

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

   3. Simplified 1.8%

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

      1. flip-+ 0.1%

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

   5. Applied egg-rr 1.6%

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

      1. associate--r- 4.3%

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

      2. +-inverses 27.4%

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

      3. associate-/r* 28.6%

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

      4. associate-/r* 29.5%

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

   7. Simplified 29.5%

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

   8. Taylor expanded in c0 around -inf 25.5%

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

      1. mul-1-neg 25.5%

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

      2. associate-/r* 25.4%

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

      3. distribute-neg-frac 25.4%

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

      4. unpow2 25.4%

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

      5. sub-neg 25.4%

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

      6. mul-1-neg 25.4%

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

      7. distribute-rgt-out 25.4%

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

   10. Simplified 36.0%

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

   11. Taylor expanded in c0 around 0 39.5%

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

       1. *-commutative 39.5%

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

       2. associate-/l* 40.1%

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

       3. *-commutative 40.1%

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

       4. unpow2 40.1%

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

       5. associate-*r* 40.7%

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

       6. unpow2 40.7%

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

       7. unpow2 40.7%

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

       8. times-frac 58.3%

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

       9. unpow2 58.3%

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

   13. Simplified 58.3%

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

4. Final simplification 65.5%

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

Alternative 2: 41.8% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := t_0 \cdot \left(2 \cdot \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{h \cdot \left(w \cdot D\right)}\right)\right)\\ t_2 := 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\\ \mathbf{if}\;h \leq -2.9 \cdot 10^{-290}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;h \leq 4 \cdot 10^{-295}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;h \leq 1.7 \cdot 10^{-154}:\\ \;\;\;\;\left(\frac{c0}{D} \cdot \frac{c0}{D}\right) \cdot \left(\frac{d}{w} \cdot \frac{\frac{d}{w}}{h}\right)\\ \mathbf{elif}\;h \leq 2.8 \cdot 10^{+35}:\\ \;\;\;\;0\\ \mathbf{elif}\;h \leq 1.01 \cdot 10^{+180}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;h \leq 2.55 \cdot 10^{+210}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \left(\frac{\frac{d}{D}}{w \cdot h} \cdot \left(d \cdot \frac{c0}{D}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w)))
        (t_1 (* t_0 (* 2.0 (* (/ (* c0 d) D) (/ d (* h (* w D)))))))
        (t_2 (* 0.25 (/ (* D D) (/ (* d d) (* h (* M M)))))))
   (if (<= h -2.9e-290)
     t_1
     (if (<= h 4e-295)
       t_2
       (if (<= h 1.7e-154)
         (* (* (/ c0 D) (/ c0 D)) (* (/ d w) (/ (/ d w) h)))
         (if (<= h 2.8e+35)
           0.0
           (if (<= h 1.01e+180)
             t_1
             (if (<= h 2.55e+210)
               t_2
               (* t_0 (* 2.0 (* (/ (/ d D) (* w h)) (* d (/ c0 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 = t_0 * (2.0 * (((c0 * d) / D) * (d / (h * (w * D)))));
	double t_2 = 0.25 * ((D * D) / ((d * d) / (h * (M * M))));
	double tmp;
	if (h <= -2.9e-290) {
		tmp = t_1;
	} else if (h <= 4e-295) {
		tmp = t_2;
	} else if (h <= 1.7e-154) {
		tmp = ((c0 / D) * (c0 / D)) * ((d / w) * ((d / w) / h));
	} else if (h <= 2.8e+35) {
		tmp = 0.0;
	} else if (h <= 1.01e+180) {
		tmp = t_1;
	} else if (h <= 2.55e+210) {
		tmp = t_2;
	} else {
		tmp = t_0 * (2.0 * (((d / D) / (w * h)) * (d * (c0 / 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = c0 / (2.0d0 * w)
    t_1 = t_0 * (2.0d0 * (((c0 * d_1) / d) * (d_1 / (h * (w * d)))))
    t_2 = 0.25d0 * ((d * d) / ((d_1 * d_1) / (h * (m * m))))
    if (h <= (-2.9d-290)) then
        tmp = t_1
    else if (h <= 4d-295) then
        tmp = t_2
    else if (h <= 1.7d-154) then
        tmp = ((c0 / d) * (c0 / d)) * ((d_1 / w) * ((d_1 / w) / h))
    else if (h <= 2.8d+35) then
        tmp = 0.0d0
    else if (h <= 1.01d+180) then
        tmp = t_1
    else if (h <= 2.55d+210) then
        tmp = t_2
    else
        tmp = t_0 * (2.0d0 * (((d_1 / d) / (w * h)) * (d_1 * (c0 / d))))
    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 t_1 = t_0 * (2.0 * (((c0 * d) / D) * (d / (h * (w * D)))));
	double t_2 = 0.25 * ((D * D) / ((d * d) / (h * (M * M))));
	double tmp;
	if (h <= -2.9e-290) {
		tmp = t_1;
	} else if (h <= 4e-295) {
		tmp = t_2;
	} else if (h <= 1.7e-154) {
		tmp = ((c0 / D) * (c0 / D)) * ((d / w) * ((d / w) / h));
	} else if (h <= 2.8e+35) {
		tmp = 0.0;
	} else if (h <= 1.01e+180) {
		tmp = t_1;
	} else if (h <= 2.55e+210) {
		tmp = t_2;
	} else {
		tmp = t_0 * (2.0 * (((d / D) / (w * h)) * (d * (c0 / D))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	t_1 = t_0 * (2.0 * (((c0 * d) / D) * (d / (h * (w * D)))))
	t_2 = 0.25 * ((D * D) / ((d * d) / (h * (M * M))))
	tmp = 0
	if h <= -2.9e-290:
		tmp = t_1
	elif h <= 4e-295:
		tmp = t_2
	elif h <= 1.7e-154:
		tmp = ((c0 / D) * (c0 / D)) * ((d / w) * ((d / w) / h))
	elif h <= 2.8e+35:
		tmp = 0.0
	elif h <= 1.01e+180:
		tmp = t_1
	elif h <= 2.55e+210:
		tmp = t_2
	else:
		tmp = t_0 * (2.0 * (((d / D) / (w * h)) * (d * (c0 / D))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(t_0 * Float64(2.0 * Float64(Float64(Float64(c0 * d) / D) * Float64(d / Float64(h * Float64(w * D))))))
	t_2 = Float64(0.25 * Float64(Float64(D * D) / Float64(Float64(d * d) / Float64(h * Float64(M * M)))))
	tmp = 0.0
	if (h <= -2.9e-290)
		tmp = t_1;
	elseif (h <= 4e-295)
		tmp = t_2;
	elseif (h <= 1.7e-154)
		tmp = Float64(Float64(Float64(c0 / D) * Float64(c0 / D)) * Float64(Float64(d / w) * Float64(Float64(d / w) / h)));
	elseif (h <= 2.8e+35)
		tmp = 0.0;
	elseif (h <= 1.01e+180)
		tmp = t_1;
	elseif (h <= 2.55e+210)
		tmp = t_2;
	else
		tmp = Float64(t_0 * Float64(2.0 * Float64(Float64(Float64(d / D) / Float64(w * h)) * Float64(d * Float64(c0 / D)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	t_1 = t_0 * (2.0 * (((c0 * d) / D) * (d / (h * (w * D)))));
	t_2 = 0.25 * ((D * D) / ((d * d) / (h * (M * M))));
	tmp = 0.0;
	if (h <= -2.9e-290)
		tmp = t_1;
	elseif (h <= 4e-295)
		tmp = t_2;
	elseif (h <= 1.7e-154)
		tmp = ((c0 / D) * (c0 / D)) * ((d / w) * ((d / w) / h));
	elseif (h <= 2.8e+35)
		tmp = 0.0;
	elseif (h <= 1.01e+180)
		tmp = t_1;
	elseif (h <= 2.55e+210)
		tmp = t_2;
	else
		tmp = t_0 * (2.0 * (((d / D) / (w * h)) * (d * (c0 / 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[(t$95$0 * N[(2.0 * N[(N[(N[(c0 * d), $MachinePrecision] / D), $MachinePrecision] * N[(d / N[(h * N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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[h, -2.9e-290], t$95$1, If[LessEqual[h, 4e-295], t$95$2, If[LessEqual[h, 1.7e-154], N[(N[(N[(c0 / D), $MachinePrecision] * N[(c0 / D), $MachinePrecision]), $MachinePrecision] * N[(N[(d / w), $MachinePrecision] * N[(N[(d / w), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 2.8e+35], 0.0, If[LessEqual[h, 1.01e+180], t$95$1, If[LessEqual[h, 2.55e+210], t$95$2, N[(t$95$0 * N[(2.0 * N[(N[(N[(d / D), $MachinePrecision] / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[(d * N[(c0 / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if h < -2.89999999999999994e-290 or 2.79999999999999999e35 < h < 1.01000000000000003e180

