Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.8% → 72.8%

Time: 25.8s

Alternatives: 11

Speedup: 151.0×

• could not determine a ground truth

  1. c0 = 4.719341322985997e+232
  2. w = -3.138963701496694e-229
  3. h = 5.635924704999175e-189
  4. D = -1.6448620345760636e-295
  5. d = 5.503438724783655e+239
  6. M = -2.092293131782178e+191

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 24.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 72.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\\ t_2 := t_0 \cdot \left(t_1 + \sqrt{t_1 \cdot t_1 - M \cdot M}\right)\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{-242}:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \frac{c0 \cdot \frac{d}{D}}{\left(w \cdot h\right) \cdot \frac{D}{d}}\right)\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(\frac{h \cdot M}{d} \cdot \frac{M}{d}\right)\right)\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{\frac{d}{D}}{\frac{D}{d}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{D \cdot M}{d}\right)}^{2} \cdot \left(h \cdot 0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w)))
        (t_1 (/ (* c0 (* d d)) (* (* D D) (* w h))))
        (t_2 (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))))
   (if (<= t_2 -1e-242)
     (* t_0 (* 2.0 (/ (* c0 (/ d D)) (* (* w h) (/ D d)))))
     (if (<= t_2 0.0)
       (* 0.25 (* (* D D) (* (/ (* h M) d) (/ M d))))
       (if (<= t_2 INFINITY)
         (* t_0 (* 2.0 (* (/ c0 (* w h)) (/ (/ d D) (/ D d)))))
         (* (pow (/ (* D M) d) 2.0) (* h 0.25)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((D * D) * (w * h));
	double t_2 = t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= -1e-242) {
		tmp = t_0 * (2.0 * ((c0 * (d / D)) / ((w * h) * (D / d))));
	} else if (t_2 <= 0.0) {
		tmp = 0.25 * ((D * D) * (((h * M) / d) * (M / d)));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_0 * (2.0 * ((c0 / (w * h)) * ((d / D) / (D / d))));
	} else {
		tmp = pow(((D * M) / d), 2.0) * (h * 0.25);
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((D * D) * (w * h));
	double t_2 = t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= -1e-242) {
		tmp = t_0 * (2.0 * ((c0 * (d / D)) / ((w * h) * (D / d))));
	} else if (t_2 <= 0.0) {
		tmp = 0.25 * ((D * D) * (((h * M) / d) * (M / d)));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * (2.0 * ((c0 / (w * h)) * ((d / D) / (D / d))));
	} else {
		tmp = Math.pow(((D * M) / d), 2.0) * (h * 0.25);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	t_1 = (c0 * (d * d)) / ((D * D) * (w * h))
	t_2 = t_0 * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))
	tmp = 0
	if t_2 <= -1e-242:
		tmp = t_0 * (2.0 * ((c0 * (d / D)) / ((w * h) * (D / d))))
	elif t_2 <= 0.0:
		tmp = 0.25 * ((D * D) * (((h * M) / d) * (M / d)))
	elif t_2 <= math.inf:
		tmp = t_0 * (2.0 * ((c0 / (w * h)) * ((d / D) / (D / d))))
	else:
		tmp = math.pow(((D * M) / d), 2.0) * (h * 0.25)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(D * D) * Float64(w * h)))
	t_2 = Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M)))))
	tmp = 0.0
	if (t_2 <= -1e-242)
		tmp = Float64(t_0 * Float64(2.0 * Float64(Float64(c0 * Float64(d / D)) / Float64(Float64(w * h) * Float64(D / d)))));
	elseif (t_2 <= 0.0)
		tmp = Float64(0.25 * Float64(Float64(D * D) * Float64(Float64(Float64(h * M) / d) * Float64(M / d))));
	elseif (t_2 <= Inf)
		tmp = Float64(t_0 * Float64(2.0 * Float64(Float64(c0 / Float64(w * h)) * Float64(Float64(d / D) / Float64(D / d)))));
	else
		tmp = Float64((Float64(Float64(D * M) / d) ^ 2.0) * Float64(h * 0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	t_1 = (c0 * (d * d)) / ((D * D) * (w * h));
	t_2 = t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	tmp = 0.0;
	if (t_2 <= -1e-242)
		tmp = t_0 * (2.0 * ((c0 * (d / D)) / ((w * h) * (D / d))));
	elseif (t_2 <= 0.0)
		tmp = 0.25 * ((D * D) * (((h * M) / d) * (M / d)));
	elseif (t_2 <= Inf)
		tmp = t_0 * (2.0 * ((c0 / (w * h)) * ((d / D) / (D / d))));
	else
		tmp = (((D * M) / d) ^ 2.0) * (h * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-242], N[(t$95$0 * N[(2.0 * N[(N[(c0 * N[(d / D), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(0.25 * N[(N[(D * D), $MachinePrecision] * N[(N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(t$95$0 * N[(2.0 * N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[(N[(d / D), $MachinePrecision] / N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * N[(h * 0.25), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < -1e-242

   1. Initial program 81.5%

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

   3. Simplified 81.1%

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

   4. Taylor expanded in c0 around inf 84.2%

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

      1. unpow2 84.2%

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

      2. *-commutative 84.2%

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

      3. associate-/l/ 84.2%

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

      4. unpow2 84.2%

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

      5. unpow2 84.2%

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

      6. associate-*l/ 83.9%

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

      7. times-frac 84.0%

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

      8. associate-*l/ 84.0%

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

      9. unpow2 84.0%

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

      10. associate-*l/ 84.0%

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

      11. associate-*r/ 83.9%

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

      12. *-commutative 83.9%

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

      13. associate-*r/ 83.9%

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

      14. unpow2 83.9%

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

   6. Simplified 83.9%

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

      1. associate-/r* 78.3%

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

      2. div-inv 78.3%

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

   8. Applied egg-rr 78.3%

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

      1. pow2 78.3%

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

   10. Applied egg-rr 78.3%

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

       1. un-div-inv 78.3%

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

       2. associate-/r* 83.9%

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

       3. clear-num 83.9%

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

       4. div-inv 83.9%

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

       5. frac-times 86.8%

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

   12. Applied egg-rr 86.8%

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

   if -1e-242 < (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < 0.0

   1. Initial program 65.5%

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

   3. Simplified 63.7%

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

   4. Taylor expanded in c0 around -inf 63.1%

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

      1. fma-def 63.1%

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

      2. distribute-lft1-in 63.1%

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

      3. metadata-eval 63.1%

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

      4. unpow2 63.1%

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

      5. unpow2 63.1%

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

      6. associate-/l* 63.1%

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

      7. unpow2 63.1%

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

      8. unpow2 63.1%

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

      9. *-commutative 63.1%

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

      10. unpow2 63.1%

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

   6. Simplified 63.1%

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

   7. Taylor expanded in c0 around 0 87.3%

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

      1. associate-/l* 87.3%

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

      2. unpow2 87.3%

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

      3. unpow2 87.3%

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

      4. *-commutative 87.3%

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

      5. unpow2 87.3%

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

   9. Simplified 87.3%

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

       1. div-inv 87.1%

          \[\leadsto 0.25 \cdot \color{blue}{\left(\left(D \cdot D\right) \cdot \frac{1}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\right)} \]

   11. Applied egg-rr 87.1%

       \[\leadsto 0.25 \cdot \color{blue}{\left(\left(D \cdot D\right) \cdot \frac{1}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\right)} \]

   12. Taylor expanded in d around 0 87.3%

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

       1. unpow2 87.3%

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

       2. *-commutative 87.3%

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

       3. associate-*r* 87.1%

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

       4. unpow2 87.1%

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

       5. times-frac 99.8%

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

   14. Simplified 99.8%

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

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

   1. Initial program 74.0%

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

   3. Simplified 76.8%

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

   4. Taylor expanded in c0 around inf 74.2%

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

      1. unpow2 74.2%

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

      2. *-commutative 74.2%

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

      3. associate-/l/ 77.1%

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

      4. unpow2 77.1%

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

      5. unpow2 77.1%

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

      6. associate-*l/ 77.1%

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

      7. times-frac 79.9%

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

      8. associate-*l/ 80.0%

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

      9. unpow2 80.0%

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

      10. associate-*l/ 79.9%

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

      11. associate-*r/ 82.8%

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

      12. *-commutative 82.8%

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

      13. associate-*r/ 85.5%

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

      14. unpow2 85.5%

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

   6. Simplified 85.5%

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

      1. pow2 85.5%

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

      2. associate-*r/ 82.8%

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

   8. Applied egg-rr 82.8%

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

      1. *-un-lft-identity 82.8%

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

      2. associate-/l* 85.5%

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

   10. Applied egg-rr 85.5%

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

   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 2.7%

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

   4. Taylor expanded in c0 around -inf 2.8%

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

      1. fma-def 2.8%

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

      2. distribute-lft1-in 2.8%

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

      3. metadata-eval 2.8%

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

      4. unpow2 2.8%

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

      5. unpow2 2.8%

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

      6. associate-/l* 2.8%

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

      7. unpow2 2.8%

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

      8. unpow2 2.8%

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

      9. *-commutative 2.8%

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

      10. unpow2 2.8%

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

   6. Simplified 2.8%

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

   7. Taylor expanded in c0 around 0 42.6%

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

      1. associate-/l* 43.1%

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

      2. unpow2 43.1%

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

      3. unpow2 43.1%

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

      4. *-commutative 43.1%

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

      5. unpow2 43.1%

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

   9. Simplified 43.1%

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

   10. Taylor expanded in D around 0 42.6%

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

       1. unpow2 42.6%

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

       2. *-commutative 42.6%

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

       3. associate-*l/ 41.4%

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

       4. unpow2 41.4%

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

       5. unpow2 41.4%

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

       6. *-commutative 41.4%

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

       7. times-frac 56.3%

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

   12. Simplified 56.3%

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

       1. expm1-log1p-u 52.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.25 \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\right)\right)} \]

       2. expm1-udef 45.6%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.25 \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\right)} - 1} \]

       3. *-commutative 45.6%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot 0.25}\right)} - 1 \]

       4. associate-*l* 46.8%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(h \cdot \left(\left(M \cdot M\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\right)} \cdot 0.25\right)} - 1 \]

       5. pow2 46.8%

          \[\leadsto e^{\mathsf{log1p}\left(\left(h \cdot \left(\color{blue}{{M}^{2}} \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\right) \cdot 0.25\right)} - 1 \]

       6. pow2 46.8%

          \[\leadsto e^{\mathsf{log1p}\left(\left(h \cdot \left({M}^{2} \cdot \color{blue}{{\left(\frac{D}{d}\right)}^{2}}\right)\right) \cdot 0.25\right)} - 1 \]

       7. pow-prod-down 53.3%

          \[\leadsto e^{\mathsf{log1p}\left(\left(h \cdot \color{blue}{{\left(M \cdot \frac{D}{d}\right)}^{2}}\right) \cdot 0.25\right)} - 1 \]

   14. Applied egg-rr 53.3%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(h \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right) \cdot 0.25\right)} - 1} \]
   15. Step-by-step derivation

       1. expm1-def 64.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(h \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right) \cdot 0.25\right)\right)} \]

       2. expm1-log1p 69.9%

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

       3. *-commutative 69.9%

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

       4. associate-*l* 69.9%

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

       5. *-commutative 69.9%

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

       6. associate-*l/ 72.0%

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

   16. Simplified 72.0%

       \[\leadsto \color{blue}{{\left(\frac{D \cdot M}{d}\right)}^{2} \cdot \left(h \cdot 0.25\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq -1 \cdot 10^{-242}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot \frac{d}{D}}{\left(w \cdot h\right) \cdot \frac{D}{d}}\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq 0:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(\frac{h \cdot M}{d} \cdot \frac{M}{d}\right)\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{\frac{d}{D}}{\frac{D}{d}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{D \cdot M}{d}\right)}^{2} \cdot \left(h \cdot 0.25\right)\\ \end{array} \]

