Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.8% → 63.3%

Time: 27.7s

Alternatives: 8

Speedup: 30.2×

• could not determine a ground truth

  1. c0 = -2.5397722409060955e+268
  2. w = -2.4534338651002628e-269
  3. h = -4.349814633676886e-12
  4. D = -2.919336369064098e-308
  5. d = -1.9918071723060505e+301
  6. M = -1.5431613889505762e+286

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: 63.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 74.8%

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

      1. times-frac 70.3%

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

      2. fma-def 69.2%

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

      3. associate-/r* 69.4%

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

      4. difference-of-squares 69.4%

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

   3. Simplified 70.3%

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

   4. Taylor expanded in c0 around inf 73.6%

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

      1. *-commutative 73.6%

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

      2. unpow2 73.6%

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

      3. associate-/l/ 71.4%

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

      4. associate-/r* 72.5%

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

      5. associate-/r* 71.3%

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

      6. unpow2 71.3%

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

      7. associate-/l/ 72.3%

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

      8. unpow2 72.3%

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

      9. *-commutative 72.3%

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

      10. unpow2 72.3%

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

      11. associate-*r* 73.3%

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

      12. *-commutative 73.3%

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

   6. Simplified 73.3%

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

      1. *-un-lft-identity 73.3%

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

      2. times-frac 69.9%

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

      3. *-commutative 69.9%

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

   8. Applied egg-rr 69.9%

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

      1. *-lft-identity 69.9%

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

      2. times-frac 76.2%

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

   10. Simplified 76.2%

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

       1. associate-*l/ 75.2%

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

       2. *-commutative 75.2%

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

       3. associate-*l* 78.5%

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

       4. *-commutative 78.5%

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

   12. Applied egg-rr 78.5%

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

       1. associate-*l/ 80.5%

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

       2. *-commutative 80.5%

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

       3. *-commutative 80.5%

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

       4. *-commutative 80.5%

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

   14. Simplified 80.5%

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

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

   1. Initial program 0.0%

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

      1. times-frac 0.6%

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

      2. fma-def 0.6%

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

      3. associate-/r* 0.6%

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

      4. difference-of-squares 14.0%

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

   3. Simplified 16.6%

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

      1. fma-udef 17.2%

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

      2. associate-/l/ 15.9%

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

      3. times-frac 13.4%

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

      4. fma-udef 13.4%

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

      5. associate-/l/ 13.4%

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

      6. times-frac 13.4%

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

      7. associate-/l/ 13.4%

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

      8. times-frac 13.4%

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

   5. Applied egg-rr 3.2%

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

      1. unpow2 3.2%

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

      2. associate--r- 6.2%

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

      3. +-inverses 32.5%

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

      4. unpow2 32.5%

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

   7. Simplified 32.5%

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

   8. Taylor expanded in c0 around -inf 21.3%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{0 + M \cdot M}{\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2} - \color{blue}{\left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left(h \cdot {M}^{2}\right)\right)}{{d}^{2} \cdot c0} + -1 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}} \]
   9. Step-by-step derivation

      1. mul-1-neg 21.3%

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

      2. unsub-neg 21.3%

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

      3. associate-*r/ 21.3%

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

      4. associate-*r* 22.0%

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

      5. unpow2 22.0%

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

      6. unpow2 22.0%

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

      7. unpow2 22.0%

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

      8. associate-*r* 22.0%

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

      9. times-frac 22.6%

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

      10. unpow2 22.6%

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

      11. unpow2 22.6%

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

      12. times-frac 23.2%

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

      13. unpow2 23.2%

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

      14. *-commutative 23.2%

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

   10. Simplified 23.3%

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

   11. Taylor expanded in c0 around inf 44.0%

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

       1. unpow2 44.0%

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

       2. *-commutative 44.0%

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

       3. unpow2 44.0%

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

       4. associate-/l* 44.0%

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

       5. unpow2 44.0%

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

       6. unpow2 44.0%

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

       7. unpow2 44.0%

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

       8. *-commutative 44.0%

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

   13. Simplified 44.0%

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

   14. Taylor expanded in d around 0 44.0%

       \[\leadsto 0.25 \cdot \frac{D \cdot D}{\color{blue}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]
   15. Step-by-step derivation