   1. Initial program 26.6%

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

   3. Simplified 28.6%

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

   4. Taylor expanded in c0 around inf 35.5%

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

      1. unpow2 35.5%

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

      2. *-commutative 35.5%

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

      3. associate-/l/ 36.2%

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

      4. associate-*r* 40.9%

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

      5. associate-*r/ 43.6%

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

      6. unpow2 43.6%

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

      7. times-frac 51.7%

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

      8. *-commutative 51.7%

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

   6. Simplified 51.7%

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

      1. add-cbrt-cube 46.9%

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

      2. associate-/l/ 46.9%

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

      3. associate-/l/ 46.9%

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

      4. associate-/l/ 46.9%

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

   8. Applied egg-rr 46.9%

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

      1. associate-*l* 46.9%

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

      2. associate-/r* 46.2%

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

      3. associate-/r* 46.2%

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

      4. associate-/r* 46.2%

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

   10. Simplified 46.2%

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

   11. Taylor expanded in d around 0 53.8%

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

       1. associate-*r* 55.4%

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

   13. Simplified 55.4%

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

   14. Taylor expanded in d around 0 53.8%

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

       1. associate-*r* 55.4%

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

       2. *-commutative 55.4%

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

       3. associate-*l* 56.1%

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

   16. Simplified 56.1%

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

   if -2.89999999999999994e-290 < h < 4.00000000000000024e-295 or 1.01000000000000003e180 < h < 2.55e210

   1. Initial program 5.6%

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

   3. Simplified 5.6%

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

      1. flip-+ 0.0%

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

   5. Applied egg-rr 0.0%

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

      1. associate--r- 0.0%

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

      2. +-inverses 61.1%

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

      3. associate-/r* 61.4%

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

      4. associate-/r* 66.9%

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

   7. Simplified 66.9%

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

   8. Taylor expanded in c0 around -inf 61.1%

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

      1. mul-1-neg 61.1%

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

      2. associate-/r* 61.1%

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

      3. distribute-neg-frac 61.1%

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

      4. unpow2 61.1%

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

      5. sub-neg 61.1%

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

      6. mul-1-neg 61.1%

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

      7. distribute-rgt-out 61.1%

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

   10. Simplified 61.1%

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

   11. Taylor expanded in c0 around 0 88.9%

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

       1. associate-/l* 88.8%

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

       2. unpow2 88.8%

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

       3. unpow2 88.8%

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

       4. *-commutative 88.8%

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

       5. unpow2 88.8%

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

   13. Simplified 88.8%

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

   if 4.00000000000000024e-295 < h < 1.6999999999999999e-154

   1. Initial program 30.8%

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

   3. Simplified 33.3%

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

   4. Taylor expanded in c0 around inf 50.4%

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

      1. unpow2 50.4%

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

      2. *-commutative 50.4%

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

      3. associate-/l/ 50.6%

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

      4. associate-*r* 50.6%

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

      5. associate-*r/ 58.9%

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

      6. unpow2 58.9%

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

      7. times-frac 64.7%

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

      8. *-commutative 64.7%

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

   6. Simplified 64.7%

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

   7. Taylor expanded in c0 around 0 34.0%

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

      1. times-frac 35.8%

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

      2. unpow2 35.8%

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

      3. unpow2 35.8%

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

      4. unpow2 35.8%

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

      5. unpow2 35.8%

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

   9. Simplified 35.8%

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

       1. times-frac 47.0%

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

   11. Applied egg-rr 47.0%

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

   12. Taylor expanded in d around 0 47.0%

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

       1. unpow2 47.0%

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

       2. unpow2 47.0%

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

       3. associate-*r* 60.6%

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

       4. times-frac 68.8%

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

       5. *-commutative 68.8%

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

       6. associate-/r* 68.7%

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

   14. Simplified 68.7%

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

   if 1.6999999999999999e-154 < h < 2.79999999999999999e35

   1. Initial program 12.9%

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

   3. Simplified 12.8%

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

   4. Taylor expanded in c0 around -inf 5.5%

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

      1. mul-1-neg 5.5%

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

      2. distribute-rgt-in 3.0%

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

   6. Simplified 48.6%

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

   7. Taylor expanded in c0 around 0 51.4%

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

   if 2.55e210 < h

   1. Initial program 21.3%

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

   3. Simplified 21.1%

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

   4. Taylor expanded in c0 around inf 32.1%

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

      1. unpow2 32.1%

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

      2. *-commutative 32.1%

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

      3. associate-/l/ 32.0%

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

      4. associate-*r* 32.3%

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

      5. associate-*r/ 32.9%

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

      6. unpow2 32.9%

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

      7. times-frac 49.0%

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

      8. *-commutative 49.0%

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

   6. Simplified 49.0%

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

      1. pow1 49.0%

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

      2. *-commutative 49.0%

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

      3. associate-/l/ 43.7%

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

      4. associate-/l* 48.7%

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

   8. Applied egg-rr 48.7%

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

      1. unpow1 48.7%

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

      2. associate-/r* 53.7%

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

      3. associate-/r/ 53.6%

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

   10. Simplified 53.6%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2.9 \cdot 10^{-290}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{h \cdot \left(w \cdot D\right)}\right)\right)\\ \mathbf{elif}\;h \leq 4 \cdot 10^{-295}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\\ \mathbf{elif}\;h \leq 1.7 \cdot 10^{-154}:\\ \;\;\;\;\left(\frac{c0}{D} \cdot \frac{c0}{D}\right) \cdot \left(\frac{d}{w} \cdot \frac{\frac{d}{w}}{h}\right)\\ \mathbf{elif}\;h \leq 2.8 \cdot 10^{+35}:\\ \;\;\;\;0\\ \mathbf{elif}\;h \leq 1.01 \cdot 10^{+180}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{h \cdot \left(w \cdot D\right)}\right)\right)\\ \mathbf{elif}\;h \leq 2.55 \cdot 10^{+210}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\frac{d}{D}}{w \cdot h} \cdot \left(d \cdot \frac{c0}{D}\right)\right)\right)\\ \end{array} \]