Alternative 2: 49.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d}{D} \cdot \frac{d}{D}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \left(h \cdot 0.25\right) \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\\ \mathbf{if}\;d \leq 5.2 \cdot 10^{-109}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq 8200000000000:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \left(\left(\frac{c0}{h} \cdot \frac{1}{w}\right) \cdot t_0\right)\right)\\ \mathbf{elif}\;d \leq 8.8 \cdot 10^{+131}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{+151}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \left(t_0 \cdot \frac{\frac{c0}{h}}{w}\right)\right)\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{+273}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{\frac{d}{D}}{\frac{D}{d}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ d D) (/ d D)))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (* (* h 0.25) (pow (* M (/ D d)) 2.0))))
   (if (<= d 5.2e-109)
     t_2
     (if (<= d 8200000000000.0)
       (* t_1 (* 2.0 (* (* (/ c0 h) (/ 1.0 w)) t_0)))
       (if (<= d 8.8e+131)
         t_2
         (if (<= d 1.35e+151)
           (* t_1 (* 2.0 (* t_0 (/ (/ c0 h) w))))
           (if (<= d 1.4e+273)
             t_2
             (* t_1 (* 2.0 (* (/ c0 (* w h)) (/ (/ d D) (/ D d))))))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (d / D) * (d / D);
	double t_1 = c0 / (2.0 * w);
	double t_2 = (h * 0.25) * pow((M * (D / d)), 2.0);
	double tmp;
	if (d <= 5.2e-109) {
		tmp = t_2;
	} else if (d <= 8200000000000.0) {
		tmp = t_1 * (2.0 * (((c0 / h) * (1.0 / w)) * t_0));
	} else if (d <= 8.8e+131) {
		tmp = t_2;
	} else if (d <= 1.35e+151) {
		tmp = t_1 * (2.0 * (t_0 * ((c0 / h) / w)));
	} else if (d <= 1.4e+273) {
		tmp = t_2;
	} else {
		tmp = t_1 * (2.0 * ((c0 / (w * h)) * ((d / D) / (D / 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 = (d_1 / d) * (d_1 / d)
    t_1 = c0 / (2.0d0 * w)
    t_2 = (h * 0.25d0) * ((m * (d / d_1)) ** 2.0d0)
    if (d_1 <= 5.2d-109) then
        tmp = t_2
    else if (d_1 <= 8200000000000.0d0) then
        tmp = t_1 * (2.0d0 * (((c0 / h) * (1.0d0 / w)) * t_0))
    else if (d_1 <= 8.8d+131) then
        tmp = t_2
    else if (d_1 <= 1.35d+151) then
        tmp = t_1 * (2.0d0 * (t_0 * ((c0 / h) / w)))
    else if (d_1 <= 1.4d+273) then
        tmp = t_2
    else
        tmp = t_1 * (2.0d0 * ((c0 / (w * h)) * ((d_1 / d) / (d / d_1))))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (d / D) * (d / D);
	double t_1 = c0 / (2.0 * w);
	double t_2 = (h * 0.25) * Math.pow((M * (D / d)), 2.0);
	double tmp;
	if (d <= 5.2e-109) {
		tmp = t_2;
	} else if (d <= 8200000000000.0) {
		tmp = t_1 * (2.0 * (((c0 / h) * (1.0 / w)) * t_0));
	} else if (d <= 8.8e+131) {
		tmp = t_2;
	} else if (d <= 1.35e+151) {
		tmp = t_1 * (2.0 * (t_0 * ((c0 / h) / w)));
	} else if (d <= 1.4e+273) {
		tmp = t_2;
	} else {
		tmp = t_1 * (2.0 * ((c0 / (w * h)) * ((d / D) / (D / d))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (d / D) * (d / D)
	t_1 = c0 / (2.0 * w)
	t_2 = (h * 0.25) * math.pow((M * (D / d)), 2.0)
	tmp = 0
	if d <= 5.2e-109:
		tmp = t_2
	elif d <= 8200000000000.0:
		tmp = t_1 * (2.0 * (((c0 / h) * (1.0 / w)) * t_0))
	elif d <= 8.8e+131:
		tmp = t_2
	elif d <= 1.35e+151:
		tmp = t_1 * (2.0 * (t_0 * ((c0 / h) / w)))
	elif d <= 1.4e+273:
		tmp = t_2
	else:
		tmp = t_1 * (2.0 * ((c0 / (w * h)) * ((d / D) / (D / d))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(d / D) * Float64(d / D))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(Float64(h * 0.25) * (Float64(M * Float64(D / d)) ^ 2.0))
	tmp = 0.0
	if (d <= 5.2e-109)
		tmp = t_2;
	elseif (d <= 8200000000000.0)
		tmp = Float64(t_1 * Float64(2.0 * Float64(Float64(Float64(c0 / h) * Float64(1.0 / w)) * t_0)));
	elseif (d <= 8.8e+131)
		tmp = t_2;
	elseif (d <= 1.35e+151)
		tmp = Float64(t_1 * Float64(2.0 * Float64(t_0 * Float64(Float64(c0 / h) / w))));
	elseif (d <= 1.4e+273)
		tmp = t_2;
	else
		tmp = Float64(t_1 * Float64(2.0 * Float64(Float64(c0 / Float64(w * h)) * Float64(Float64(d / D) / Float64(D / d)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (d / D) * (d / D);
	t_1 = c0 / (2.0 * w);
	t_2 = (h * 0.25) * ((M * (D / d)) ^ 2.0);
	tmp = 0.0;
	if (d <= 5.2e-109)
		tmp = t_2;
	elseif (d <= 8200000000000.0)
		tmp = t_1 * (2.0 * (((c0 / h) * (1.0 / w)) * t_0));
	elseif (d <= 8.8e+131)
		tmp = t_2;
	elseif (d <= 1.35e+151)
		tmp = t_1 * (2.0 * (t_0 * ((c0 / h) / w)));
	elseif (d <= 1.4e+273)
		tmp = t_2;
	else
		tmp = t_1 * (2.0 * ((c0 / (w * h)) * ((d / D) / (D / d))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(h * 0.25), $MachinePrecision] * N[Power[N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, 5.2e-109], t$95$2, If[LessEqual[d, 8200000000000.0], N[(t$95$1 * N[(2.0 * N[(N[(N[(c0 / h), $MachinePrecision] * N[(1.0 / w), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 8.8e+131], t$95$2, If[LessEqual[d, 1.35e+151], N[(t$95$1 * N[(2.0 * N[(t$95$0 * N[(N[(c0 / h), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.4e+273], t$95$2, N[(t$95$1 * N[(2.0 * N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[(N[(d / D), $MachinePrecision] / N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < 5.1999999999999997e-109 or 8.2e12 < d < 8.7999999999999995e131 or 1.3500000000000001e151 < d < 1.40000000000000009e273

   1. Initial program 21.0%

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

   3. Simplified 22.6%

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

   4. Taylor expanded in c0 around -inf 5.1%

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

      1. fma-def 5.1%

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

      2. distribute-lft1-in 5.1%

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

      3. metadata-eval 5.1%

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

      4. unpow2 5.1%

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

      5. unpow2 5.1%

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

      6. associate-/l* 5.1%

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

      7. unpow2 5.1%

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

      8. unpow2 5.1%

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

      9. *-commutative 5.1%

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

      10. unpow2 5.1%

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

   6. Simplified 5.1%

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

   7. Taylor expanded in c0 around 0 36.3%

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

      1. associate-/l* 35.5%

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

      2. unpow2 35.5%

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

      3. unpow2 35.5%

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

      4. *-commutative 35.5%

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

      5. unpow2 35.5%

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

   9. Simplified 35.5%

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

   10. Taylor expanded in D around 0 36.3%

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

       1. unpow2 36.3%

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

       2. *-commutative 36.3%

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

       3. associate-*l/ 34.4%

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

       4. unpow2 34.4%

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

       5. unpow2 34.4%

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

       6. *-commutative 34.4%

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

       7. times-frac 45.9%

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

   12. Simplified 45.9%

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

       1. expm1-log1p-u 42.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.25 \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\right)\right)} \]

       2. expm1-udef 36.6%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.25 \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\right)} - 1} \]

       3. *-commutative 36.6%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot 0.25}\right)} - 1 \]

       4. associate-*l* 37.6%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(h \cdot \left(\left(M \cdot M\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\right)} \cdot 0.25\right)} - 1 \]

       5. pow2 37.6%

          \[\leadsto e^{\mathsf{log1p}\left(\left(h \cdot \left(\color{blue}{{M}^{2}} \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\right) \cdot 0.25\right)} - 1 \]

       6. pow2 37.6%

          \[\leadsto e^{\mathsf{log1p}\left(\left(h \cdot \left({M}^{2} \cdot \color{blue}{{\left(\frac{D}{d}\right)}^{2}}\right)\right) \cdot 0.25\right)} - 1 \]

       7. pow-prod-down 42.4%

          \[\leadsto e^{\mathsf{log1p}\left(\left(h \cdot \color{blue}{{\left(M \cdot \frac{D}{d}\right)}^{2}}\right) \cdot 0.25\right)} - 1 \]

   14. Applied egg-rr 42.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(h \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right) \cdot 0.25\right)} - 1} \]
   15. Step-by-step derivation

       1. expm1-def 52.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(h \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right) \cdot 0.25\right)\right)} \]

       2. expm1-log1p 56.9%

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

       3. *-commutative 56.9%

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

       4. associate-*l* 56.9%

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

   16. Simplified 56.9%

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

   if 5.1999999999999997e-109 < d < 8.2e12

   1. Initial program 45.5%

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

   3. Simplified 55.8%

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

   4. Taylor expanded in c0 around inf 51.2%

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

      1. unpow2 51.2%

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

      2. *-commutative 51.2%

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

      3. associate-/l/ 56.3%

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

      4. unpow2 56.3%

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

      5. unpow2 56.3%

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

      6. associate-*l/ 61.5%

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

      7. times-frac 61.6%

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

      8. associate-*l/ 61.6%

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

      9. unpow2 61.6%

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

      10. associate-*l/ 61.6%

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

      11. associate-*r/ 61.5%

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

      12. *-commutative 61.5%

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

      13. associate-*r/ 61.4%

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

      14. unpow2 61.4%

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

   6. Simplified 61.4%

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

      1. associate-/r* 66.4%

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

      2. div-inv 66.3%

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

   8. Applied egg-rr 66.3%

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

      1. pow2 66.3%

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

   10. Applied egg-rr 66.3%

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

   if 8.7999999999999995e131 < d < 1.3500000000000001e151

   1. Initial program 50.0%

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

   3. Simplified 50.0%

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

   4. Taylor expanded in c0 around inf 100.0%

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

      1. unpow2 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-/l/ 100.0%

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

      4. unpow2 100.0%

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

      5. unpow2 100.0%

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

      6. associate-*l/ 100.0%

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

      7. times-frac 100.0%

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

      8. associate-*l/ 100.0%

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

      9. unpow2 100.0%

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

      10. associate-*l/ 100.0%

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

      11. associate-*r/ 100.0%

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

      12. *-commutative 100.0%

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

      13. associate-*r/ 100.0%

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

      14. unpow2 100.0%

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

   6. Simplified 100.0%

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

      1. associate-/r* 100.0%

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

      2. div-inv 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. pow2 100.0%

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

   10. Applied egg-rr 100.0%

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

       1. un-div-inv 100.0%

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

   12. Applied egg-rr 100.0%

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

   if 1.40000000000000009e273 < d

   1. Initial program 18.2%

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

   3. Simplified 18.2%

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

   4. Taylor expanded in c0 around inf 55.0%

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

      1. unpow2 55.0%

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

      2. *-commutative 55.0%

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

      3. associate-/l/ 55.0%

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

      4. unpow2 55.0%

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

      5. unpow2 55.0%

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

      6. associate-*l/ 55.0%

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

      7. times-frac 56.1%

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

      8. associate-*l/ 56.1%

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

      9. unpow2 56.1%

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

      10. associate-*l/ 56.1%

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

      11. associate-*r/ 64.2%

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

      12. *-commutative 64.2%

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

      13. associate-*r/ 64.2%

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

      14. unpow2 64.2%

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

   6. Simplified 64.2%

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

      1. pow2 64.2%

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

      2. associate-*r/ 64.2%

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

   8. Applied egg-rr 64.2%

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

      1. *-un-lft-identity 64.2%

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

      2. associate-/l* 64.2%

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

   10. Applied egg-rr 64.2%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{h \cdot w} \cdot \color{blue}{\left(1 \cdot \frac{\frac{d}{D}}{\frac{D}{d}}\right)}\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 5.2 \cdot 10^{-109}:\\ \;\;\;\;\left(h \cdot 0.25\right) \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\\ \mathbf{elif}\;d \leq 8200000000000:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\left(\frac{c0}{h} \cdot \frac{1}{w}\right) \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\right)\\ \mathbf{elif}\;d \leq 8.8 \cdot 10^{+131}:\\ \;\;\;\;\left(h \cdot 0.25\right) \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{+151}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right)\right)\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{+273}:\\ \;\;\;\;\left(h \cdot 0.25\right) \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{\frac{d}{D}}{\frac{D}{d}}\right)\right)\\ \end{array} \]