       1. unpow2 44.0%

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

       2. unpow2 44.0%

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

       3. associate-*r* 47.0%

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

       4. times-frac 58.1%

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

   16. Simplified 58.1%

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

4. Final simplification 66.1%

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

Alternative 2: 48.5% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \cdot M \leq 2 \cdot 10^{+117}:\\ \;\;\;\;0.25 \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{d}{D} \cdot \left(\frac{d}{D} \cdot \frac{c0}{w \cdot h}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= (* M M) 2e+117)
   (* 0.25 (* (* h (* M M)) (* (/ D d) (/ D d))))
   (* (/ c0 (* 2.0 w)) (* 2.0 (* (/ d D) (* (/ d D) (/ c0 (* w h))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((M * M) <= 2e+117) {
		tmp = 0.25 * ((h * (M * M)) * ((D / d) * (D / d)));
	} else {
		tmp = (c0 / (2.0 * w)) * (2.0 * ((d / D) * ((d / D) * (c0 / (w * h)))));
	}
	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) <= 2d+117) then
        tmp = 0.25d0 * ((h * (m * m)) * ((d / d_1) * (d / d_1)))
    else
        tmp = (c0 / (2.0d0 * w)) * (2.0d0 * ((d_1 / d) * ((d_1 / d) * (c0 / (w * h)))))
    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) <= 2e+117) {
		tmp = 0.25 * ((h * (M * M)) * ((D / d) * (D / d)));
	} else {
		tmp = (c0 / (2.0 * w)) * (2.0 * ((d / D) * ((d / D) * (c0 / (w * h)))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (M * M) <= 2e+117:
		tmp = 0.25 * ((h * (M * M)) * ((D / d) * (D / d)))
	else:
		tmp = (c0 / (2.0 * w)) * (2.0 * ((d / D) * ((d / D) * (c0 / (w * h)))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (Float64(M * M) <= 2e+117)
		tmp = Float64(0.25 * Float64(Float64(h * Float64(M * M)) * Float64(Float64(D / d) * Float64(D / d))));
	else
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(2.0 * Float64(Float64(d / D) * Float64(Float64(d / D) * Float64(c0 / Float64(w * h))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((M * M) <= 2e+117)
		tmp = 0.25 * ((h * (M * M)) * ((D / d) * (D / d)));
	else
		tmp = (c0 / (2.0 * w)) * (2.0 * ((d / D) * ((d / D) * (c0 / (w * h)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[N[(M * M), $MachinePrecision], 2e+117], 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[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[(d / D), $MachinePrecision] * N[(N[(d / D), $MachinePrecision] * N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M M) < 2.0000000000000001e117

   1. Initial program 31.1%

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

      1. times-frac 28.7%

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

      2. fma-def 28.2%

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

      3. associate-/r* 28.3%

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

      4. difference-of-squares 28.3%

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

   3. Simplified 30.7%

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

      1. fma-udef 32.5%

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

      2. associate-/l/ 30.1%

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

      3. times-frac 29.4%

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

      4. fma-udef 29.4%

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

      5. associate-/l/ 29.3%

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

      6. times-frac 29.3%

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

      7. associate-/l/ 29.3%

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

      8. times-frac 31.1%

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

   5. Applied egg-rr 5.5%

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

      1. unpow2 5.5%

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

      2. associate--r- 9.0%

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

      3. +-inverses 32.5%

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

      4. unpow2 32.5%

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

   7. Simplified 32.5%

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

   8. Taylor expanded in c0 around -inf 26.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{0 + M \cdot M}{\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2} - \color{blue}{\left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left(h \cdot {M}^{2}\right)\right)}{{d}^{2} \cdot c0} + -1 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}} \]
   9. Step-by-step derivation