Alternative 3: 41.8% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := t_0 \cdot \left(2 \cdot \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{h \cdot \left(w \cdot D\right)}\right)\right)\\ t_2 := 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\\ \mathbf{if}\;h \leq -2.6 \cdot 10^{-290}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;h \leq 2.1 \cdot 10^{-295}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;h \leq 2.6 \cdot 10^{-155}:\\ \;\;\;\;\left(\frac{c0}{D} \cdot \frac{c0}{D}\right) \cdot \left(\frac{d}{w} \cdot \frac{\frac{d}{w}}{h}\right)\\ \mathbf{elif}\;h \leq 2.7 \cdot 10^{+34}:\\ \;\;\;\;0\\ \mathbf{elif}\;h \leq 2.9 \cdot 10^{+179}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;h \leq 1.55 \cdot 10^{+212}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \frac{\frac{c0}{\frac{D}{d}} \cdot \frac{d}{w \cdot h}}{D}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w)))
        (t_1 (* t_0 (* 2.0 (* (/ (* c0 d) D) (/ d (* h (* w D)))))))
        (t_2 (* 0.25 (/ (* D D) (/ (* d d) (* h (* M M)))))))
   (if (<= h -2.6e-290)
     t_1
     (if (<= h 2.1e-295)
       t_2
       (if (<= h 2.6e-155)
         (* (* (/ c0 D) (/ c0 D)) (* (/ d w) (/ (/ d w) h)))
         (if (<= h 2.7e+34)
           0.0
           (if (<= h 2.9e+179)
             t_1
             (if (<= h 1.55e+212)
               t_2
               (* 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 t_1 = t_0 * (2.0 * (((c0 * d) / D) * (d / (h * (w * D)))));
	double t_2 = 0.25 * ((D * D) / ((d * d) / (h * (M * M))));
	double tmp;
	if (h <= -2.6e-290) {
		tmp = t_1;
	} else if (h <= 2.1e-295) {
		tmp = t_2;
	} else if (h <= 2.6e-155) {
		tmp = ((c0 / D) * (c0 / D)) * ((d / w) * ((d / w) / h));
	} else if (h <= 2.7e+34) {
		tmp = 0.0;
	} else if (h <= 2.9e+179) {
		tmp = t_1;
	} else if (h <= 1.55e+212) {
		tmp = t_2;
	} 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = c0 / (2.0d0 * w)
    t_1 = t_0 * (2.0d0 * (((c0 * d_1) / d) * (d_1 / (h * (w * d)))))
    t_2 = 0.25d0 * ((d * d) / ((d_1 * d_1) / (h * (m * m))))
    if (h <= (-2.6d-290)) then
        tmp = t_1
    else if (h <= 2.1d-295) then
        tmp = t_2
    else if (h <= 2.6d-155) then
        tmp = ((c0 / d) * (c0 / d)) * ((d_1 / w) * ((d_1 / w) / h))
    else if (h <= 2.7d+34) then
        tmp = 0.0d0
    else if (h <= 2.9d+179) then
        tmp = t_1
    else if (h <= 1.55d+212) then
        tmp = t_2
    else
        tmp = t_0 * (2.0d0 * (((c0 / (d / d_1)) * (d_1 / (w * h))) / d))
    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 t_1 = t_0 * (2.0 * (((c0 * d) / D) * (d / (h * (w * D)))));
	double t_2 = 0.25 * ((D * D) / ((d * d) / (h * (M * M))));
	double tmp;
	if (h <= -2.6e-290) {
		tmp = t_1;
	} else if (h <= 2.1e-295) {
		tmp = t_2;
	} else if (h <= 2.6e-155) {
		tmp = ((c0 / D) * (c0 / D)) * ((d / w) * ((d / w) / h));
	} else if (h <= 2.7e+34) {
		tmp = 0.0;
	} else if (h <= 2.9e+179) {
		tmp = t_1;
	} else if (h <= 1.55e+212) {
		tmp = t_2;
	} 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)
	t_1 = t_0 * (2.0 * (((c0 * d) / D) * (d / (h * (w * D)))))
	t_2 = 0.25 * ((D * D) / ((d * d) / (h * (M * M))))
	tmp = 0
	if h <= -2.6e-290:
		tmp = t_1
	elif h <= 2.1e-295:
		tmp = t_2
	elif h <= 2.6e-155:
		tmp = ((c0 / D) * (c0 / D)) * ((d / w) * ((d / w) / h))
	elif h <= 2.7e+34:
		tmp = 0.0
	elif h <= 2.9e+179:
		tmp = t_1
	elif h <= 1.55e+212:
		tmp = t_2
	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))
	t_1 = Float64(t_0 * Float64(2.0 * Float64(Float64(Float64(c0 * d) / D) * Float64(d / Float64(h * Float64(w * D))))))
	t_2 = Float64(0.25 * Float64(Float64(D * D) / Float64(Float64(d * d) / Float64(h * Float64(M * M)))))
	tmp = 0.0
	if (h <= -2.6e-290)
		tmp = t_1;
	elseif (h <= 2.1e-295)
		tmp = t_2;
	elseif (h <= 2.6e-155)
		tmp = Float64(Float64(Float64(c0 / D) * Float64(c0 / D)) * Float64(Float64(d / w) * Float64(Float64(d / w) / h)));
	elseif (h <= 2.7e+34)
		tmp = 0.0;
	elseif (h <= 2.9e+179)
		tmp = t_1;
	elseif (h <= 1.55e+212)
		tmp = t_2;
	else
		tmp = Float64(t_0 * Float64(2.0 * Float64(Float64(Float64(c0 / Float64(D / d)) * Float64(d / 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);
	t_1 = t_0 * (2.0 * (((c0 * d) / D) * (d / (h * (w * D)))));
	t_2 = 0.25 * ((D * D) / ((d * d) / (h * (M * M))));
	tmp = 0.0;
	if (h <= -2.6e-290)
		tmp = t_1;
	elseif (h <= 2.1e-295)
		tmp = t_2;
	elseif (h <= 2.6e-155)
		tmp = ((c0 / D) * (c0 / D)) * ((d / w) * ((d / w) / h));
	elseif (h <= 2.7e+34)
		tmp = 0.0;
	elseif (h <= 2.9e+179)
		tmp = t_1;
	elseif (h <= 1.55e+212)
		tmp = t_2;
	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]}, Block[{t$95$1 = N[(t$95$0 * N[(2.0 * N[(N[(N[(c0 * d), $MachinePrecision] / D), $MachinePrecision] * N[(d / N[(h * N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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[h, -2.6e-290], t$95$1, If[LessEqual[h, 2.1e-295], t$95$2, If[LessEqual[h, 2.6e-155], N[(N[(N[(c0 / D), $MachinePrecision] * N[(c0 / D), $MachinePrecision]), $MachinePrecision] * N[(N[(d / w), $MachinePrecision] * N[(N[(d / w), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 2.7e+34], 0.0, If[LessEqual[h, 2.9e+179], t$95$1, If[LessEqual[h, 1.55e+212], t$95$2, N[(t$95$0 * N[(2.0 * N[(N[(N[(c0 / N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(d / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if h < -2.60000000000000001e-290 or 2.7e34 < h < 2.90000000000000019e179

   1. Initial program 26.6%

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

   3. Simplified 28.6%

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

   4. Taylor expanded in c0 around inf 35.5%

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

      1. unpow2 35.5%

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

      2. *-commutative 35.5%

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

      3. associate-/l/ 36.2%

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

      4. associate-*r* 40.9%

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

      5. associate-*r/ 43.6%

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

      6. unpow2 43.6%

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

      7. times-frac 51.7%

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

      8. *-commutative 51.7%

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

   6. Simplified 51.7%

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

      1. add-cbrt-cube 46.9%

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

      2. associate-/l/ 46.9%

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

      3. associate-/l/ 46.9%

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

      4. associate-/l/ 46.9%

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

   8. Applied egg-rr 46.9%

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

      1. associate-*l* 46.9%

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

      2. associate-/r* 46.2%

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

      3. associate-/r* 46.2%

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

      4. associate-/r* 46.2%

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

   10. Simplified 46.2%

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

   11. Taylor expanded in d around 0 53.8%

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

       1. associate-*r* 55.4%

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

   13. Simplified 55.4%

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

   14. Taylor expanded in d around 0 53.8%

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

       1. associate-*r* 55.4%

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

       2. *-commutative 55.4%

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

       3. associate-*l* 56.1%

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

   16. Simplified 56.1%

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

   if -2.60000000000000001e-290 < h < 2.09999999999999993e-295 or 2.90000000000000019e179 < h < 1.54999999999999999e212