Alternative 3: 45.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d}{D} \cdot \frac{d}{D}\\ t_1 := \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w \cdot h} \cdot t_0\right)\right)\\ t_2 := h \cdot \left(M \cdot M\right)\\ \mathbf{if}\;M \cdot M \leq 4 \cdot 10^{-221}:\\ \;\;\;\;0.25 \cdot \frac{t_2}{t_0}\\ \mathbf{elif}\;M \cdot M \leq 5 \cdot 10^{-165}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;M \cdot M \leq 10^{-48}:\\ \;\;\;\;0.25 \cdot \left(t_2 \cdot \frac{D \cdot \frac{D}{d}}{d}\right)\\ \mathbf{elif}\;M \cdot M \leq 500 \lor \neg \left(M \cdot M \leq 2 \cdot 10^{+144}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(\frac{h \cdot M}{d} \cdot \frac{M}{d}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ d D) (/ d D)))
        (t_1 (* (/ c0 (* 2.0 w)) (* 2.0 (* (/ c0 (* w h)) t_0))))
        (t_2 (* h (* M M))))
   (if (<= (* M M) 4e-221)
     (* 0.25 (/ t_2 t_0))
     (if (<= (* M M) 5e-165)
       t_1
       (if (<= (* M M) 1e-48)
         (* 0.25 (* t_2 (/ (* D (/ D d)) d)))
         (if (or (<= (* M M) 500.0) (not (<= (* M M) 2e+144)))
           t_1
           (* 0.25 (* (* D D) (* (/ (* h M) d) (/ M d))))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (d / D) * (d / D);
	double t_1 = (c0 / (2.0 * w)) * (2.0 * ((c0 / (w * h)) * t_0));
	double t_2 = h * (M * M);
	double tmp;
	if ((M * M) <= 4e-221) {
		tmp = 0.25 * (t_2 / t_0);
	} else if ((M * M) <= 5e-165) {
		tmp = t_1;
	} else if ((M * M) <= 1e-48) {
		tmp = 0.25 * (t_2 * ((D * (D / d)) / d));
	} else if (((M * M) <= 500.0) || !((M * M) <= 2e+144)) {
		tmp = t_1;
	} else {
		tmp = 0.25 * ((D * D) * (((h * M) / d) * (M / 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 = (d_1 / d) * (d_1 / d)
    t_1 = (c0 / (2.0d0 * w)) * (2.0d0 * ((c0 / (w * h)) * t_0))
    t_2 = h * (m * m)
    if ((m * m) <= 4d-221) then
        tmp = 0.25d0 * (t_2 / t_0)
    else if ((m * m) <= 5d-165) then
        tmp = t_1
    else if ((m * m) <= 1d-48) then
        tmp = 0.25d0 * (t_2 * ((d * (d / d_1)) / d_1))
    else if (((m * m) <= 500.0d0) .or. (.not. ((m * m) <= 2d+144))) then
        tmp = t_1
    else
        tmp = 0.25d0 * ((d * d) * (((h * m) / d_1) * (m / d_1)))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (d / D) * (d / D);
	double t_1 = (c0 / (2.0 * w)) * (2.0 * ((c0 / (w * h)) * t_0));
	double t_2 = h * (M * M);
	double tmp;
	if ((M * M) <= 4e-221) {
		tmp = 0.25 * (t_2 / t_0);
	} else if ((M * M) <= 5e-165) {
		tmp = t_1;
	} else if ((M * M) <= 1e-48) {
		tmp = 0.25 * (t_2 * ((D * (D / d)) / d));
	} else if (((M * M) <= 500.0) || !((M * M) <= 2e+144)) {
		tmp = t_1;
	} else {
		tmp = 0.25 * ((D * D) * (((h * M) / d) * (M / d)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (d / D) * (d / D)
	t_1 = (c0 / (2.0 * w)) * (2.0 * ((c0 / (w * h)) * t_0))
	t_2 = h * (M * M)
	tmp = 0
	if (M * M) <= 4e-221:
		tmp = 0.25 * (t_2 / t_0)
	elif (M * M) <= 5e-165:
		tmp = t_1
	elif (M * M) <= 1e-48:
		tmp = 0.25 * (t_2 * ((D * (D / d)) / d))
	elif ((M * M) <= 500.0) or not ((M * M) <= 2e+144):
		tmp = t_1
	else:
		tmp = 0.25 * ((D * D) * (((h * M) / d) * (M / d)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(d / D) * Float64(d / D))
	t_1 = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(2.0 * Float64(Float64(c0 / Float64(w * h)) * t_0)))
	t_2 = Float64(h * Float64(M * M))
	tmp = 0.0
	if (Float64(M * M) <= 4e-221)
		tmp = Float64(0.25 * Float64(t_2 / t_0));
	elseif (Float64(M * M) <= 5e-165)
		tmp = t_1;
	elseif (Float64(M * M) <= 1e-48)
		tmp = Float64(0.25 * Float64(t_2 * Float64(Float64(D * Float64(D / d)) / d)));
	elseif ((Float64(M * M) <= 500.0) || !(Float64(M * M) <= 2e+144))
		tmp = t_1;
	else
		tmp = Float64(0.25 * Float64(Float64(D * D) * Float64(Float64(Float64(h * M) / d) * Float64(M / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (d / D) * (d / D);
	t_1 = (c0 / (2.0 * w)) * (2.0 * ((c0 / (w * h)) * t_0));
	t_2 = h * (M * M);
	tmp = 0.0;
	if ((M * M) <= 4e-221)
		tmp = 0.25 * (t_2 / t_0);
	elseif ((M * M) <= 5e-165)
		tmp = t_1;
	elseif ((M * M) <= 1e-48)
		tmp = 0.25 * (t_2 * ((D * (D / d)) / d));
	elseif (((M * M) <= 500.0) || ~(((M * M) <= 2e+144)))
		tmp = t_1;
	else
		tmp = 0.25 * ((D * D) * (((h * M) / d) * (M / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(M * M), $MachinePrecision], 4e-221], N[(0.25 * N[(t$95$2 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(M * M), $MachinePrecision], 5e-165], t$95$1, If[LessEqual[N[(M * M), $MachinePrecision], 1e-48], N[(0.25 * N[(t$95$2 * N[(N[(D * N[(D / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(M * M), $MachinePrecision], 500.0], N[Not[LessEqual[N[(M * M), $MachinePrecision], 2e+144]], $MachinePrecision]], t$95$1, N[(0.25 * N[(N[(D * D), $MachinePrecision] * N[(N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 M M) < 4.00000000000000007e-221

   1. Initial program 27.7%

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

   3. Simplified 29.8%

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

   4. Taylor expanded in c0 around -inf 7.5%

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

      1. fma-def 7.5%

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

      2. distribute-lft1-in 7.5%

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

      3. metadata-eval 7.5%

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

      4. unpow2 7.5%

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

      5. unpow2 7.5%

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

      6. associate-/l* 7.5%

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

      7. unpow2 7.5%

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

      8. unpow2 7.5%

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

      9. *-commutative 7.5%

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

      10. unpow2 7.5%

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

   6. Simplified 7.5%

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

   7. Taylor expanded in c0 around 0 42.6%

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

      1. associate-/l* 42.6%

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

      2. unpow2 42.6%

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

      3. unpow2 42.6%

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

      4. *-commutative 42.6%

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

      5. unpow2 42.6%

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

   9. Simplified 42.6%

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

   10. Taylor expanded in D around 0 42.6%

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

       1. *-commutative 42.6%

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

       2. unpow2 42.6%

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

       3. *-commutative 42.6%

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

       4. associate-/l* 41.6%

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

       5. unpow2 41.6%

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

       6. unpow2 41.6%

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

       7. times-frac 58.7%

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

       8. unpow2 58.7%

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

   12. Simplified 58.7%

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

       1. pow2 39.2%

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

   14. Applied egg-rr 58.7%

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

   if 4.00000000000000007e-221 < (*.f64 M M) < 4.99999999999999981e-165 or 9.9999999999999997e-49 < (*.f64 M M) < 500 or 2.00000000000000005e144 < (*.f64 M M)

   1. Initial program 22.2%

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

   3. Simplified 24.9%

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

   4. Taylor expanded in c0 around inf 45.9%

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

      1. unpow2 45.9%

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

      2. *-commutative 45.9%

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

      3. associate-/l/ 47.0%

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

      4. unpow2 47.0%

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

      5. unpow2 47.0%

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

      6. associate-*l/ 48.6%

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

      7. times-frac 55.0%

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

      8. associate-*l/ 55.0%

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

      9. unpow2 55.0%

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

      10. associate-*l/ 59.2%

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

      11. associate-*r/ 59.0%

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

      12. *-commutative 59.0%

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

      13. associate-*r/ 61.0%

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

      14. unpow2 61.0%

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

   6. Simplified 61.0%

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

      1. pow2 60.3%

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

   8. Applied egg-rr 61.0%

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

   if 4.99999999999999981e-165 < (*.f64 M M) < 9.9999999999999997e-49

   1. Initial program 20.4%

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

   3. Simplified 20.7%

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

   4. Taylor expanded in c0 around -inf 0.3%

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

      1. fma-def 0.3%

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

      2. distribute-lft1-in 0.3%

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

      3. metadata-eval 0.3%

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

      4. unpow2 0.3%

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

      5. unpow2 0.3%

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

      6. associate-/l* 0.3%

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

      7. unpow2 0.3%

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

      8. unpow2 0.3%

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

      9. *-commutative 0.3%

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

      10. unpow2 0.3%

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

   6. Simplified 0.3%

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

   7. Taylor expanded in c0 around 0 48.1%

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

      1. associate-/l* 48.3%

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

      2. unpow2 48.3%

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

      3. unpow2 48.3%

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

      4. *-commutative 48.3%

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

      5. unpow2 48.3%

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

   9. Simplified 48.3%

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

   10. Taylor expanded in D around 0 48.1%

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

       1. unpow2 48.1%

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

       2. *-commutative 48.1%

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

       3. associate-*l/ 51.6%

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

       4. unpow2 51.6%

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

       5. unpow2 51.6%

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

       6. *-commutative 51.6%

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

       7. times-frac 54.7%

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

   12. Simplified 54.7%

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

       1. associate-*r/ 54.7%

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

   14. Applied egg-rr 54.7%

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

   if 500 < (*.f64 M M) < 2.00000000000000005e144

   1. Initial program 14.9%

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

   3. Simplified 17.5%

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

   4. Taylor expanded in c0 around -inf 8.9%

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

      1. fma-def 8.9%

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

      2. distribute-lft1-in 8.9%

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

      3. metadata-eval 8.9%

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

      4. unpow2 8.9%

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

      5. unpow2 8.9%

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

      6. associate-/l* 8.8%

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

      7. unpow2 8.8%

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

      8. unpow2 8.8%

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

      9. *-commutative 8.8%

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

      10. unpow2 8.8%

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

   6. Simplified 8.8%

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

   7. Taylor expanded in c0 around 0 47.5%

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

      1. associate-/l* 50.0%

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

      2. unpow2 50.0%

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

      3. unpow2 50.0%

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

      4. *-commutative 50.0%

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

      5. unpow2 50.0%

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

   9. Simplified 50.0%

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

       1. div-inv 49.9%

          \[\leadsto 0.25 \cdot \color{blue}{\left(\left(D \cdot D\right) \cdot \frac{1}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\right)} \]

   11. Applied egg-rr 49.9%

       \[\leadsto 0.25 \cdot \color{blue}{\left(\left(D \cdot D\right) \cdot \frac{1}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\right)} \]

   12. Taylor expanded in d around 0 50.0%

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

       1. unpow2 50.0%

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

       2. *-commutative 50.0%

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

       3. associate-*r* 49.9%

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

       4. unpow2 49.9%

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

       5. times-frac 56.2%

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

   14. Simplified 56.2%

       \[\leadsto 0.25 \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(\frac{h \cdot M}{d} \cdot \frac{M}{d}\right)}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot M \leq 4 \cdot 10^{-221}:\\ \;\;\;\;0.25 \cdot \frac{h \cdot \left(M \cdot M\right)}{\frac{d}{D} \cdot \frac{d}{D}}\\ \mathbf{elif}\;M \cdot M \leq 5 \cdot 10^{-165}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\right)\\ \mathbf{elif}\;M \cdot M \leq 10^{-48}:\\ \;\;\;\;0.25 \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \frac{D \cdot \frac{D}{d}}{d}\right)\\ \mathbf{elif}\;M \cdot M \leq 500 \lor \neg \left(M \cdot M \leq 2 \cdot 10^{+144}\right):\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(\frac{h \cdot M}{d} \cdot \frac{M}{d}\right)\right)\\ \end{array} \]