      1. mul-1-neg 26.5%

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

      2. unsub-neg 26.5%

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

      3. associate-*r/ 26.5%

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

      4. associate-*r* 25.4%

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

      5. unpow2 25.4%

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

      6. unpow2 25.4%

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

      7. unpow2 25.4%

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

      8. associate-*r* 25.4%

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

      9. times-frac 24.8%

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

      10. unpow2 24.8%

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

      11. unpow2 24.8%

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

      12. times-frac 25.4%

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

      13. unpow2 25.4%

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

      14. *-commutative 25.4%

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

   10. Simplified 26.0%

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

   11. Taylor expanded in c0 around inf 43.9%

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

       1. unpow2 43.9%

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

       2. *-commutative 43.9%

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

       3. unpow2 43.9%

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

       4. associate-/l* 43.6%

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

       5. unpow2 43.6%

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

       6. unpow2 43.6%

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

       7. unpow2 43.6%

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

       8. *-commutative 43.6%

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

   13. Simplified 43.6%

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

   14. Taylor expanded in D around 0 43.9%

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

       1. *-commutative 43.9%

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

       2. unpow2 43.9%

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

       3. associate-*r/ 42.2%

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

       4. unpow2 42.2%

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

       5. unpow2 42.2%

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

       6. times-frac 51.7%

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

   16. Simplified 51.7%

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

   if 2.0000000000000001e117 < (*.f64 M M)

   1. Initial program 17.6%

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

      1. times-frac 18.8%

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

      2. fma-def 18.8%

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

      3. associate-/r* 18.8%

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

      4. difference-of-squares 44.4%

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

   3. Simplified 45.6%

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

   4. Taylor expanded in c0 around inf 49.8%

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

      1. *-commutative 49.8%

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

      2. unpow2 49.8%

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

      3. associate-/l/ 52.3%

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

      4. associate-/r* 54.6%

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

      5. associate-/r* 53.5%

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

      6. unpow2 53.5%

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

      7. associate-/l/ 50.9%

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

      8. unpow2 50.9%

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

      9. *-commutative 50.9%

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

      10. unpow2 50.9%

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

      11. associate-*r* 52.1%

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

      12. *-commutative 52.1%

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

   6. Simplified 52.1%

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

      1. *-un-lft-identity 52.1%

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

      2. times-frac 53.6%

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

      3. *-commutative 53.6%

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

   8. Applied egg-rr 53.6%

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

      1. *-lft-identity 53.6%

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

      2. times-frac 55.1%

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

   10. Simplified 55.1%

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

       1. associate-*l/ 55.1%

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

       2. *-commutative 55.1%

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

       3. associate-*l* 57.3%

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

       4. *-commutative 57.3%

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

   12. Applied egg-rr 57.3%

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

       1. associate-*l/ 58.4%

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

       2. *-commutative 58.4%

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

       3. *-commutative 58.4%

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

       4. *-commutative 58.4%

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

   14. Simplified 58.4%

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

   15. Taylor expanded in d around 0 58.4%

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

       1. times-frac 58.4%

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

   17. Simplified 58.4%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot M \leq 2 \cdot 10^{+117}:\\ \;\;\;\;0.25 \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{d}{D} \cdot \left(\frac{d}{D} \cdot \frac{c0}{w \cdot h}\right)\right)\right)\\ \end{array} \]