   1. Initial program 5.6%

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

   3. Simplified 5.6%

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

      1. flip-+ 0.0%

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

   5. Applied egg-rr 0.0%

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

      1. associate--r- 0.0%

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

      2. +-inverses 61.1%

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

      3. associate-/r* 61.4%

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

      4. associate-/r* 66.9%

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

   7. Simplified 66.9%

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

   8. Taylor expanded in c0 around -inf 61.1%

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

      1. mul-1-neg 61.1%

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

      2. associate-/r* 61.1%

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

      3. distribute-neg-frac 61.1%

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

      4. unpow2 61.1%

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

      5. sub-neg 61.1%

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

      6. mul-1-neg 61.1%

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

      7. distribute-rgt-out 61.1%

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

   10. Simplified 61.1%

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

   11. Taylor expanded in c0 around 0 88.9%

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

       1. associate-/l* 88.8%

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

       2. unpow2 88.8%

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

       3. unpow2 88.8%

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

       4. *-commutative 88.8%

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

       5. unpow2 88.8%

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

   13. Simplified 88.8%

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

   if 2.09999999999999993e-295 < h < 2.60000000000000008e-155

   1. Initial program 30.8%

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

   3. Simplified 33.3%

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

   4. Taylor expanded in c0 around inf 50.4%

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

      1. unpow2 50.4%

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

      2. *-commutative 50.4%

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

      3. associate-/l/ 50.6%

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

      4. associate-*r* 50.6%

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

      5. associate-*r/ 58.9%

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

      6. unpow2 58.9%

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

      7. times-frac 64.7%

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

      8. *-commutative 64.7%

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

   6. Simplified 64.7%

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

   7. Taylor expanded in c0 around 0 34.0%

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

      1. times-frac 35.8%

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

      2. unpow2 35.8%

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

      3. unpow2 35.8%

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

      4. unpow2 35.8%

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

      5. unpow2 35.8%

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

   9. Simplified 35.8%

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

       1. times-frac 47.0%

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

   11. Applied egg-rr 47.0%

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

   12. Taylor expanded in d around 0 47.0%

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

       1. unpow2 47.0%

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

       2. unpow2 47.0%

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

       3. associate-*r* 60.6%

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

       4. times-frac 68.8%

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

       5. *-commutative 68.8%

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

       6. associate-/r* 68.7%

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

   14. Simplified 68.7%

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

   if 2.60000000000000008e-155 < h < 2.7e34

   1. Initial program 12.9%

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

   3. Simplified 12.8%

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

   4. Taylor expanded in c0 around -inf 5.5%

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

      1. mul-1-neg 5.5%

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

      2. distribute-rgt-in 3.0%

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

   6. Simplified 48.6%

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

   7. Taylor expanded in c0 around 0 51.4%

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

   if 1.54999999999999999e212 < h

   1. Initial program 21.3%

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

   3. Simplified 21.1%

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

   4. Taylor expanded in c0 around inf 32.1%

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

      1. unpow2 32.1%

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

      2. *-commutative 32.1%

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

      3. associate-/l/ 32.0%

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

      4. associate-*r* 32.3%

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

      5. associate-*r/ 32.9%

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

      6. unpow2 32.9%

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

      7. times-frac 49.0%

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

      8. *-commutative 49.0%

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

   6. Simplified 49.0%

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

      1. associate-*r/ 48.9%

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

      2. associate-/l* 54.0%

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

   8. Applied egg-rr 54.0%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2.6 \cdot 10^{-290}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{h \cdot \left(w \cdot D\right)}\right)\right)\\ \mathbf{elif}\;h \leq 2.1 \cdot 10^{-295}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\\ \mathbf{elif}\;h \leq 2.6 \cdot 10^{-155}:\\ \;\;\;\;\left(\frac{c0}{D} \cdot \frac{c0}{D}\right) \cdot \left(\frac{d}{w} \cdot \frac{\frac{d}{w}}{h}\right)\\ \mathbf{elif}\;h \leq 2.7 \cdot 10^{+34}:\\ \;\;\;\;0\\ \mathbf{elif}\;h \leq 2.9 \cdot 10^{+179}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{h \cdot \left(w \cdot D\right)}\right)\right)\\ \mathbf{elif}\;h \leq 1.55 \cdot 10^{+212}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{\frac{D}{d}} \cdot \frac{d}{w \cdot h}}{D}\right)\\ \end{array} \]

Alternative 4: 41.9% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\\ t_2 := \frac{c0}{\frac{D}{d}}\\ \mathbf{if}\;h \leq -3.8 \cdot 10^{-290}:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{h \cdot \left(w \cdot D\right)}\right)\right)\\ \mathbf{elif}\;h \leq 3.6 \cdot 10^{-295}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;h \leq 2.05 \cdot 10^{-155}:\\ \;\;\;\;\left(\frac{c0}{D} \cdot \frac{c0}{D}\right) \cdot \left(\frac{d}{w} \cdot \frac{\frac{d}{w}}{h}\right)\\ \mathbf{elif}\;h \leq 2.55 \cdot 10^{+33}:\\ \;\;\;\;0\\ \mathbf{elif}\;h \leq 2.2 \cdot 10^{+186}:\\ \;\;\;\;\frac{c0 \cdot \left(2 \cdot \left(\frac{d}{w \cdot \left(h \cdot D\right)} \cdot t_2\right)\right)}{2 \cdot w}\\ \mathbf{elif}\;h \leq 3.4 \cdot 10^{+212}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \frac{t_2 \cdot \frac{d}{w \cdot h}}{D}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w)))
        (t_1 (* 0.25 (/ (* D D) (/ (* d d) (* h (* M M))))))
        (t_2 (/ c0 (/ D d))))
   (if (<= h -3.8e-290)
     (* t_0 (* 2.0 (* (/ (* c0 d) D) (/ d (* h (* w D))))))
     (if (<= h 3.6e-295)
       t_1
       (if (<= h 2.05e-155)
         (* (* (/ c0 D) (/ c0 D)) (* (/ d w) (/ (/ d w) h)))
         (if (<= h 2.55e+33)
           0.0
           (if (<= h 2.2e+186)
             (/ (* c0 (* 2.0 (* (/ d (* w (* h D))) t_2))) (* 2.0 w))
             (if (<= h 3.4e+212)
               t_1
               (* t_0 (* 2.0 (/ (* t_2 (/ 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 t_1 = 0.25 * ((D * D) / ((d * d) / (h * (M * M))));
	double t_2 = c0 / (D / d);
	double tmp;
	if (h <= -3.8e-290) {
		tmp = t_0 * (2.0 * (((c0 * d) / D) * (d / (h * (w * D)))));
	} else if (h <= 3.6e-295) {
		tmp = t_1;
	} else if (h <= 2.05e-155) {
		tmp = ((c0 / D) * (c0 / D)) * ((d / w) * ((d / w) / h));
	} else if (h <= 2.55e+33) {
		tmp = 0.0;
	} else if (h <= 2.2e+186) {
		tmp = (c0 * (2.0 * ((d / (w * (h * D))) * t_2))) / (2.0 * w);
	} else if (h <= 3.4e+212) {
		tmp = t_1;
	} else {
		tmp = t_0 * (2.0 * ((t_2 * (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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = c0 / (2.0d0 * w)
    t_1 = 0.25d0 * ((d * d) / ((d_1 * d_1) / (h * (m * m))))
    t_2 = c0 / (d / d_1)
    if (h <= (-3.8d-290)) then
        tmp = t_0 * (2.0d0 * (((c0 * d_1) / d) * (d_1 / (h * (w * d)))))
    else if (h <= 3.6d-295) then
        tmp = t_1
    else if (h <= 2.05d-155) then
        tmp = ((c0 / d) * (c0 / d)) * ((d_1 / w) * ((d_1 / w) / h))
    else if (h <= 2.55d+33) then
        tmp = 0.0d0
    else if (h <= 2.2d+186) then
        tmp = (c0 * (2.0d0 * ((d_1 / (w * (h * d))) * t_2))) / (2.0d0 * w)
    else if (h <= 3.4d+212) then
        tmp = t_1
    else
        tmp = t_0 * (2.0d0 * ((t_2 * (d_1 / (w * h))) / d))
    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 t_1 = 0.25 * ((D * D) / ((d * d) / (h * (M * M))));
	double t_2 = c0 / (D / d);
	double tmp;
	if (h <= -3.8e-290) {
		tmp = t_0 * (2.0 * (((c0 * d) / D) * (d / (h * (w * D)))));
	} else if (h <= 3.6e-295) {
		tmp = t_1;
	} else if (h <= 2.05e-155) {
		tmp = ((c0 / D) * (c0 / D)) * ((d / w) * ((d / w) / h));
	} else if (h <= 2.55e+33) {
		tmp = 0.0;
	} else if (h <= 2.2e+186) {
		tmp = (c0 * (2.0 * ((d / (w * (h * D))) * t_2))) / (2.0 * w);
	} else if (h <= 3.4e+212) {
		tmp = t_1;
	} else {
		tmp = t_0 * (2.0 * ((t_2 * (d / (w * h))) / D));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	t_1 = 0.25 * ((D * D) / ((d * d) / (h * (M * M))))
	t_2 = c0 / (D / d)
	tmp = 0
	if h <= -3.8e-290:
		tmp = t_0 * (2.0 * (((c0 * d) / D) * (d / (h * (w * D)))))
	elif h <= 3.6e-295:
		tmp = t_1
	elif h <= 2.05e-155:
		tmp = ((c0 / D) * (c0 / D)) * ((d / w) * ((d / w) / h))
	elif h <= 2.55e+33:
		tmp = 0.0
	elif h <= 2.2e+186:
		tmp = (c0 * (2.0 * ((d / (w * (h * D))) * t_2))) / (2.0 * w)
	elif h <= 3.4e+212:
		tmp = t_1
	else:
		tmp = t_0 * (2.0 * ((t_2 * (d / (w * h))) / D))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(0.25 * Float64(Float64(D * D) / Float64(Float64(d * d) / Float64(h * Float64(M * M)))))
	t_2 = Float64(c0 / Float64(D / d))
	tmp = 0.0
	if (h <= -3.8e-290)
		tmp = Float64(t_0 * Float64(2.0 * Float64(Float64(Float64(c0 * d) / D) * Float64(d / Float64(h * Float64(w * D))))));
	elseif (h <= 3.6e-295)
		tmp = t_1;
	elseif (h <= 2.05e-155)
		tmp = Float64(Float64(Float64(c0 / D) * Float64(c0 / D)) * Float64(Float64(d / w) * Float64(Float64(d / w) / h)));
	elseif (h <= 2.55e+33)
		tmp = 0.0;
	elseif (h <= 2.2e+186)
		tmp = Float64(Float64(c0 * Float64(2.0 * Float64(Float64(d / Float64(w * Float64(h * D))) * t_2))) / Float64(2.0 * w));
	elseif (h <= 3.4e+212)
		tmp = t_1;
	else
		tmp = Float64(t_0 * Float64(2.0 * Float64(Float64(t_2 * Float64(d / 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);
	t_1 = 0.25 * ((D * D) / ((d * d) / (h * (M * M))));
	t_2 = c0 / (D / d);
	tmp = 0.0;
	if (h <= -3.8e-290)
		tmp = t_0 * (2.0 * (((c0 * d) / D) * (d / (h * (w * D)))));
	elseif (h <= 3.6e-295)
		tmp = t_1;
	elseif (h <= 2.05e-155)
		tmp = ((c0 / D) * (c0 / D)) * ((d / w) * ((d / w) / h));
	elseif (h <= 2.55e+33)
		tmp = 0.0;
	elseif (h <= 2.2e+186)
		tmp = (c0 * (2.0 * ((d / (w * (h * D))) * t_2))) / (2.0 * w);
	elseif (h <= 3.4e+212)
		tmp = t_1;
	else
		tmp = t_0 * (2.0 * ((t_2 * (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]}, Block[{t$95$1 = N[(0.25 * N[(N[(D * D), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] / N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(c0 / N[(D / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -3.8e-290], N[(t$95$0 * N[(2.0 * N[(N[(N[(c0 * d), $MachinePrecision] / D), $MachinePrecision] * N[(d / N[(h * N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 3.6e-295], t$95$1, If[LessEqual[h, 2.05e-155], N[(N[(N[(c0 / D), $MachinePrecision] * N[(c0 / D), $MachinePrecision]), $MachinePrecision] * N[(N[(d / w), $MachinePrecision] * N[(N[(d / w), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 2.55e+33], 0.0, If[LessEqual[h, 2.2e+186], N[(N[(c0 * N[(2.0 * N[(N[(d / N[(w * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 3.4e+212], t$95$1, N[(t$95$0 * N[(2.0 * N[(N[(t$95$2 * N[(d / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if h < -3.79999999999999975e-290