Alternative 4: 45.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d}{D} \cdot \frac{d}{D}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := t_1 \cdot \left(2 \cdot \left(\frac{c0}{w \cdot h} \cdot t_0\right)\right)\\ t_3 := h \cdot \left(M \cdot M\right)\\ \mathbf{if}\;M \cdot M \leq 4 \cdot 10^{-221}:\\ \;\;\;\;0.25 \cdot \frac{t_3}{t_0}\\ \mathbf{elif}\;M \cdot M \leq 5 \cdot 10^{-165}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;M \cdot M \leq 10^{-48}:\\ \;\;\;\;0.25 \cdot \left(t_3 \cdot \frac{D \cdot \frac{D}{d}}{d}\right)\\ \mathbf{elif}\;M \cdot M \leq 500:\\ \;\;\;\;t_2\\ \mathbf{elif}\;M \cdot M \leq 2 \cdot 10^{+144}:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(\frac{h \cdot M}{d} \cdot \frac{M}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \left(t_0 \cdot \frac{\frac{c0}{h}}{w}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ d D) (/ d D)))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (* t_1 (* 2.0 (* (/ c0 (* w h)) t_0))))
        (t_3 (* h (* M M))))
   (if (<= (* M M) 4e-221)
     (* 0.25 (/ t_3 t_0))
     (if (<= (* M M) 5e-165)
       t_2
       (if (<= (* M M) 1e-48)
         (* 0.25 (* t_3 (/ (* D (/ D d)) d)))
         (if (<= (* M M) 500.0)
           t_2
           (if (<= (* M M) 2e+144)
             (* 0.25 (* (* D D) (* (/ (* h M) d) (/ M d))))
             (* t_1 (* 2.0 (* t_0 (/ (/ c0 h) w)))))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (d / D) * (d / D);
	double t_1 = c0 / (2.0 * w);
	double t_2 = t_1 * (2.0 * ((c0 / (w * h)) * t_0));
	double t_3 = h * (M * M);
	double tmp;
	if ((M * M) <= 4e-221) {
		tmp = 0.25 * (t_3 / t_0);
	} else if ((M * M) <= 5e-165) {
		tmp = t_2;
	} else if ((M * M) <= 1e-48) {
		tmp = 0.25 * (t_3 * ((D * (D / d)) / d));
	} else if ((M * M) <= 500.0) {
		tmp = t_2;
	} else if ((M * M) <= 2e+144) {
		tmp = 0.25 * ((D * D) * (((h * M) / d) * (M / d)));
	} else {
		tmp = t_1 * (2.0 * (t_0 * ((c0 / h) / w)));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (d_1 / d) * (d_1 / d)
    t_1 = c0 / (2.0d0 * w)
    t_2 = t_1 * (2.0d0 * ((c0 / (w * h)) * t_0))
    t_3 = h * (m * m)
    if ((m * m) <= 4d-221) then
        tmp = 0.25d0 * (t_3 / t_0)
    else if ((m * m) <= 5d-165) then
        tmp = t_2
    else if ((m * m) <= 1d-48) then
        tmp = 0.25d0 * (t_3 * ((d * (d / d_1)) / d_1))
    else if ((m * m) <= 500.0d0) then
        tmp = t_2
    else if ((m * m) <= 2d+144) then
        tmp = 0.25d0 * ((d * d) * (((h * m) / d_1) * (m / d_1)))
    else
        tmp = t_1 * (2.0d0 * (t_0 * ((c0 / h) / w)))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (d / D) * (d / D);
	double t_1 = c0 / (2.0 * w);
	double t_2 = t_1 * (2.0 * ((c0 / (w * h)) * t_0));
	double t_3 = h * (M * M);
	double tmp;
	if ((M * M) <= 4e-221) {
		tmp = 0.25 * (t_3 / t_0);
	} else if ((M * M) <= 5e-165) {
		tmp = t_2;
	} else if ((M * M) <= 1e-48) {
		tmp = 0.25 * (t_3 * ((D * (D / d)) / d));
	} else if ((M * M) <= 500.0) {
		tmp = t_2;
	} else if ((M * M) <= 2e+144) {
		tmp = 0.25 * ((D * D) * (((h * M) / d) * (M / d)));
	} else {
		tmp = t_1 * (2.0 * (t_0 * ((c0 / h) / w)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (d / D) * (d / D)
	t_1 = c0 / (2.0 * w)
	t_2 = t_1 * (2.0 * ((c0 / (w * h)) * t_0))
	t_3 = h * (M * M)
	tmp = 0
	if (M * M) <= 4e-221:
		tmp = 0.25 * (t_3 / t_0)
	elif (M * M) <= 5e-165:
		tmp = t_2
	elif (M * M) <= 1e-48:
		tmp = 0.25 * (t_3 * ((D * (D / d)) / d))
	elif (M * M) <= 500.0:
		tmp = t_2
	elif (M * M) <= 2e+144:
		tmp = 0.25 * ((D * D) * (((h * M) / d) * (M / d)))
	else:
		tmp = t_1 * (2.0 * (t_0 * ((c0 / h) / w)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(d / D) * Float64(d / D))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(t_1 * Float64(2.0 * Float64(Float64(c0 / Float64(w * h)) * t_0)))
	t_3 = Float64(h * Float64(M * M))
	tmp = 0.0
	if (Float64(M * M) <= 4e-221)
		tmp = Float64(0.25 * Float64(t_3 / t_0));
	elseif (Float64(M * M) <= 5e-165)
		tmp = t_2;
	elseif (Float64(M * M) <= 1e-48)
		tmp = Float64(0.25 * Float64(t_3 * Float64(Float64(D * Float64(D / d)) / d)));
	elseif (Float64(M * M) <= 500.0)
		tmp = t_2;
	elseif (Float64(M * M) <= 2e+144)
		tmp = Float64(0.25 * Float64(Float64(D * D) * Float64(Float64(Float64(h * M) / d) * Float64(M / d))));
	else
		tmp = Float64(t_1 * Float64(2.0 * Float64(t_0 * Float64(Float64(c0 / h) / w))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (d / D) * (d / D);
	t_1 = c0 / (2.0 * w);
	t_2 = t_1 * (2.0 * ((c0 / (w * h)) * t_0));
	t_3 = h * (M * M);
	tmp = 0.0;
	if ((M * M) <= 4e-221)
		tmp = 0.25 * (t_3 / t_0);
	elseif ((M * M) <= 5e-165)
		tmp = t_2;
	elseif ((M * M) <= 1e-48)
		tmp = 0.25 * (t_3 * ((D * (D / d)) / d));
	elseif ((M * M) <= 500.0)
		tmp = t_2;
	elseif ((M * M) <= 2e+144)
		tmp = 0.25 * ((D * D) * (((h * M) / d) * (M / d)));
	else
		tmp = t_1 * (2.0 * (t_0 * ((c0 / h) / w)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(2.0 * N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(M * M), $MachinePrecision], 4e-221], N[(0.25 * N[(t$95$3 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(M * M), $MachinePrecision], 5e-165], t$95$2, If[LessEqual[N[(M * M), $MachinePrecision], 1e-48], N[(0.25 * N[(t$95$3 * N[(N[(D * N[(D / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(M * M), $MachinePrecision], 500.0], t$95$2, If[LessEqual[N[(M * M), $MachinePrecision], 2e+144], N[(0.25 * N[(N[(D * D), $MachinePrecision] * N[(N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(2.0 * N[(t$95$0 * N[(N[(c0 / h), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 M M) < 4.00000000000000007e-221

   1. Initial program 27.7%

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

   3. Simplified 29.8%

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

   4. Taylor expanded in c0 around -inf 7.5%

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

      1. fma-def 7.5%

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

      2. distribute-lft1-in 7.5%

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

      3. metadata-eval 7.5%

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

      4. unpow2 7.5%

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

      5. unpow2 7.5%

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

      6. associate-/l* 7.5%

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

      7. unpow2 7.5%

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

      8. unpow2 7.5%

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

      9. *-commutative 7.5%

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

      10. unpow2 7.5%

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

   6. Simplified 7.5%

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

   7. Taylor expanded in c0 around 0 42.6%

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

      1. associate-/l* 42.6%

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

      2. unpow2 42.6%

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

      3. unpow2 42.6%

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

      4. *-commutative 42.6%

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

      5. unpow2 42.6%

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

   9. Simplified 42.6%

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

   10. Taylor expanded in D around 0 42.6%

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

       1. *-commutative 42.6%

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

       2. unpow2 42.6%

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

       3. *-commutative 42.6%

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

       4. associate-/l* 41.6%

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

       5. unpow2 41.6%

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

       6. unpow2 41.6%

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

       7. times-frac 58.7%

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

       8. unpow2 58.7%

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

   12. Simplified 58.7%

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

       1. pow2 39.2%

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

   14. Applied egg-rr 58.7%

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

   if 4.00000000000000007e-221 < (*.f64 M M) < 4.99999999999999981e-165 or 9.9999999999999997e-49 < (*.f64 M M) < 500

   1. Initial program 44.3%

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

   3. Simplified 51.0%

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

   4. Taylor expanded in c0 around inf 44.7%

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

      1. unpow2 44.7%

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

      2. *-commutative 44.7%

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

      3. associate-/l/ 44.9%

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

      4. unpow2 44.9%

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

      5. unpow2 44.9%

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

      6. associate-*l/ 51.4%

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

      7. times-frac 51.7%

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

      8. associate-*l/ 51.7%

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

      9. unpow2 51.7%

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

      10. associate-*l/ 63.4%

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

      11. associate-*r/ 67.2%

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

      12. *-commutative 67.2%

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

      13. associate-*r/ 67.2%

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

      14. unpow2 67.2%

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

   6. Simplified 67.2%

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

      1. pow2 60.6%

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

   8. Applied egg-rr 67.2%

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

   if 4.99999999999999981e-165 < (*.f64 M M) < 9.9999999999999997e-49

   1. Initial program 20.4%

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

   3. Simplified 20.7%

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

   4. Taylor expanded in c0 around -inf 0.3%

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

      1. fma-def 0.3%

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

      2. distribute-lft1-in 0.3%

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

      3. metadata-eval 0.3%

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

      4. unpow2 0.3%

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

      5. unpow2 0.3%

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

      6. associate-/l* 0.3%

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

      7. unpow2 0.3%

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

      8. unpow2 0.3%

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

      9. *-commutative 0.3%

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

      10. unpow2 0.3%

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

   6. Simplified 0.3%

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

   7. Taylor expanded in c0 around 0 48.1%

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

      1. associate-/l* 48.3%

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

      2. unpow2 48.3%

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

      3. unpow2 48.3%

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

      4. *-commutative 48.3%

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

      5. unpow2 48.3%

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

   9. Simplified 48.3%

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

   10. Taylor expanded in D around 0 48.1%

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

       1. unpow2 48.1%

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

       2. *-commutative 48.1%

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

       3. associate-*l/ 51.6%

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

       4. unpow2 51.6%

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

       5. unpow2 51.6%

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

       6. *-commutative 51.6%

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

       7. times-frac 54.7%

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

   12. Simplified 54.7%

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

       1. associate-*r/ 54.7%

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

   14. Applied egg-rr 54.7%

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

   if 500 < (*.f64 M M) < 2.00000000000000005e144

   1. Initial program 14.9%

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

   3. Simplified 17.5%

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

   4. Taylor expanded in c0 around -inf 8.9%

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

      1. fma-def 8.9%

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

      2. distribute-lft1-in 8.9%

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

      3. metadata-eval 8.9%

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

      4. unpow2 8.9%

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

      5. unpow2 8.9%

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

      6. associate-/l* 8.8%

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

      7. unpow2 8.8%

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

      8. unpow2 8.8%

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

      9. *-commutative 8.8%

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

      10. unpow2 8.8%

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

   6. Simplified 8.8%

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

   7. Taylor expanded in c0 around 0 47.5%

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

      1. associate-/l* 50.0%

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

      2. unpow2 50.0%

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

      3. unpow2 50.0%

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

      4. *-commutative 50.0%

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

      5. unpow2 50.0%

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

   9. Simplified 50.0%

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

       1. div-inv 49.9%

          \[\leadsto 0.25 \cdot \color{blue}{\left(\left(D \cdot D\right) \cdot \frac{1}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\right)} \]

   11. Applied egg-rr 49.9%

       \[\leadsto 0.25 \cdot \color{blue}{\left(\left(D \cdot D\right) \cdot \frac{1}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\right)} \]

   12. Taylor expanded in d around 0 50.0%

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

       1. unpow2 50.0%

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

       2. *-commutative 50.0%

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

       3. associate-*r* 49.9%

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

       4. unpow2 49.9%

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

       5. times-frac 56.2%

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

   14. Simplified 56.2%

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

   if 2.00000000000000005e144 < (*.f64 M M)