Alternative 3: 42.9% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := h \cdot \left(M \cdot M\right)\\ \mathbf{if}\;D \leq 7.2 \cdot 10^{-146}:\\ \;\;\;\;0.25 \cdot \frac{D}{\frac{\frac{d \cdot d}{t_0}}{D}}\\ \mathbf{elif}\;D \leq 6.6 \cdot 10^{-137}:\\ \;\;\;\;\frac{d \cdot d}{\frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(w \cdot w\right)\right)}{c0 \cdot c0}}\\ \mathbf{elif}\;D \leq 3.8 \cdot 10^{+75}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot D}{\frac{d}{h \cdot M} \cdot \frac{d}{M}}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(t_0 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* h (* M M))))
   (if (<= D 7.2e-146)
     (* 0.25 (/ D (/ (/ (* d d) t_0) D)))
     (if (<= D 6.6e-137)
       (/ (* d d) (/ (* h (* (* D D) (* w w))) (* c0 c0)))
       (if (<= D 3.8e+75)
         (* 0.25 (/ (* D D) (* (/ d (* h M)) (/ d M))))
         (* 0.25 (* t_0 (* (/ D d) (/ D d)))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = h * (M * M);
	double tmp;
	if (D <= 7.2e-146) {
		tmp = 0.25 * (D / (((d * d) / t_0) / D));
	} else if (D <= 6.6e-137) {
		tmp = (d * d) / ((h * ((D * D) * (w * w))) / (c0 * c0));
	} else if (D <= 3.8e+75) {
		tmp = 0.25 * ((D * D) / ((d / (h * M)) * (d / M)));
	} else {
		tmp = 0.25 * (t_0 * ((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) :: tmp
    t_0 = h * (m * m)
    if (d <= 7.2d-146) then
        tmp = 0.25d0 * (d / (((d_1 * d_1) / t_0) / d))
    else if (d <= 6.6d-137) then
        tmp = (d_1 * d_1) / ((h * ((d * d) * (w * w))) / (c0 * c0))
    else if (d <= 3.8d+75) then
        tmp = 0.25d0 * ((d * d) / ((d_1 / (h * m)) * (d_1 / m)))
    else
        tmp = 0.25d0 * (t_0 * ((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 t_0 = h * (M * M);
	double tmp;
	if (D <= 7.2e-146) {
		tmp = 0.25 * (D / (((d * d) / t_0) / D));
	} else if (D <= 6.6e-137) {
		tmp = (d * d) / ((h * ((D * D) * (w * w))) / (c0 * c0));
	} else if (D <= 3.8e+75) {
		tmp = 0.25 * ((D * D) / ((d / (h * M)) * (d / M)));
	} else {
		tmp = 0.25 * (t_0 * ((D / d) * (D / d)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = h * (M * M)
	tmp = 0
	if D <= 7.2e-146:
		tmp = 0.25 * (D / (((d * d) / t_0) / D))
	elif D <= 6.6e-137:
		tmp = (d * d) / ((h * ((D * D) * (w * w))) / (c0 * c0))
	elif D <= 3.8e+75:
		tmp = 0.25 * ((D * D) / ((d / (h * M)) * (d / M)))
	else:
		tmp = 0.25 * (t_0 * ((D / d) * (D / d)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(h * Float64(M * M))
	tmp = 0.0
	if (D <= 7.2e-146)
		tmp = Float64(0.25 * Float64(D / Float64(Float64(Float64(d * d) / t_0) / D)));
	elseif (D <= 6.6e-137)
		tmp = Float64(Float64(d * d) / Float64(Float64(h * Float64(Float64(D * D) * Float64(w * w))) / Float64(c0 * c0)));
	elseif (D <= 3.8e+75)
		tmp = Float64(0.25 * Float64(Float64(D * D) / Float64(Float64(d / Float64(h * M)) * Float64(d / M))));
	else
		tmp = Float64(0.25 * Float64(t_0 * Float64(Float64(D / d) * Float64(D / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = h * (M * M);
	tmp = 0.0;
	if (D <= 7.2e-146)
		tmp = 0.25 * (D / (((d * d) / t_0) / D));
	elseif (D <= 6.6e-137)
		tmp = (d * d) / ((h * ((D * D) * (w * w))) / (c0 * c0));
	elseif (D <= 3.8e+75)
		tmp = 0.25 * ((D * D) / ((d / (h * M)) * (d / M)));
	else
		tmp = 0.25 * (t_0 * ((D / d) * (D / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[D, 7.2e-146], N[(0.25 * N[(D / N[(N[(N[(d * d), $MachinePrecision] / t$95$0), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[D, 6.6e-137], N[(N[(d * d), $MachinePrecision] / N[(N[(h * N[(N[(D * D), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c0 * c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[D, 3.8e+75], N[(0.25 * N[(N[(D * D), $MachinePrecision] / N[(N[(d / N[(h * M), $MachinePrecision]), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(t$95$0 * N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if D < 7.19999999999999957e-146

   1. Initial program 24.7%

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

      1. times-frac 22.9%

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

      2. fma-def 22.9%

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

      3. associate-/r* 23.0%

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

      4. difference-of-squares 32.0%

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

   3. Simplified 34.9%

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

      1. fma-udef 36.0%

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

      2. associate-/l/ 33.8%

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

      3. times-frac 32.6%

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

      4. fma-udef 32.6%

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

      5. associate-/l/ 32.5%

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

      6. times-frac 32.5%

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

      7. associate-/l/ 32.5%

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

      8. times-frac 33.7%

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

   5. Applied egg-rr 3.6%

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

      1. unpow2 3.6%

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

      2. associate--r- 6.3%

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

      3. +-inverses 28.1%

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

      4. unpow2 28.1%

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

   7. Simplified 28.1%

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

   8. Taylor expanded in c0 around -inf 20.7%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{0 + M \cdot M}{\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2} - \color{blue}{\left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left(h \cdot {M}^{2}\right)\right)}{{d}^{2} \cdot c0} + -1 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}} \]
   9. Step-by-step derivation