   1. Initial program 26.8%

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

   3. Simplified 28.3%

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

   4. Taylor expanded in c0 around inf 34.7%

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

      1. unpow2 34.7%

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

      2. *-commutative 34.7%

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

      3. associate-/l/ 35.5%

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

      4. associate-*r* 40.4%

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

      5. associate-*r/ 42.8%

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

      6. unpow2 42.8%

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

      7. times-frac 52.5%

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

      8. *-commutative 52.5%

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

   6. Simplified 52.5%

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

      1. add-cbrt-cube 47.5%

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

      2. associate-/l/ 47.5%

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

      3. associate-/l/ 47.5%

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

      4. associate-/l/ 47.5%

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

   8. Applied egg-rr 47.5%

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

      1. associate-*l* 47.5%

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

      2. associate-/r* 46.7%

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

      3. associate-/r* 46.7%

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

      4. associate-/r* 46.7%

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

   10. Simplified 46.7%

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

   11. Taylor expanded in d around 0 54.2%

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

       1. associate-*r* 55.0%

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

   13. Simplified 55.0%

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

   14. Taylor expanded in d around 0 54.2%

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

       1. associate-*r* 55.0%

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

       2. *-commutative 55.0%

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

       3. associate-*l* 55.8%

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

   16. Simplified 55.8%

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

   if -3.79999999999999975e-290 < h < 3.6000000000000001e-295 or 2.1999999999999998e186 < h < 3.40000000000000037e212

   1. Initial program 5.9%

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

   3. Simplified 5.9%

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

      1. flip-+ 0.0%

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

   5. Applied egg-rr 0.0%

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

      1. associate--r- 0.0%

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

      2. +-inverses 64.7%

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

      3. associate-/r* 64.7%

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

      4. associate-/r* 70.6%

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

   7. Simplified 70.6%

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

   8. Taylor expanded in c0 around -inf 64.7%

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

      1. mul-1-neg 64.7%

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

      2. associate-/r* 64.7%

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

      3. distribute-neg-frac 64.7%

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

      4. unpow2 64.7%

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

      5. sub-neg 64.7%

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

      6. mul-1-neg 64.7%

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

      7. distribute-rgt-out 64.7%

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

   10. Simplified 64.7%

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

   11. Taylor expanded in c0 around 0 94.1%

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

       1. associate-/l* 94.0%

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

       2. unpow2 94.0%

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

       3. unpow2 94.0%

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

       4. *-commutative 94.0%

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

       5. unpow2 94.0%

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

   13. Simplified 94.0%

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

   if 3.6000000000000001e-295 < h < 2.0499999999999999e-155

   1. Initial program 30.8%

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

   3. Simplified 33.3%

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

   4. Taylor expanded in c0 around inf 50.4%

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

      1. unpow2 50.4%

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

      2. *-commutative 50.4%

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

      3. associate-/l/ 50.6%

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

      4. associate-*r* 50.6%

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

      5. associate-*r/ 58.9%

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

      6. unpow2 58.9%

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

      7. times-frac 64.7%

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

      8. *-commutative 64.7%

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

   6. Simplified 64.7%

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

   7. Taylor expanded in c0 around 0 34.0%

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

      1. times-frac 35.8%

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

      2. unpow2 35.8%

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

      3. unpow2 35.8%

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

      4. unpow2 35.8%

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

      5. unpow2 35.8%

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

   9. Simplified 35.8%

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

       1. times-frac 47.0%

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

   11. Applied egg-rr 47.0%

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

   12. Taylor expanded in d around 0 47.0%

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

       1. unpow2 47.0%

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

       2. unpow2 47.0%

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

       3. associate-*r* 60.6%

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

       4. times-frac 68.8%

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

       5. *-commutative 68.8%

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

       6. associate-/r* 68.7%

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

   14. Simplified 68.7%

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

   if 2.0499999999999999e-155 < h < 2.5499999999999999e33

   1. Initial program 12.9%

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

   3. Simplified 12.8%

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

   4. Taylor expanded in c0 around -inf 5.5%

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

      1. mul-1-neg 5.5%

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

      2. distribute-rgt-in 3.0%

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

   6. Simplified 48.6%

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

   7. Taylor expanded in c0 around 0 51.4%

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

   if 2.5499999999999999e33 < h < 2.1999999999999998e186

   1. Initial program 25.0%

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

   3. Simplified 29.0%

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

   4. Taylor expanded in c0 around inf 38.0%

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

      1. unpow2 38.0%

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

      2. *-commutative 38.0%

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

      3. associate-/l/ 37.9%

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

      4. associate-*r* 41.7%

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

      5. associate-*r/ 45.7%

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

      6. unpow2 45.7%

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

      7. times-frac 45.9%

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

      8. *-commutative 45.9%

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

   6. Simplified 45.9%

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

      1. associate-*l/ 45.9%

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

      2. *-commutative 45.9%

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

      3. associate-/l/ 49.8%

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

      4. associate-/l* 50.0%

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

      5. *-commutative 50.0%

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

   8. Applied egg-rr 50.0%

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

      1. pow1 50.0%

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

   10. Applied egg-rr 50.0%

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

       1. unpow1 50.0%

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

       2. associate-*r* 55.2%

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

   12. Simplified 55.2%

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

   if 3.40000000000000037e212 < h

   1. Initial program 21.3%

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

   3. Simplified 21.1%

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

   4. Taylor expanded in c0 around inf 32.1%

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

      1. unpow2 32.1%

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

      2. *-commutative 32.1%

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

      3. associate-/l/ 32.0%

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

      4. associate-*r* 32.3%

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

      5. associate-*r/ 32.9%

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

      6. unpow2 32.9%

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

      7. times-frac 49.0%

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

      8. *-commutative 49.0%

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

   6. Simplified 49.0%

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

      1. associate-*r/ 48.9%

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

      2. associate-/l* 54.0%

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

   8. Applied egg-rr 54.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{\frac{c0}{\frac{D}{d}} \cdot \frac{d}{h \cdot w}}{D}}\right) \]
3. Recombined 6 regimes into one program.