   1. Initial program 14.4%

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

   3. Simplified 15.7%

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

   4. Taylor expanded in c0 around inf 46.3%

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

      1. unpow2 46.3%

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

      2. *-commutative 46.3%

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

      3. associate-/l/ 47.7%

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

      4. unpow2 47.7%

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

      5. unpow2 47.7%

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

      6. associate-*l/ 47.6%

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

      7. times-frac 56.1%

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

      8. associate-*l/ 56.2%

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

      9. unpow2 56.2%

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

      10. associate-*l/ 57.7%

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

      11. associate-*r/ 56.1%

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

      12. *-commutative 56.1%

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

      13. associate-*r/ 58.8%

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

      14. unpow2 58.8%

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

   6. Simplified 58.8%

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

      1. associate-/r* 60.2%

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

      2. div-inv 60.2%

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

   8. Applied egg-rr 60.2%

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

      1. pow2 60.2%

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

   10. Applied egg-rr 60.2%

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

       1. un-div-inv 60.2%

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

   12. Applied egg-rr 60.2%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot M \leq 4 \cdot 10^{-221}:\\ \;\;\;\;0.25 \cdot \frac{h \cdot \left(M \cdot M\right)}{\frac{d}{D} \cdot \frac{d}{D}}\\ \mathbf{elif}\;M \cdot M \leq 5 \cdot 10^{-165}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\right)\\ \mathbf{elif}\;M \cdot M \leq 10^{-48}:\\ \;\;\;\;0.25 \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \frac{D \cdot \frac{D}{d}}{d}\right)\\ \mathbf{elif}\;M \cdot M \leq 500:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\right)\\ \mathbf{elif}\;M \cdot M \leq 2 \cdot 10^{+144}:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(\frac{h \cdot M}{d} \cdot \frac{M}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right)\right)\\ \end{array} \]

Alternative 5: 46.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d}{D} \cdot \frac{d}{D}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := t_1 \cdot \left(2 \cdot \left(\frac{c0}{w \cdot h} \cdot t_0\right)\right)\\ t_3 := h \cdot \left(M \cdot M\right)\\ \mathbf{if}\;M \cdot M \leq 4 \cdot 10^{-221}:\\ \;\;\;\;0.25 \cdot \frac{t_3}{t_0}\\ \mathbf{elif}\;M \cdot M \leq 5 \cdot 10^{-165}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;M \cdot M \leq 10^{-48}:\\ \;\;\;\;0.25 \cdot \left(t_3 \cdot \frac{D \cdot \frac{D}{d}}{d}\right)\\ \mathbf{elif}\;M \cdot M \leq 500:\\ \;\;\;\;t_2\\ \mathbf{elif}\;M \cdot M \leq 2 \cdot 10^{+144}:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(\frac{h \cdot M}{d} \cdot \frac{M}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \frac{c0 \cdot \frac{d}{D}}{\left(w \cdot h\right) \cdot \frac{D}{d}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ d D) (/ d D)))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (* t_1 (* 2.0 (* (/ c0 (* w h)) t_0))))
        (t_3 (* h (* M M))))
   (if (<= (* M M) 4e-221)
     (* 0.25 (/ t_3 t_0))
     (if (<= (* M M) 5e-165)
       t_2
       (if (<= (* M M) 1e-48)
         (* 0.25 (* t_3 (/ (* D (/ D d)) d)))
         (if (<= (* M M) 500.0)
           t_2
           (if (<= (* M M) 2e+144)
             (* 0.25 (* (* D D) (* (/ (* h M) d) (/ M d))))
             (* t_1 (* 2.0 (/ (* c0 (/ d D)) (* (* w h) (/ D d))))))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (d / D) * (d / D);
	double t_1 = c0 / (2.0 * w);
	double t_2 = t_1 * (2.0 * ((c0 / (w * h)) * t_0));
	double t_3 = h * (M * M);
	double tmp;
	if ((M * M) <= 4e-221) {
		tmp = 0.25 * (t_3 / t_0);
	} else if ((M * M) <= 5e-165) {
		tmp = t_2;
	} else if ((M * M) <= 1e-48) {
		tmp = 0.25 * (t_3 * ((D * (D / d)) / d));
	} else if ((M * M) <= 500.0) {
		tmp = t_2;
	} else if ((M * M) <= 2e+144) {
		tmp = 0.25 * ((D * D) * (((h * M) / d) * (M / d)));
	} else {
		tmp = t_1 * (2.0 * ((c0 * (d / D)) / ((w * h) * (D / 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) :: t_3
    real(8) :: tmp
    t_0 = (d_1 / d) * (d_1 / d)
    t_1 = c0 / (2.0d0 * w)
    t_2 = t_1 * (2.0d0 * ((c0 / (w * h)) * t_0))
    t_3 = h * (m * m)
    if ((m * m) <= 4d-221) then
        tmp = 0.25d0 * (t_3 / t_0)
    else if ((m * m) <= 5d-165) then
        tmp = t_2
    else if ((m * m) <= 1d-48) then
        tmp = 0.25d0 * (t_3 * ((d * (d / d_1)) / d_1))
    else if ((m * m) <= 500.0d0) then
        tmp = t_2
    else if ((m * m) <= 2d+144) then
        tmp = 0.25d0 * ((d * d) * (((h * m) / d_1) * (m / d_1)))
    else
        tmp = t_1 * (2.0d0 * ((c0 * (d_1 / d)) / ((w * h) * (d / d_1))))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (d / D) * (d / D);
	double t_1 = c0 / (2.0 * w);
	double t_2 = t_1 * (2.0 * ((c0 / (w * h)) * t_0));
	double t_3 = h * (M * M);
	double tmp;
	if ((M * M) <= 4e-221) {
		tmp = 0.25 * (t_3 / t_0);
	} else if ((M * M) <= 5e-165) {
		tmp = t_2;
	} else if ((M * M) <= 1e-48) {
		tmp = 0.25 * (t_3 * ((D * (D / d)) / d));
	} else if ((M * M) <= 500.0) {
		tmp = t_2;
	} else if ((M * M) <= 2e+144) {
		tmp = 0.25 * ((D * D) * (((h * M) / d) * (M / d)));
	} else {
		tmp = t_1 * (2.0 * ((c0 * (d / D)) / ((w * h) * (D / d))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (d / D) * (d / D)
	t_1 = c0 / (2.0 * w)
	t_2 = t_1 * (2.0 * ((c0 / (w * h)) * t_0))
	t_3 = h * (M * M)
	tmp = 0
	if (M * M) <= 4e-221:
		tmp = 0.25 * (t_3 / t_0)
	elif (M * M) <= 5e-165:
		tmp = t_2
	elif (M * M) <= 1e-48:
		tmp = 0.25 * (t_3 * ((D * (D / d)) / d))
	elif (M * M) <= 500.0:
		tmp = t_2
	elif (M * M) <= 2e+144:
		tmp = 0.25 * ((D * D) * (((h * M) / d) * (M / d)))
	else:
		tmp = t_1 * (2.0 * ((c0 * (d / D)) / ((w * h) * (D / d))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(d / D) * Float64(d / D))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(t_1 * Float64(2.0 * Float64(Float64(c0 / Float64(w * h)) * t_0)))
	t_3 = Float64(h * Float64(M * M))
	tmp = 0.0
	if (Float64(M * M) <= 4e-221)
		tmp = Float64(0.25 * Float64(t_3 / t_0));
	elseif (Float64(M * M) <= 5e-165)
		tmp = t_2;
	elseif (Float64(M * M) <= 1e-48)
		tmp = Float64(0.25 * Float64(t_3 * Float64(Float64(D * Float64(D / d)) / d)));
	elseif (Float64(M * M) <= 500.0)
		tmp = t_2;
	elseif (Float64(M * M) <= 2e+144)
		tmp = Float64(0.25 * Float64(Float64(D * D) * Float64(Float64(Float64(h * M) / d) * Float64(M / d))));
	else
		tmp = Float64(t_1 * Float64(2.0 * Float64(Float64(c0 * Float64(d / D)) / Float64(Float64(w * h) * Float64(D / d)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (d / D) * (d / D);
	t_1 = c0 / (2.0 * w);
	t_2 = t_1 * (2.0 * ((c0 / (w * h)) * t_0));
	t_3 = h * (M * M);
	tmp = 0.0;
	if ((M * M) <= 4e-221)
		tmp = 0.25 * (t_3 / t_0);
	elseif ((M * M) <= 5e-165)
		tmp = t_2;
	elseif ((M * M) <= 1e-48)
		tmp = 0.25 * (t_3 * ((D * (D / d)) / d));
	elseif ((M * M) <= 500.0)
		tmp = t_2;
	elseif ((M * M) <= 2e+144)
		tmp = 0.25 * ((D * D) * (((h * M) / d) * (M / d)));
	else
		tmp = t_1 * (2.0 * ((c0 * (d / D)) / ((w * h) * (D / d))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(2.0 * N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(M * M), $MachinePrecision], 4e-221], N[(0.25 * N[(t$95$3 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(M * M), $MachinePrecision], 5e-165], t$95$2, If[LessEqual[N[(M * M), $MachinePrecision], 1e-48], N[(0.25 * N[(t$95$3 * N[(N[(D * N[(D / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(M * M), $MachinePrecision], 500.0], t$95$2, If[LessEqual[N[(M * M), $MachinePrecision], 2e+144], N[(0.25 * N[(N[(D * D), $MachinePrecision] * N[(N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(2.0 * N[(N[(c0 * N[(d / D), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 M M) < 4.00000000000000007e-221

   1. Initial program 27.7%

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

   3. Simplified 29.8%

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

   4. Taylor expanded in c0 around -inf 7.5%

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

      1. fma-def 7.5%

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

      2. distribute-lft1-in 7.5%

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

      3. metadata-eval 7.5%

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

      4. unpow2 7.5%

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

      5. unpow2 7.5%

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

      6. associate-/l* 7.5%

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

      7. unpow2 7.5%

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

      8. unpow2 7.5%

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

      9. *-commutative 7.5%

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

      10. unpow2 7.5%

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

   6. Simplified 7.5%

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

   7. Taylor expanded in c0 around 0 42.6%

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

      1. associate-/l* 42.6%

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

      2. unpow2 42.6%

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

      3. unpow2 42.6%

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

      4. *-commutative 42.6%

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

      5. unpow2 42.6%

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

   9. Simplified 42.6%

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

   10. Taylor expanded in D around 0 42.6%

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

       1. *-commutative 42.6%

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

       2. unpow2 42.6%

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

       3. *-commutative 42.6%

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

       4. associate-/l* 41.6%

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

       5. unpow2 41.6%

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

       6. unpow2 41.6%

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

       7. times-frac 58.7%

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

       8. unpow2 58.7%

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

   12. Simplified 58.7%

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

       1. pow2 39.2%

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

   14. Applied egg-rr 58.7%

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

   if 4.00000000000000007e-221 < (*.f64 M M) < 4.99999999999999981e-165 or 9.9999999999999997e-49 < (*.f64 M M) < 500

   1. Initial program 44.3%

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

   3. Simplified 51.0%

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

   4. Taylor expanded in c0 around inf 44.7%

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

      1. unpow2 44.7%

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

      2. *-commutative 44.7%

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

      3. associate-/l/ 44.9%

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

      4. unpow2 44.9%

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

      5. unpow2 44.9%

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

      6. associate-*l/ 51.4%

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

      7. times-frac 51.7%

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

      8. associate-*l/ 51.7%

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

      9. unpow2 51.7%

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

      10. associate-*l/ 63.4%

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

      11. associate-*r/ 67.2%

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

      12. *-commutative 67.2%

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

      13. associate-*r/ 67.2%

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

      14. unpow2 67.2%

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

   6. Simplified 67.2%

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

      1. pow2 60.6%

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

   8. Applied egg-rr 67.2%

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

   if 4.99999999999999981e-165 < (*.f64 M M) < 9.9999999999999997e-49

   1. Initial program 20.4%

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

   3. Simplified 20.7%

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

   4. Taylor expanded in c0 around -inf 0.3%

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

      1. fma-def 0.3%

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

      2. distribute-lft1-in 0.3%

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

      3. metadata-eval 0.3%

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

      4. unpow2 0.3%

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

      5. unpow2 0.3%

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

      6. associate-/l* 0.3%

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

      7. unpow2 0.3%

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

      8. unpow2 0.3%

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

      9. *-commutative 0.3%

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

      10. unpow2 0.3%

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

   6. Simplified 0.3%

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

   7. Taylor expanded in c0 around 0 48.1%

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

      1. associate-/l* 48.3%

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

      2. unpow2 48.3%

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

      3. unpow2 48.3%

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

      4. *-commutative 48.3%

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

      5. unpow2 48.3%

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

   9. Simplified 48.3%

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

   10. Taylor expanded in D around 0 48.1%

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

       1. unpow2 48.1%

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

       2. *-commutative 48.1%

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

       3. associate-*l/ 51.6%

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

       4. unpow2 51.6%

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

       5. unpow2 51.6%

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

       6. *-commutative 51.6%

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

       7. times-frac 54.7%

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

   12. Simplified 54.7%

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

       1. associate-*r/ 54.7%

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

   14. Applied egg-rr 54.7%

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

   if 500 < (*.f64 M M) < 2.00000000000000005e144

   1. Initial program 14.9%

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

   3. Simplified 17.5%

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

   4. Taylor expanded in c0 around -inf 8.9%

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

      1. fma-def 8.9%

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

      2. distribute-lft1-in 8.9%

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

      3. metadata-eval 8.9%

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

      4. unpow2 8.9%

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

      5. unpow2 8.9%

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

      6. associate-/l* 8.8%

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

      7. unpow2 8.8%

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

      8. unpow2 8.8%

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

      9. *-commutative 8.8%

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

      10. unpow2 8.8%

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

   6. Simplified 8.8%

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

   7. Taylor expanded in c0 around 0 47.5%

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

      1. associate-/l* 50.0%

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

      2. unpow2 50.0%

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

      3. unpow2 50.0%

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

      4. *-commutative 50.0%

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

      5. unpow2 50.0%

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

   9. Simplified 50.0%

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

       1. div-inv 49.9%

          \[\leadsto 0.25 \cdot \color{blue}{\left(\left(D \cdot D\right) \cdot \frac{1}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\right)} \]