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -3.8 \cdot 10^{-290}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{h \cdot \left(w \cdot D\right)}\right)\right)\\ \mathbf{elif}\;h \leq 3.6 \cdot 10^{-295}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\\ \mathbf{elif}\;h \leq 2.05 \cdot 10^{-155}:\\ \;\;\;\;\left(\frac{c0}{D} \cdot \frac{c0}{D}\right) \cdot \left(\frac{d}{w} \cdot \frac{\frac{d}{w}}{h}\right)\\ \mathbf{elif}\;h \leq 2.55 \cdot 10^{+33}:\\ \;\;\;\;0\\ \mathbf{elif}\;h \leq 2.2 \cdot 10^{+186}:\\ \;\;\;\;\frac{c0 \cdot \left(2 \cdot \left(\frac{d}{w \cdot \left(h \cdot D\right)} \cdot \frac{c0}{\frac{D}{d}}\right)\right)}{2 \cdot w}\\ \mathbf{elif}\;h \leq 3.4 \cdot 10^{+212}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{\frac{D}{d}} \cdot \frac{d}{w \cdot h}}{D}\right)\\ \end{array} \]

Alternative 5: 41.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{c0}{D} \cdot \frac{c0}{D}\right) \cdot \left(\frac{d}{w} \cdot \frac{\frac{d}{w}}{h}\right)\\ t_1 := \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{h \cdot \left(w \cdot D\right)}\right)\right)\\ t_2 := 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\\ \mathbf{if}\;h \leq -4.8 \cdot 10^{-289}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;h \leq 1.65 \cdot 10^{-294}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;h \leq 3.5 \cdot 10^{-155}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;h \leq 3.9 \cdot 10^{+33}:\\ \;\;\;\;0\\ \mathbf{elif}\;h \leq 2.45 \cdot 10^{+179}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;h \leq 4.6 \cdot 10^{+212}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (* (/ c0 D) (/ c0 D)) (* (/ d w) (/ (/ d w) h))))
        (t_1
         (* (/ c0 (* 2.0 w)) (* 2.0 (* (/ (* c0 d) D) (/ d (* h (* w D)))))))
        (t_2 (* 0.25 (/ (* D D) (/ (* d d) (* h (* M M)))))))
   (if (<= h -4.8e-289)
     t_1
     (if (<= h 1.65e-294)
       t_2
       (if (<= h 3.5e-155)
         t_0
         (if (<= h 3.9e+33)
           0.0
           (if (<= h 2.45e+179) t_1 (if (<= h 4.6e+212) t_2 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 / w) * ((d / w) / h));
	double t_1 = (c0 / (2.0 * w)) * (2.0 * (((c0 * d) / D) * (d / (h * (w * D)))));
	double t_2 = 0.25 * ((D * D) / ((d * d) / (h * (M * M))));
	double tmp;
	if (h <= -4.8e-289) {
		tmp = t_1;
	} else if (h <= 1.65e-294) {
		tmp = t_2;
	} else if (h <= 3.5e-155) {
		tmp = t_0;
	} else if (h <= 3.9e+33) {
		tmp = 0.0;
	} else if (h <= 2.45e+179) {
		tmp = t_1;
	} else if (h <= 4.6e+212) {
		tmp = t_2;
	} 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = ((c0 / d) * (c0 / d)) * ((d_1 / w) * ((d_1 / w) / h))
    t_1 = (c0 / (2.0d0 * w)) * (2.0d0 * (((c0 * d_1) / d) * (d_1 / (h * (w * d)))))
    t_2 = 0.25d0 * ((d * d) / ((d_1 * d_1) / (h * (m * m))))
    if (h <= (-4.8d-289)) then
        tmp = t_1
    else if (h <= 1.65d-294) then
        tmp = t_2
    else if (h <= 3.5d-155) then
        tmp = t_0
    else if (h <= 3.9d+33) then
        tmp = 0.0d0
    else if (h <= 2.45d+179) then
        tmp = t_1
    else if (h <= 4.6d+212) then
        tmp = t_2
    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 / w) * ((d / w) / h));
	double t_1 = (c0 / (2.0 * w)) * (2.0 * (((c0 * d) / D) * (d / (h * (w * D)))));
	double t_2 = 0.25 * ((D * D) / ((d * d) / (h * (M * M))));
	double tmp;
	if (h <= -4.8e-289) {
		tmp = t_1;
	} else if (h <= 1.65e-294) {
		tmp = t_2;
	} else if (h <= 3.5e-155) {
		tmp = t_0;
	} else if (h <= 3.9e+33) {
		tmp = 0.0;
	} else if (h <= 2.45e+179) {
		tmp = t_1;
	} else if (h <= 4.6e+212) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = ((c0 / D) * (c0 / D)) * ((d / w) * ((d / w) / h))
	t_1 = (c0 / (2.0 * w)) * (2.0 * (((c0 * d) / D) * (d / (h * (w * D)))))
	t_2 = 0.25 * ((D * D) / ((d * d) / (h * (M * M))))
	tmp = 0
	if h <= -4.8e-289:
		tmp = t_1
	elif h <= 1.65e-294:
		tmp = t_2
	elif h <= 3.5e-155:
		tmp = t_0
	elif h <= 3.9e+33:
		tmp = 0.0
	elif h <= 2.45e+179:
		tmp = t_1
	elif h <= 4.6e+212:
		tmp = t_2
	else:
		tmp = t_0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(Float64(c0 / D) * Float64(c0 / D)) * Float64(Float64(d / w) * Float64(Float64(d / w) / h)))
	t_1 = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(2.0 * Float64(Float64(Float64(c0 * d) / D) * Float64(d / Float64(h * Float64(w * D))))))
	t_2 = Float64(0.25 * Float64(Float64(D * D) / Float64(Float64(d * d) / Float64(h * Float64(M * M)))))
	tmp = 0.0
	if (h <= -4.8e-289)
		tmp = t_1;
	elseif (h <= 1.65e-294)
		tmp = t_2;
	elseif (h <= 3.5e-155)
		tmp = t_0;
	elseif (h <= 3.9e+33)
		tmp = 0.0;
	elseif (h <= 2.45e+179)
		tmp = t_1;
	elseif (h <= 4.6e+212)
		tmp = t_2;
	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 / w) * ((d / w) / h));
	t_1 = (c0 / (2.0 * w)) * (2.0 * (((c0 * d) / D) * (d / (h * (w * D)))));
	t_2 = 0.25 * ((D * D) / ((d * d) / (h * (M * M))));
	tmp = 0.0;
	if (h <= -4.8e-289)
		tmp = t_1;
	elseif (h <= 1.65e-294)
		tmp = t_2;
	elseif (h <= 3.5e-155)
		tmp = t_0;
	elseif (h <= 3.9e+33)
		tmp = 0.0;
	elseif (h <= 2.45e+179)
		tmp = t_1;
	elseif (h <= 4.6e+212)
		tmp = t_2;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(N[(c0 / D), $MachinePrecision] * N[(c0 / D), $MachinePrecision]), $MachinePrecision] * N[(N[(d / w), $MachinePrecision] * N[(N[(d / w), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[(N[(c0 * d), $MachinePrecision] / D), $MachinePrecision] * N[(d / N[(h * N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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[h, -4.8e-289], t$95$1, If[LessEqual[h, 1.65e-294], t$95$2, If[LessEqual[h, 3.5e-155], t$95$0, If[LessEqual[h, 3.9e+33], 0.0, If[LessEqual[h, 2.45e+179], t$95$1, If[LessEqual[h, 4.6e+212], t$95$2, t$95$0]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if h < -4.79999999999999988e-289 or 3.9000000000000002e33 < h < 2.4499999999999999e179

   1. Initial program 26.6%

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

   3. Simplified 28.6%

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

   4. Taylor expanded in c0 around inf 35.5%

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

      1. unpow2 35.5%

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

      2. *-commutative 35.5%

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

      3. associate-/l/ 36.2%

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

      4. associate-*r* 40.9%

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

      5. associate-*r/ 43.6%

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

      6. unpow2 43.6%

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

      7. times-frac 51.7%

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

      8. *-commutative 51.7%

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

   6. Simplified 51.7%

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

      1. add-cbrt-cube 46.9%

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

      2. associate-/l/ 46.9%

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

      3. associate-/l/ 46.9%

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

      4. associate-/l/ 46.9%

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

   8. Applied egg-rr 46.9%

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

      1. associate-*l* 46.9%

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

      2. associate-/r* 46.2%

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

      3. associate-/r* 46.2%

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

      4. associate-/r* 46.2%

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

   10. Simplified 46.2%

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

   11. Taylor expanded in d around 0 53.8%

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

       1. associate-*r* 55.4%

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

   13. Simplified 55.4%

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

   14. Taylor expanded in d around 0 53.8%

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

       1. associate-*r* 55.4%

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

       2. *-commutative 55.4%

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

       3. associate-*l* 56.1%

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

   16. Simplified 56.1%

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

   if -4.79999999999999988e-289 < h < 1.65e-294 or 2.4499999999999999e179 < h < 4.5999999999999997e212

   1. Initial program 5.6%

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

   3. Simplified 5.6%

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

      1. flip-+ 0.0%

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

   5. Applied egg-rr 0.0%

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

      1. associate--r- 0.0%

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

      2. +-inverses 61.1%

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

      3. associate-/r* 61.4%

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

      4. associate-/r* 66.9%

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

   7. Simplified 66.9%

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

   8. Taylor expanded in c0 around -inf 61.1%

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

      1. mul-1-neg 61.1%

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

      2. associate-/r* 61.1%

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

      3. distribute-neg-frac 61.1%

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

      4. unpow2 61.1%

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

      5. sub-neg 61.1%

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

      6. mul-1-neg 61.1%

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

      7. distribute-rgt-out 61.1%

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

   10. Simplified 61.1%

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

   11. Taylor expanded in c0 around 0 88.9%

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

       1. associate-/l* 88.8%

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

       2. unpow2 88.8%

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

       3. unpow2 88.8%

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

       4. *-commutative 88.8%

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

       5. unpow2 88.8%

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

   13. Simplified 88.8%

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

   if 1.65e-294 < h < 3.50000000000000015e-155 or 4.5999999999999997e212 < h

   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 29.1%

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

   4. Taylor expanded in c0 around inf 44.1%

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

      1. unpow2 44.1%

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

      2. *-commutative 44.1%

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

      3. associate-/l/ 44.2%

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

      4. associate-*r* 44.3%

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

      5. associate-*r/ 49.9%

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

      6. unpow2 49.9%

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

      7. times-frac 59.2%

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

      8. *-commutative 59.2%

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

   6. Simplified 59.2%

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

   7. Taylor expanded in c0 around 0 33.3%

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

      1. times-frac 34.4%

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

      2. unpow2 34.4%

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

      3. unpow2 34.4%

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

      4. unpow2 34.4%

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

      5. unpow2 34.4%

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

   9. Simplified 34.4%

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

       1. times-frac 43.6%

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

   11. Applied egg-rr 43.6%

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

   12. Taylor expanded in d around 0 43.6%

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

       1. unpow2 43.6%

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

       2. unpow2 43.6%

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

       3. associate-*r* 52.5%

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

       4. times-frac 63.5%

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

       5. *-commutative 63.5%

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

       6. associate-/r* 63.5%

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

   14. Simplified 63.5%

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

   if 3.50000000000000015e-155 < h < 3.9000000000000002e33

   1. Initial program 12.9%

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

   3. Simplified 12.8%

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

   4. Taylor expanded in c0 around -inf 5.5%

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

      1. mul-1-neg 5.5%

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

      2. distribute-rgt-in 3.0%

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

   6. Simplified 48.6%

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

   7. Taylor expanded in c0 around 0 51.4%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -4.8 \cdot 10^{-289}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{h \cdot \left(w \cdot D\right)}\right)\right)\\ \mathbf{elif}\;h \leq 1.65 \cdot 10^{-294}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\\ \mathbf{elif}\;h \leq 3.5 \cdot 10^{-155}:\\ \;\;\;\;\left(\frac{c0}{D} \cdot \frac{c0}{D}\right) \cdot \left(\frac{d}{w} \cdot \frac{\frac{d}{w}}{h}\right)\\ \mathbf{elif}\;h \leq 3.9 \cdot 10^{+33}:\\ \;\;\;\;0\\ \mathbf{elif}\;h \leq 2.45 \cdot 10^{+179}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0 \cdot d}{D} \cdot \frac{d}{h \cdot \left(w \cdot D\right)}\right)\right)\\ \mathbf{elif}\;h \leq 4.6 \cdot 10^{+212}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c0}{D} \cdot \frac{c0}{D}\right) \cdot \left(\frac{d}{w} \cdot \frac{\frac{d}{w}}{h}\right)\\ \end{array} \]

Alternative 6: 39.6% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c0 \leq -1.25 \cdot 10^{-107} \lor \neg \left(c0 \leq 1.8 \cdot 10^{-144}\right):\\ \;\;\;\;\left(\frac{c0}{D} \cdot \frac{c0}{D}\right) \cdot \left(\frac{d}{h} \cdot \frac{d}{w \cdot w}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (or (<= c0 -1.25e-107) (not (<= c0 1.8e-144)))
   (* (* (/ c0 D) (/ c0 D)) (* (/ d h) (/ d (* w w))))
   0.0))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((c0 <= -1.25e-107) || !(c0 <= 1.8e-144)) {
		tmp = ((c0 / D) * (c0 / D)) * ((d / h) * (d / (w * w)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((c0 <= (-1.25d-107)) .or. (.not. (c0 <= 1.8d-144))) then
        tmp = ((c0 / d) * (c0 / d)) * ((d_1 / h) * (d_1 / (w * w)))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((c0 <= -1.25e-107) || !(c0 <= 1.8e-144)) {
		tmp = ((c0 / D) * (c0 / D)) * ((d / h) * (d / (w * w)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (c0 <= -1.25e-107) or not (c0 <= 1.8e-144):
		tmp = ((c0 / D) * (c0 / D)) * ((d / h) * (d / (w * w)))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if ((c0 <= -1.25e-107) || !(c0 <= 1.8e-144))
		tmp = Float64(Float64(Float64(c0 / D) * Float64(c0 / D)) * Float64(Float64(d / h) * Float64(d / Float64(w * w))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((c0 <= -1.25e-107) || ~((c0 <= 1.8e-144)))
		tmp = ((c0 / D) * (c0 / D)) * ((d / h) * (d / (w * w)));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[Or[LessEqual[c0, -1.25e-107], N[Not[LessEqual[c0, 1.8e-144]], $MachinePrecision]], N[(N[(N[(c0 / D), $MachinePrecision] * N[(c0 / D), $MachinePrecision]), $MachinePrecision] * N[(N[(d / h), $MachinePrecision] * N[(d / N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if c0 < -1.24999999999999993e-107 or 1.8e-144 < c0