   11. Applied egg-rr 49.9%

       \[\leadsto 0.25 \cdot \color{blue}{\left(\left(D \cdot D\right) \cdot \frac{1}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\right)} \]

   12. Taylor expanded in d around 0 50.0%

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

       1. unpow2 50.0%

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

       2. *-commutative 50.0%

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

       3. associate-*r* 49.9%

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

       4. unpow2 49.9%

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

       5. times-frac 56.2%

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

   14. Simplified 56.2%

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

   if 2.00000000000000005e144 < (*.f64 M M)

   1. Initial program 14.4%

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

   3. Simplified 15.7%

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

   4. Taylor expanded in c0 around inf 46.3%

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

      1. unpow2 46.3%

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

      2. *-commutative 46.3%

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

      3. associate-/l/ 47.7%

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

      4. unpow2 47.7%

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

      5. unpow2 47.7%

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

      6. associate-*l/ 47.6%

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

      7. times-frac 56.1%

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

      8. associate-*l/ 56.2%

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

      9. unpow2 56.2%

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

      10. associate-*l/ 57.7%

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

      11. associate-*r/ 56.1%

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

      12. *-commutative 56.1%

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

      13. associate-*r/ 58.8%

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

      14. unpow2 58.8%

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

   6. Simplified 58.8%

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

      1. associate-/r* 60.2%

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

      2. div-inv 60.2%

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

   8. Applied egg-rr 60.2%

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

      1. pow2 60.2%

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

   10. Applied egg-rr 60.2%

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

       1. un-div-inv 60.2%

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

       2. associate-/r* 58.8%

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

       3. clear-num 58.8%

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

       4. div-inv 58.8%

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

       5. frac-times 61.7%

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

   12. Applied egg-rr 61.7%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot M \leq 4 \cdot 10^{-221}:\\ \;\;\;\;0.25 \cdot \frac{h \cdot \left(M \cdot M\right)}{\frac{d}{D} \cdot \frac{d}{D}}\\ \mathbf{elif}\;M \cdot M \leq 5 \cdot 10^{-165}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\right)\\ \mathbf{elif}\;M \cdot M \leq 10^{-48}:\\ \;\;\;\;0.25 \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \frac{D \cdot \frac{D}{d}}{d}\right)\\ \mathbf{elif}\;M \cdot M \leq 500:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\right)\\ \mathbf{elif}\;M \cdot M \leq 2 \cdot 10^{+144}:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(\frac{h \cdot M}{d} \cdot \frac{M}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot \frac{d}{D}}{\left(w \cdot h\right) \cdot \frac{D}{d}}\right)\\ \end{array} \]

Alternative 6: 43.4% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \cdot M \leq 10^{+190}:\\ \;\;\;\;0.25 \cdot \frac{h \cdot \left(M \cdot M\right)}{\frac{d}{D} \cdot \frac{d}{D}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot c0}{D \cdot D} \cdot \frac{d \cdot d}{h \cdot \left(w \cdot w\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= (* M M) 1e+190)
   (* 0.25 (/ (* h (* M M)) (* (/ d D) (/ d D))))
   (* (/ (* c0 c0) (* D D)) (/ (* d d) (* h (* w w))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((M * M) <= 1e+190) {
		tmp = 0.25 * ((h * (M * M)) / ((d / D) * (d / D)));
	} else {
		tmp = ((c0 * c0) / (D * D)) * ((d * d) / (h * (w * w)));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((m * m) <= 1d+190) then
        tmp = 0.25d0 * ((h * (m * m)) / ((d_1 / d) * (d_1 / d)))
    else
        tmp = ((c0 * c0) / (d * d)) * ((d_1 * d_1) / (h * (w * w)))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((M * M) <= 1e+190) {
		tmp = 0.25 * ((h * (M * M)) / ((d / D) * (d / D)));
	} else {
		tmp = ((c0 * c0) / (D * D)) * ((d * d) / (h * (w * w)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (M * M) <= 1e+190:
		tmp = 0.25 * ((h * (M * M)) / ((d / D) * (d / D)))
	else:
		tmp = ((c0 * c0) / (D * D)) * ((d * d) / (h * (w * w)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (Float64(M * M) <= 1e+190)
		tmp = Float64(0.25 * Float64(Float64(h * Float64(M * M)) / Float64(Float64(d / D) * Float64(d / D))));
	else
		tmp = Float64(Float64(Float64(c0 * c0) / Float64(D * D)) * Float64(Float64(d * d) / Float64(h * Float64(w * w))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((M * M) <= 1e+190)
		tmp = 0.25 * ((h * (M * M)) / ((d / D) * (d / D)));
	else
		tmp = ((c0 * c0) / (D * D)) * ((d * d) / (h * (w * w)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[N[(M * M), $MachinePrecision], 1e+190], N[(0.25 * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c0 * c0), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / N[(h * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M M) < 1.0000000000000001e190

   1. Initial program 25.4%

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

   3. Simplified 28.4%

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

   4. Taylor expanded in c0 around -inf 5.4%

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

      1. fma-def 5.4%

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

      2. distribute-lft1-in 5.4%

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

      3. metadata-eval 5.4%

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

      4. unpow2 5.4%

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

      5. unpow2 5.4%

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

      6. associate-/l* 5.4%

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

      7. unpow2 5.4%

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

      8. unpow2 5.4%

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

      9. *-commutative 5.4%

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

      10. unpow2 5.4%

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

   6. Simplified 5.4%

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

   7. Taylor expanded in c0 around 0 40.6%

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

      1. associate-/l* 41.1%

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

      2. unpow2 41.1%

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

      3. unpow2 41.1%

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

      4. *-commutative 41.1%

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

      5. unpow2 41.1%

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

   9. Simplified 41.1%

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

   10. Taylor expanded in D around 0 40.6%

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

       1. *-commutative 40.6%

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

       2. unpow2 40.6%

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

       3. *-commutative 40.6%

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

       4. associate-/l* 40.2%

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

       5. unpow2 40.2%

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

       6. unpow2 40.2%

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

       7. times-frac 51.9%

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

       8. unpow2 51.9%

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

   12. Simplified 51.9%

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

       1. pow2 39.8%

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

   14. Applied egg-rr 51.9%

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

   if 1.0000000000000001e190 < (*.f64 M M)

   1. Initial program 15.8%

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

   3. Simplified 15.8%

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

   4. Taylor expanded in c0 around inf 49.7%

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

      1. unpow2 49.7%

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

      2. *-commutative 49.7%

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

      3. associate-/l/ 49.7%

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

      4. unpow2 49.7%

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

      5. unpow2 49.7%

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

      6. associate-*l/ 49.6%

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

      7. times-frac 57.4%

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

      8. associate-*l/ 57.4%

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

      9. unpow2 57.4%

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

      10. associate-*l/ 59.0%

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

      11. associate-*r/ 55.9%

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

      12. *-commutative 55.9%

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

      13. associate-*r/ 58.8%

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

      14. unpow2 58.8%

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

   6. Simplified 58.8%

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

      1. associate-/r* 60.4%

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

      2. div-inv 60.4%

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

   8. Applied egg-rr 60.4%

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

   9. Taylor expanded in c0 around 0 46.2%

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

       1. times-frac 46.1%

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

       2. unpow2 46.1%

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

       3. unpow2 46.1%

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

       4. unpow2 46.1%

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

       5. unpow2 46.1%

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

   11. Simplified 46.1%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot M \leq 10^{+190}:\\ \;\;\;\;0.25 \cdot \frac{h \cdot \left(M \cdot M\right)}{\frac{d}{D} \cdot \frac{d}{D}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot c0}{D \cdot D} \cdot \frac{d \cdot d}{h \cdot \left(w \cdot w\right)}\\ \end{array} \]

Alternative 7: 43.4% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \cdot M \leq 10^{+190}:\\ \;\;\;\;0.25 \cdot \frac{h \cdot \left(M \cdot M\right)}{\frac{d}{D} \cdot \frac{d}{D}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= (* M M) 1e+190)
   (* 0.25 (/ (* h (* M M)) (* (/ d D) (/ d D))))
   (/ (* (* d d) (* c0 c0)) (* (* D D) (* h (* w w))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((M * M) <= 1e+190) {
		tmp = 0.25 * ((h * (M * M)) / ((d / D) * (d / D)));
	} else {
		tmp = ((d * d) * (c0 * c0)) / ((D * D) * (h * (w * w)));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((m * m) <= 1d+190) then
        tmp = 0.25d0 * ((h * (m * m)) / ((d_1 / d) * (d_1 / d)))
    else
        tmp = ((d_1 * d_1) * (c0 * c0)) / ((d * d) * (h * (w * w)))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((M * M) <= 1e+190) {
		tmp = 0.25 * ((h * (M * M)) / ((d / D) * (d / D)));
	} else {
		tmp = ((d * d) * (c0 * c0)) / ((D * D) * (h * (w * w)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (M * M) <= 1e+190:
		tmp = 0.25 * ((h * (M * M)) / ((d / D) * (d / D)))
	else:
		tmp = ((d * d) * (c0 * c0)) / ((D * D) * (h * (w * w)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (Float64(M * M) <= 1e+190)
		tmp = Float64(0.25 * Float64(Float64(h * Float64(M * M)) / Float64(Float64(d / D) * Float64(d / D))));
	else
		tmp = Float64(Float64(Float64(d * d) * Float64(c0 * c0)) / Float64(Float64(D * D) * Float64(h * Float64(w * w))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((M * M) <= 1e+190)
		tmp = 0.25 * ((h * (M * M)) / ((d / D) * (d / D)));
	else
		tmp = ((d * d) * (c0 * c0)) / ((D * D) * (h * (w * w)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[N[(M * M), $MachinePrecision], 1e+190], N[(0.25 * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(d * d), $MachinePrecision] * N[(c0 * c0), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(h * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M M) < 1.0000000000000001e190

   1. Initial program 25.4%

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

   3. Simplified 28.4%

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

   4. Taylor expanded in c0 around -inf 5.4%

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

      1. fma-def 5.4%

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

      2. distribute-lft1-in 5.4%

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

      3. metadata-eval 5.4%

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

      4. unpow2 5.4%

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

      5. unpow2 5.4%

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

      6. associate-/l* 5.4%

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

      7. unpow2 5.4%

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

      8. unpow2 5.4%

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

      9. *-commutative 5.4%

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

      10. unpow2 5.4%

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

   6. Simplified 5.4%

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

   7. Taylor expanded in c0 around 0 40.6%

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

      1. associate-/l* 41.1%

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

      2. unpow2 41.1%

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

      3. unpow2 41.1%

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

      4. *-commutative 41.1%

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

      5. unpow2 41.1%

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

   9. Simplified 41.1%

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

   10. Taylor expanded in D around 0 40.6%

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

       1. *-commutative 40.6%

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

       2. unpow2 40.6%

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

       3. *-commutative 40.6%

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

       4. associate-/l* 40.2%

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

       5. unpow2 40.2%

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

       6. unpow2 40.2%

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

       7. times-frac 51.9%

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

       8. unpow2 51.9%

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

   12. Simplified 51.9%

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

       1. pow2 39.8%

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

   14. Applied egg-rr 51.9%

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

   if 1.0000000000000001e190 < (*.f64 M M)

   1. Initial program 15.8%

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

   3. Simplified 15.8%

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

   4. Taylor expanded in c0 around inf 49.7%

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

      1. unpow2 49.7%

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

      2. *-commutative 49.7%

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

      3. associate-/l/ 49.7%

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

      4. unpow2 49.7%

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

      5. unpow2 49.7%

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

      6. associate-*l/ 49.6%

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

      7. times-frac 57.4%

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

      8. associate-*l/ 57.4%

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

      9. unpow2 57.4%

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

      10. associate-*l/ 59.0%

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

      11. associate-*r/ 55.9%

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

      12. *-commutative 55.9%

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

      13. associate-*r/ 58.8%

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

      14. unpow2 58.8%

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

   6. Simplified 58.8%

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

   7. Taylor expanded in c0 around 0 46.2%

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

      1. *-commutative 46.2%

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

      2. unpow2 46.2%

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

      3. unpow2 46.2%

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

      4. unpow2 46.2%

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

      5. unpow2 46.2%

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

   9. Simplified 46.2%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot M \leq 10^{+190}:\\ \;\;\;\;0.25 \cdot \frac{h \cdot \left(M \cdot M\right)}{\frac{d}{D} \cdot \frac{d}{D}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}\\ \end{array} \]

Alternative 8: 45.4% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \cdot D \leq 2 \cdot 10^{+127}:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(\frac{h \cdot M}{d} \cdot \frac{M}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= (* D D) 2e+127)
   (* 0.25 (* (* D D) (* (/ (* h M) d) (/ M d))))
   (* 0.25 (* (* h (* M M)) (* (/ D d) (/ D d))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((D * D) <= 2e+127) {
		tmp = 0.25 * ((D * D) * (((h * M) / d) * (M / d)));
	} else {
		tmp = 0.25 * ((h * (M * M)) * ((D / d) * (D / 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) :: tmp
    if ((d * d) <= 2d+127) then
        tmp = 0.25d0 * ((d * d) * (((h * m) / d_1) * (m / d_1)))
    else
        tmp = 0.25d0 * ((h * (m * m)) * ((d / d_1) * (d / d_1)))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((D * D) <= 2e+127) {
		tmp = 0.25 * ((D * D) * (((h * M) / d) * (M / d)));
	} else {
		tmp = 0.25 * ((h * (M * M)) * ((D / d) * (D / d)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (D * D) <= 2e+127:
		tmp = 0.25 * ((D * D) * (((h * M) / d) * (M / d)))
	else:
		tmp = 0.25 * ((h * (M * M)) * ((D / d) * (D / d)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (Float64(D * D) <= 2e+127)
		tmp = Float64(0.25 * Float64(Float64(D * D) * Float64(Float64(Float64(h * M) / d) * Float64(M / d))));
	else
		tmp = Float64(0.25 * Float64(Float64(h * Float64(M * M)) * Float64(Float64(D / d) * Float64(D / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((D * D) <= 2e+127)
		tmp = 0.25 * ((D * D) * (((h * M) / d) * (M / d)));
	else
		tmp = 0.25 * ((h * (M * M)) * ((D / d) * (D / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[N[(D * D), $MachinePrecision], 2e+127], N[(0.25 * N[(N[(D * D), $MachinePrecision] * N[(N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 D D) < 1.99999999999999991e127

   1. Initial program 23.5%

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

   3. Simplified 26.1%

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

   4. Taylor expanded in c0 around -inf 4.5%

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

      1. fma-def 4.5%

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

      2. distribute-lft1-in 4.5%

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

      3. metadata-eval 4.5%

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

      4. unpow2 4.5%

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

      5. unpow2 4.5%

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

      6. associate-/l* 4.5%

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

      7. unpow2 4.5%

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

      8. unpow2 4.5%

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

      9. *-commutative 4.5%

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

      10. unpow2 4.5%

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

   6. Simplified 4.5%

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

   7. Taylor expanded in c0 around 0 36.2%

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

      1. associate-/l* 37.1%

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

      2. unpow2 37.1%

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

      3. unpow2 37.1%

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

      4. *-commutative 37.1%

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

      5. unpow2 37.1%

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

   9. Simplified 37.1%

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

       1. div-inv 37.1%

          \[\leadsto 0.25 \cdot \color{blue}{\left(\left(D \cdot D\right) \cdot \frac{1}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\right)} \]

   11. Applied egg-rr 37.1%

       \[\leadsto 0.25 \cdot \color{blue}{\left(\left(D \cdot D\right) \cdot \frac{1}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\right)} \]

   12. Taylor expanded in d around 0 37.1%

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

       1. unpow2 37.1%

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

       2. *-commutative 37.1%

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

       3. associate-*r* 37.7%

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

       4. unpow2 37.7%

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

       5. times-frac 48.8%

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

   14. Simplified 48.8%

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

   if 1.99999999999999991e127 < (*.f64 D D)

   1. Initial program 21.1%

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

   3. Simplified 21.2%

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

   4. Taylor expanded in c0 around -inf 8.6%

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

      1. fma-def 8.6%

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

      2. distribute-lft1-in 8.6%

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

      3. metadata-eval 8.6%

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

      4. unpow2 8.6%

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

      5. unpow2 8.6%

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

      6. associate-/l* 8.5%

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

      7. unpow2 8.5%

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

      8. unpow2 8.5%

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

      9. *-commutative 8.5%

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

      10. unpow2 8.5%

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

   6. Simplified 8.5%

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

   7. Taylor expanded in c0 around 0 31.6%

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

      1. associate-/l* 27.6%

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

      2. unpow2 27.6%

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

      3. unpow2 27.6%

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

      4. *-commutative 27.6%

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

      5. unpow2 27.6%

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

   9. Simplified 27.6%

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

   10. Taylor expanded in D around 0 31.6%

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

       1. unpow2 31.6%

          \[\leadsto 0.25 \cdot \frac{{D}^{2} \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot h\right)}{{d}^{2}} \]

       2. *-commutative 31.6%

          \[\leadsto 0.25 \cdot \frac{{D}^{2} \cdot \color{blue}{\left(h \cdot \left(M \cdot M\right)\right)}}{{d}^{2}} \]

       3. associate-*l/ 29.7%

          \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)} \]

       4. unpow2 29.7%

          \[\leadsto 0.25 \cdot \left(\frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

       5. unpow2 29.7%

          \[\leadsto 0.25 \cdot \left(\frac{D \cdot D}{\color{blue}{d \cdot d}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

       6. *-commutative 29.7%

          \[\leadsto 0.25 \cdot \color{blue}{\left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \frac{D \cdot D}{d \cdot d}\right)} \]

       7. times-frac 53.9%

          \[\leadsto 0.25 \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right) \]

   12. Simplified 53.9%

       \[\leadsto 0.25 \cdot \color{blue}{\left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \cdot D \leq 2 \cdot 10^{+127}:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(\frac{h \cdot M}{d} \cdot \frac{M}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\\ \end{array} \]

Alternative 9: 45.3% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \cdot D \leq 2 \cdot 10^{+127}:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(\frac{h \cdot M}{d} \cdot \frac{M}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{h \cdot \left(M \cdot M\right)}{\frac{d}{D} \cdot \frac{d}{D}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= (* D D) 2e+127)
   (* 0.25 (* (* D D) (* (/ (* h M) d) (/ M d))))
   (* 0.25 (/ (* h (* M M)) (* (/ d D) (/ d D))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((D * D) <= 2e+127) {
		tmp = 0.25 * ((D * D) * (((h * M) / d) * (M / d)));
	} else {
		tmp = 0.25 * ((h * (M * M)) / ((d / D) * (d / 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) :: tmp
    if ((d * d) <= 2d+127) then
        tmp = 0.25d0 * ((d * d) * (((h * m) / d_1) * (m / d_1)))
    else
        tmp = 0.25d0 * ((h * (m * m)) / ((d_1 / d) * (d_1 / 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 tmp;
	if ((D * D) <= 2e+127) {
		tmp = 0.25 * ((D * D) * (((h * M) / d) * (M / d)));
	} else {
		tmp = 0.25 * ((h * (M * M)) / ((d / D) * (d / D)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (D * D) <= 2e+127:
		tmp = 0.25 * ((D * D) * (((h * M) / d) * (M / d)))
	else:
		tmp = 0.25 * ((h * (M * M)) / ((d / D) * (d / D)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (Float64(D * D) <= 2e+127)
		tmp = Float64(0.25 * Float64(Float64(D * D) * Float64(Float64(Float64(h * M) / d) * Float64(M / d))));
	else
		tmp = Float64(0.25 * Float64(Float64(h * Float64(M * M)) / Float64(Float64(d / D) * Float64(d / D))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((D * D) <= 2e+127)
		tmp = 0.25 * ((D * D) * (((h * M) / d) * (M / d)));
	else
		tmp = 0.25 * ((h * (M * M)) / ((d / D) * (d / D)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[N[(D * D), $MachinePrecision], 2e+127], N[(0.25 * N[(N[(D * D), $MachinePrecision] * N[(N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 D D) < 1.99999999999999991e127

   1. Initial program 23.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Simplified 26.1%

      \[\leadsto \color{blue}{\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}, \frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}, -M \cdot M\right)}\right)} \]

   4. Taylor expanded in c0 around -inf 4.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) + 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]
   5. Step-by-step derivation

      1. fma-def 4.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(-1, c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]

      2. distribute-lft1-in 4.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}, 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      3. metadata-eval 4.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      4. unpow2 4.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\color{blue}{d \cdot d}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      5. unpow2 4.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{d \cdot d}{\color{blue}{\left(D \cdot D\right)} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      6. associate-/l* 4.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{d \cdot d}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \color{blue}{\frac{{D}^{2}}{\frac{c0 \cdot {d}^{2}}{{M}^{2} \cdot \left(h \cdot w\right)}}}\right) \]

      7. unpow2 4.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{d \cdot d}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{\color{blue}{D \cdot D}}{\frac{c0 \cdot {d}^{2}}{{M}^{2} \cdot \left(h \cdot w\right)}}\right) \]

      8. unpow2 4.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{d \cdot d}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{D \cdot D}{\frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{{M}^{2} \cdot \left(h \cdot w\right)}}\right) \]

      9. *-commutative 4.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{d \cdot d}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{D \cdot D}{\frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(h \cdot w\right) \cdot {M}^{2}}}}\right) \]

      10. unpow2 4.5%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{d \cdot d}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{D \cdot D}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(h \cdot w\right) \cdot \color{blue}{\left(M \cdot M\right)}}}\right) \]

   6. Simplified 4.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{d \cdot d}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{D \cdot D}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(h \cdot w\right) \cdot \left(M \cdot M\right)}}\right)} \]

   7. Taylor expanded in c0 around 0 36.2%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   8. Step-by-step derivation

      1. associate-/l* 37.1%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{{M}^{2} \cdot h}}} \]

      2. unpow2 37.1%

         \[\leadsto 0.25 \cdot \frac{\color{blue}{D \cdot D}}{\frac{{d}^{2}}{{M}^{2} \cdot h}} \]

      3. unpow2 37.1%

         \[\leadsto 0.25 \cdot \frac{D \cdot D}{\frac{\color{blue}{d \cdot d}}{{M}^{2} \cdot h}} \]

      4. *-commutative 37.1%

         \[\leadsto 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{\color{blue}{h \cdot {M}^{2}}}} \]

      5. unpow2 37.1%

         \[\leadsto 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \color{blue}{\left(M \cdot M\right)}}} \]

   9. Simplified 37.1%

      \[\leadsto \color{blue}{0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}} \]
   10. Step-by-step derivation

       1. div-inv 37.1%

          \[\leadsto 0.25 \cdot \color{blue}{\left(\left(D \cdot D\right) \cdot \frac{1}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\right)} \]

   11. Applied egg-rr 37.1%

       \[\leadsto 0.25 \cdot \color{blue}{\left(\left(D \cdot D\right) \cdot \frac{1}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\right)} \]

   12. Taylor expanded in d around 0 37.1%

       \[\leadsto 0.25 \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\frac{{M}^{2} \cdot h}{{d}^{2}}}\right) \]
   13. Step-by-step derivation

       1. unpow2 37.1%

          \[\leadsto 0.25 \cdot \left(\left(D \cdot D\right) \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right) \]

       2. *-commutative 37.1%

          \[\leadsto 0.25 \cdot \left(\left(D \cdot D\right) \cdot \frac{\color{blue}{h \cdot \left(M \cdot M\right)}}{{d}^{2}}\right) \]

       3. associate-*r* 37.7%

          \[\leadsto 0.25 \cdot \left(\left(D \cdot D\right) \cdot \frac{\color{blue}{\left(h \cdot M\right) \cdot M}}{{d}^{2}}\right) \]