   1. Initial program 23.6%

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

   3. Simplified 24.9%

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

   4. Taylor expanded in c0 around inf 37.9%

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

      1. unpow2 37.9%

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

      2. *-commutative 37.9%

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

      3. associate-/l/ 38.3%

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

      4. associate-*r* 40.1%

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

      5. associate-*r/ 43.6%

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

      6. unpow2 43.6%

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

      7. times-frac 50.4%

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

      8. *-commutative 50.4%

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

   6. Simplified 50.4%

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

   7. Taylor expanded in c0 around 0 32.7%

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

      1. times-frac 33.8%

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

      2. unpow2 33.8%

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

      3. unpow2 33.8%

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

      4. unpow2 33.8%

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

      5. unpow2 33.8%

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

   9. Simplified 33.8%

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

       1. times-frac 37.2%

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

   11. Applied egg-rr 37.2%

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

       1. times-frac 44.4%

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

   13. Applied egg-rr 44.4%

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

   if -1.24999999999999993e-107 < c0 < 1.8e-144

   1. Initial program 21.6%

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

   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 4.2%

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

      1. mul-1-neg 4.2%

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

      2. distribute-rgt-in 4.2%

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

   6. Simplified 54.5%

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

   7. Taylor expanded in c0 around 0 54.5%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -1.25 \cdot 10^{-107} \lor \neg \left(c0 \leq 1.8 \cdot 10^{-144}\right):\\ \;\;\;\;\left(\frac{c0}{D} \cdot \frac{c0}{D}\right) \cdot \left(\frac{d}{h} \cdot \frac{d}{w \cdot w}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7: 43.8% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c0 \leq -4 \cdot 10^{-156} \lor \neg \left(c0 \leq 8.8 \cdot 10^{-145}\right):\\ \;\;\;\;\left(\frac{c0}{D} \cdot \frac{c0}{D}\right) \cdot \left(\frac{d}{w} \cdot \frac{\frac{d}{w}}{h}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (or (<= c0 -4e-156) (not (<= c0 8.8e-145)))
   (* (* (/ c0 D) (/ c0 D)) (* (/ d w) (/ (/ d w) h)))
   0.0))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((c0 <= -4e-156) || !(c0 <= 8.8e-145)) {
		tmp = ((c0 / D) * (c0 / D)) * ((d / w) * ((d / w) / h));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((c0 <= (-4d-156)) .or. (.not. (c0 <= 8.8d-145))) then
        tmp = ((c0 / d) * (c0 / d)) * ((d_1 / w) * ((d_1 / w) / h))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((c0 <= -4e-156) || !(c0 <= 8.8e-145)) {
		tmp = ((c0 / D) * (c0 / D)) * ((d / w) * ((d / w) / h));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (c0 <= -4e-156) or not (c0 <= 8.8e-145):
		tmp = ((c0 / D) * (c0 / D)) * ((d / w) * ((d / w) / h))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if ((c0 <= -4e-156) || !(c0 <= 8.8e-145))
		tmp = Float64(Float64(Float64(c0 / D) * Float64(c0 / D)) * Float64(Float64(d / w) * Float64(Float64(d / w) / h)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((c0 <= -4e-156) || ~((c0 <= 8.8e-145)))
		tmp = ((c0 / D) * (c0 / D)) * ((d / w) * ((d / w) / h));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[Or[LessEqual[c0, -4e-156], N[Not[LessEqual[c0, 8.8e-145]], $MachinePrecision]], N[(N[(N[(c0 / D), $MachinePrecision] * N[(c0 / D), $MachinePrecision]), $MachinePrecision] * N[(N[(d / w), $MachinePrecision] * N[(N[(d / w), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if c0 < -4.00000000000000016e-156 or 8.79999999999999996e-145 < c0

   1. Initial program 24.7%

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

   3. Simplified 26.4%

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

   4. Taylor expanded in c0 around inf 37.6%

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

      1. unpow2 37.6%

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

      2. *-commutative 37.6%

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

      3. associate-/l/ 38.0%

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

      4. associate-*r* 39.7%

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

      5. associate-*r/ 43.4%

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

      6. unpow2 43.4%

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

      7. times-frac 50.2%

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

      8. *-commutative 50.2%

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

   6. Simplified 50.2%

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

   7. Taylor expanded in c0 around 0 31.9%

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

      1. times-frac 33.1%

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

      2. unpow2 33.1%

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

      3. unpow2 33.1%

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

      4. unpow2 33.1%

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

      5. unpow2 33.1%

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

   9. Simplified 33.1%

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

       1. times-frac 36.8%

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

   11. Applied egg-rr 36.8%

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

   12. Taylor expanded in d around 0 36.8%

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

       1. unpow2 36.8%

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

       2. unpow2 36.8%

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

       3. associate-*r* 40.0%

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

       4. times-frac 47.7%

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

       5. *-commutative 47.7%

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

       6. associate-/r* 50.4%

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

   14. Simplified 50.4%

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

   if -4.00000000000000016e-156 < c0 < 8.79999999999999996e-145

   1. Initial program 14.1%

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

   3. Simplified 14.4%

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

   4. Taylor expanded in c0 around -inf 0.6%

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

      1. mul-1-neg 0.6%

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

      2. distribute-rgt-in 0.6%

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

   6. Simplified 61.0%

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

   7. Taylor expanded in c0 around 0 61.0%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -4 \cdot 10^{-156} \lor \neg \left(c0 \leq 8.8 \cdot 10^{-145}\right):\\ \;\;\;\;\left(\frac{c0}{D} \cdot \frac{c0}{D}\right) \cdot \left(\frac{d}{w} \cdot \frac{\frac{d}{w}}{h}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8: 37.6% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \cdot d \leq 1.22 \cdot 10^{+294}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= (* d d) 1.22e+294)
   (* 0.25 (/ (* D D) (/ (* d d) (* h (* M M)))))
   0.0))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((d * d) <= 1.22e+294) {
		tmp = 0.25 * ((D * D) / ((d * d) / (h * (M * M))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((d_1 * d_1) <= 1.22d+294) then
        tmp = 0.25d0 * ((d * d) / ((d_1 * d_1) / (h * (m * m))))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((d * d) <= 1.22e+294) {
		tmp = 0.25 * ((D * D) / ((d * d) / (h * (M * M))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (d * d) <= 1.22e+294:
		tmp = 0.25 * ((D * D) / ((d * d) / (h * (M * M))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (Float64(d * d) <= 1.22e+294)
		tmp = Float64(0.25 * Float64(Float64(D * D) / Float64(Float64(d * d) / Float64(h * Float64(M * M)))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((d * d) <= 1.22e+294)
		tmp = 0.25 * ((D * D) / ((d * d) / (h * (M * M))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[N[(d * d), $MachinePrecision], 1.22e+294], N[(0.25 * N[(N[(D * D), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] / N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 d d) < 1.2199999999999999e294

   1. Initial program 24.0%

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

   3. Simplified 26.3%

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

      1. flip-+ 2.3%

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

   5. Applied egg-rr 3.1%

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

      1. associate--r- 5.2%

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

      2. +-inverses 21.6%

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

      3. associate-/r* 22.3%

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

      4. associate-/r* 23.9%

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

   7. Simplified 23.9%

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

   8. Taylor expanded in c0 around -inf 21.2%

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

      1. mul-1-neg 21.2%

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

      2. associate-/r* 21.8%

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

      3. distribute-neg-frac 21.8%

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

      4. unpow2 21.8%

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

      5. sub-neg 21.8%

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

      6. mul-1-neg 21.8%

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

      7. distribute-rgt-out 21.8%

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

   10. Simplified 27.1%

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

   11. Taylor expanded in c0 around 0 36.2%

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

       1. associate-/l* 37.4%

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

       2. unpow2 37.4%

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

       3. unpow2 37.4%

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

       4. *-commutative 37.4%

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

       5. unpow2 37.4%

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

   13. Simplified 37.4%

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

   if 1.2199999999999999e294 < (*.f64 d d)

   1. Initial program 22.0%

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

   3. Simplified 22.1%

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

   4. Taylor expanded in c0 around -inf 0.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)} \]
   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-rgt-in 0.0%

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

   6. Simplified 31.5%

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

   7. Taylor expanded in c0 around 0 35.7%

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

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \cdot d \leq 1.22 \cdot 10^{+294}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9: 33.6% accurate, 151.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.2%

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

3. Simplified 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 3.5%

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

   1. mul-1-neg 3.5%

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

   2. distribute-rgt-in 3.1%

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

6. Simplified 27.8%

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

7. Taylor expanded in c0 around 0 30.6%

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

8. Final simplification 30.6%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023279 
(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))))))