       4. unpow2 37.7%

          \[\leadsto 0.25 \cdot \left(\left(D \cdot D\right) \cdot \frac{\left(h \cdot M\right) \cdot M}{\color{blue}{d \cdot d}}\right) \]

       5. times-frac 48.8%

          \[\leadsto 0.25 \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(\frac{h \cdot M}{d} \cdot \frac{M}{d}\right)}\right) \]

   14. Simplified 48.8%

       \[\leadsto 0.25 \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(\frac{h \cdot M}{d} \cdot \frac{M}{d}\right)}\right) \]

   if 1.99999999999999991e127 < (*.f64 D D)

   1. Initial program 21.1%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Simplified 21.2%

      \[\leadsto \color{blue}{\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}, \frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}, -M \cdot M\right)}\right)} \]

   4. Taylor expanded in c0 around -inf 8.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) + 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]
   5. Step-by-step derivation

      1. fma-def 8.6%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(-1, c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]

      2. distribute-lft1-in 8.6%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}, 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      3. metadata-eval 8.6%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      4. unpow2 8.6%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\color{blue}{d \cdot d}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      5. unpow2 8.6%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{d \cdot d}{\color{blue}{\left(D \cdot D\right)} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      6. associate-/l* 8.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{d \cdot d}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \color{blue}{\frac{{D}^{2}}{\frac{c0 \cdot {d}^{2}}{{M}^{2} \cdot \left(h \cdot w\right)}}}\right) \]

      7. unpow2 8.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{d \cdot d}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{\color{blue}{D \cdot D}}{\frac{c0 \cdot {d}^{2}}{{M}^{2} \cdot \left(h \cdot w\right)}}\right) \]

      8. unpow2 8.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{d \cdot d}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{D \cdot D}{\frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{{M}^{2} \cdot \left(h \cdot w\right)}}\right) \]

      9. *-commutative 8.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{d \cdot d}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{D \cdot D}{\frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(h \cdot w\right) \cdot {M}^{2}}}}\right) \]

      10. unpow2 8.5%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{d \cdot d}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{D \cdot D}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(h \cdot w\right) \cdot \color{blue}{\left(M \cdot M\right)}}}\right) \]

   6. Simplified 8.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{d \cdot d}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{D \cdot D}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(h \cdot w\right) \cdot \left(M \cdot M\right)}}\right)} \]

   7. Taylor expanded in c0 around 0 31.6%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   8. Step-by-step derivation

      1. associate-/l* 27.6%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{{M}^{2} \cdot h}}} \]

      2. unpow2 27.6%

         \[\leadsto 0.25 \cdot \frac{\color{blue}{D \cdot D}}{\frac{{d}^{2}}{{M}^{2} \cdot h}} \]

      3. unpow2 27.6%

         \[\leadsto 0.25 \cdot \frac{D \cdot D}{\frac{\color{blue}{d \cdot d}}{{M}^{2} \cdot h}} \]

      4. *-commutative 27.6%

         \[\leadsto 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{\color{blue}{h \cdot {M}^{2}}}} \]

      5. unpow2 27.6%

         \[\leadsto 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \color{blue}{\left(M \cdot M\right)}}} \]

   9. Simplified 27.6%

      \[\leadsto \color{blue}{0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}} \]

   10. Taylor expanded in D around 0 31.6%

       \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   11. Step-by-step derivation

       1. *-commutative 31.6%

          \[\leadsto 0.25 \cdot \frac{\color{blue}{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}}{{d}^{2}} \]

       2. unpow2 31.6%

          \[\leadsto 0.25 \cdot \frac{\left(\color{blue}{\left(M \cdot M\right)} \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \]

       3. *-commutative 31.6%

          \[\leadsto 0.25 \cdot \frac{\color{blue}{\left(h \cdot \left(M \cdot M\right)\right)} \cdot {D}^{2}}{{d}^{2}} \]

       4. associate-/l* 29.7%

          \[\leadsto 0.25 \cdot \color{blue}{\frac{h \cdot \left(M \cdot M\right)}{\frac{{d}^{2}}{{D}^{2}}}} \]

       5. unpow2 29.7%

          \[\leadsto 0.25 \cdot \frac{h \cdot \left(M \cdot M\right)}{\frac{\color{blue}{d \cdot d}}{{D}^{2}}} \]

       6. unpow2 29.7%

          \[\leadsto 0.25 \cdot \frac{h \cdot \left(M \cdot M\right)}{\frac{d \cdot d}{\color{blue}{D \cdot D}}} \]

       7. times-frac 56.0%

          \[\leadsto 0.25 \cdot \frac{h \cdot \left(M \cdot M\right)}{\color{blue}{\frac{d}{D} \cdot \frac{d}{D}}} \]

       8. unpow2 56.0%

          \[\leadsto 0.25 \cdot \frac{h \cdot \left(M \cdot M\right)}{\color{blue}{{\left(\frac{d}{D}\right)}^{2}}} \]

   12. Simplified 56.0%

       \[\leadsto 0.25 \cdot \color{blue}{\frac{h \cdot \left(M \cdot M\right)}{{\left(\frac{d}{D}\right)}^{2}}} \]
   13. Step-by-step derivation

       1. pow2 55.3%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\left(\frac{c0}{h} \cdot \frac{1}{w}\right) \cdot \color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D}\right)}\right)\right) \]

   14. Applied egg-rr 56.0%

       \[\leadsto 0.25 \cdot \frac{h \cdot \left(M \cdot M\right)}{\color{blue}{\frac{d}{D} \cdot \frac{d}{D}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \cdot D \leq 2 \cdot 10^{+127}:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(\frac{h \cdot M}{d} \cdot \frac{M}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{h \cdot \left(M \cdot M\right)}{\frac{d}{D} \cdot \frac{d}{D}}\\ \end{array} \]

Alternative 10: 38.5% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 7.8 \cdot 10^{-118}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(\frac{h \cdot M}{d} \cdot \frac{M}{d}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= M 7.8e-118) 0.0 (* 0.25 (* (* D D) (* (/ (* h M) d) (/ M d))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 7.8e-118) {
		tmp = 0.0;
	} else {
		tmp = 0.25 * ((D * D) * (((h * M) / d) * (M / 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) :: tmp
    if (m <= 7.8d-118) then
        tmp = 0.0d0
    else
        tmp = 0.25d0 * ((d * d) * (((h * m) / d_1) * (m / d_1)))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 7.8e-118) {
		tmp = 0.0;
	} else {
		tmp = 0.25 * ((D * D) * (((h * M) / d) * (M / d)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if M <= 7.8e-118:
		tmp = 0.0
	else:
		tmp = 0.25 * ((D * D) * (((h * M) / d) * (M / d)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (M <= 7.8e-118)
		tmp = 0.0;
	else
		tmp = Float64(0.25 * Float64(Float64(D * D) * Float64(Float64(Float64(h * M) / d) * Float64(M / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (M <= 7.8e-118)
		tmp = 0.0;
	else
		tmp = 0.25 * ((D * D) * (((h * M) / d) * (M / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[M, 7.8e-118], 0.0, N[(0.25 * N[(N[(D * D), $MachinePrecision] * N[(N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 7.80000000000000002e-118

   1. Initial program 23.8%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Simplified 25.9%

      \[\leadsto \color{blue}{\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}, \frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}, -M \cdot M\right)}\right)} \]

   4. Taylor expanded in c0 around -inf 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 33.8%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{0} \]

   7. Taylor expanded in c0 around 0 39.9%

      \[\leadsto \color{blue}{0} \]

   if 7.80000000000000002e-118 < M

   1. Initial program 21.4%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Simplified 23.6%

      \[\leadsto \color{blue}{\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}, \frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}, -M \cdot M\right)}\right)} \]

   4. Taylor expanded in c0 around -inf 5.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) + 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]
   5. Step-by-step derivation

      1. fma-def 5.2%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(-1, c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]

      2. distribute-lft1-in 5.2%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}, 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      3. metadata-eval 5.2%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      4. unpow2 5.2%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{\color{blue}{d \cdot d}}{{D}^{2} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      5. unpow2 5.2%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{d \cdot d}{\color{blue}{\left(D \cdot D\right)} \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right) \]

      6. associate-/l* 5.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{d \cdot d}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \color{blue}{\frac{{D}^{2}}{\frac{c0 \cdot {d}^{2}}{{M}^{2} \cdot \left(h \cdot w\right)}}}\right) \]

      7. unpow2 5.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{d \cdot d}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{\color{blue}{D \cdot D}}{\frac{c0 \cdot {d}^{2}}{{M}^{2} \cdot \left(h \cdot w\right)}}\right) \]

      8. unpow2 5.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{d \cdot d}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{D \cdot D}{\frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{{M}^{2} \cdot \left(h \cdot w\right)}}\right) \]

      9. *-commutative 5.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{d \cdot d}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{D \cdot D}{\frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(h \cdot w\right) \cdot {M}^{2}}}}\right) \]

      10. unpow2 5.1%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{d \cdot d}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{D \cdot D}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(h \cdot w\right) \cdot \color{blue}{\left(M \cdot M\right)}}}\right) \]

   6. Simplified 5.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(-1, c0 \cdot \left(0 \cdot \frac{d \cdot d}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right), 0.5 \cdot \frac{D \cdot D}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(h \cdot w\right) \cdot \left(M \cdot M\right)}}\right)} \]

   7. Taylor expanded in c0 around 0 31.1%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   8. Step-by-step derivation

      1. associate-/l* 29.9%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{{M}^{2} \cdot h}}} \]

      2. unpow2 29.9%

         \[\leadsto 0.25 \cdot \frac{\color{blue}{D \cdot D}}{\frac{{d}^{2}}{{M}^{2} \cdot h}} \]

      3. unpow2 29.9%

         \[\leadsto 0.25 \cdot \frac{D \cdot D}{\frac{\color{blue}{d \cdot d}}{{M}^{2} \cdot h}} \]

      4. *-commutative 29.9%

         \[\leadsto 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{\color{blue}{h \cdot {M}^{2}}}} \]

      5. unpow2 29.9%

         \[\leadsto 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \color{blue}{\left(M \cdot M\right)}}} \]

   9. Simplified 29.9%

      \[\leadsto \color{blue}{0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}} \]
   10. Step-by-step derivation

       1. div-inv 29.9%

          \[\leadsto 0.25 \cdot \color{blue}{\left(\left(D \cdot D\right) \cdot \frac{1}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\right)} \]

   11. Applied egg-rr 29.9%

       \[\leadsto 0.25 \cdot \color{blue}{\left(\left(D \cdot D\right) \cdot \frac{1}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\right)} \]

   12. Taylor expanded in d around 0 29.9%

       \[\leadsto 0.25 \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\frac{{M}^{2} \cdot h}{{d}^{2}}}\right) \]
   13. Step-by-step derivation

       1. unpow2 29.9%

          \[\leadsto 0.25 \cdot \left(\left(D \cdot D\right) \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right) \]

       2. *-commutative 29.9%

          \[\leadsto 0.25 \cdot \left(\left(D \cdot D\right) \cdot \frac{\color{blue}{h \cdot \left(M \cdot M\right)}}{{d}^{2}}\right) \]

       3. associate-*r* 31.3%

          \[\leadsto 0.25 \cdot \left(\left(D \cdot D\right) \cdot \frac{\color{blue}{\left(h \cdot M\right) \cdot M}}{{d}^{2}}\right) \]

       4. unpow2 31.3%

          \[\leadsto 0.25 \cdot \left(\left(D \cdot D\right) \cdot \frac{\left(h \cdot M\right) \cdot M}{\color{blue}{d \cdot d}}\right) \]

       5. times-frac 45.7%

          \[\leadsto 0.25 \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(\frac{h \cdot M}{d} \cdot \frac{M}{d}\right)}\right) \]

   14. Simplified 45.7%

       \[\leadsto 0.25 \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(\frac{h \cdot M}{d} \cdot \frac{M}{d}\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 7.8 \cdot 10^{-118}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(\frac{h \cdot M}{d} \cdot \frac{M}{d}\right)\right)\\ \end{array} \]

Alternative 11: 34.0% 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 23.0%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

3. Simplified 25.2%

   \[\leadsto \color{blue}{\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}, \frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}, -M \cdot M\right)}\right)} \]

4. Taylor expanded in c0 around -inf 2.9%

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)} \]
5. Step-by-step derivation

   1. mul-1-neg 2.9%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)} \]

   2. distribute-rgt-in 2.9%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(-\color{blue}{\left(\left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right) \cdot c0 + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} \cdot c0\right)}\right) \]

6. Simplified 31.1%

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{0} \]

7. Taylor expanded in c0 around 0 35.8%

   \[\leadsto \color{blue}{0} \]

8. Final simplification 35.8%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023271 
(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))))))