      1. mul-1-neg 20.7%

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

      2. unsub-neg 20.7%

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

      3. associate-*r/ 20.7%

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

      4. associate-*r* 21.3%

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

      5. unpow2 21.3%

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

      6. unpow2 21.3%

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

      7. unpow2 21.3%

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

      8. associate-*r* 21.3%

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

      9. times-frac 21.3%

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

      10. unpow2 21.3%

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

      11. unpow2 21.3%

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

      12. times-frac 21.8%

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

      13. unpow2 21.8%

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

      14. *-commutative 21.8%

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

   10. Simplified 21.9%

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

   11. Taylor expanded in c0 around inf 36.1%

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

       1. unpow2 36.1%

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

       2. *-commutative 36.1%

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

       3. unpow2 36.1%

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

       4. associate-/l* 36.4%

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

       5. unpow2 36.4%

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

       6. unpow2 36.4%

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

       7. unpow2 36.4%

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

       8. *-commutative 36.4%

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

   13. Simplified 36.4%

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

       1. *-un-lft-identity 36.4%

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

       2. associate-/l* 45.8%

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

   15. Applied egg-rr 45.8%

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

   if 7.19999999999999957e-146 < D < 6.6000000000000004e-137

   1. Initial program 80.0%

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

      1. associate-*l* 80.0%

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

      2. difference-of-squares 80.0%

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

      3. associate-*l* 80.0%

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

      4. associate-*l* 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in c0 around inf 80.0%

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

      1. associate-/l* 80.0%

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

      2. unpow2 80.0%

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

      3. associate-*r* 80.0%

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

      4. unpow2 80.0%

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

      5. unpow2 80.0%

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

      6. unpow2 80.0%

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

   6. Simplified 80.0%

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

   if 6.6000000000000004e-137 < D < 3.8000000000000002e75

   1. Initial program 33.5%

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

      1. times-frac 33.6%

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

      2. fma-def 31.7%

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

      3. associate-/r* 31.7%

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

      4. difference-of-squares 41.6%

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

   3. Simplified 39.6%

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

      1. fma-udef 41.5%

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

      2. associate-/l/ 41.5%

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

      3. times-frac 41.5%

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

      4. fma-udef 41.5%

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

      5. associate-/l/ 41.5%

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

      6. times-frac 41.5%

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

      7. associate-/l/ 41.5%

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

      8. times-frac 43.4%

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

   5. Applied egg-rr 3.9%

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

      1. unpow2 3.9%

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

      2. associate--r- 6.0%

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

      3. +-inverses 12.3%

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

      4. unpow2 12.3%

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

   7. Simplified 12.3%

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

   8. Taylor expanded in c0 around -inf 14.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{0 + M \cdot M}{\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2} - \color{blue}{\left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left(h \cdot {M}^{2}\right)\right)}{{d}^{2} \cdot c0} + -1 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}} \]
   9. Step-by-step derivation

      1. mul-1-neg 14.5%

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

      2. unsub-neg 14.5%

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

      3. associate-*r/ 14.5%

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

      4. associate-*r* 14.5%

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

      5. unpow2 14.5%

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

      6. unpow2 14.5%

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

      7. unpow2 14.5%

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

      8. associate-*r* 14.5%

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

      9. times-frac 10.6%

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

      10. unpow2 10.6%

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

      11. unpow2 10.6%

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

      12. times-frac 10.6%

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

      13. unpow2 10.6%

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

      14. *-commutative 10.6%

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

   10. Simplified 12.5%

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

   11. Taylor expanded in c0 around inf 40.1%

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

       1. unpow2 40.1%

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

       2. *-commutative 40.1%

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

       3. unpow2 40.1%

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

       4. associate-/l* 40.3%

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

       5. unpow2 40.3%

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

       6. unpow2 40.3%

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

       7. unpow2 40.3%

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

       8. *-commutative 40.3%

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

   13. Simplified 40.3%

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

   14. Taylor expanded in d around 0 40.3%

       \[\leadsto 0.25 \cdot \frac{D \cdot D}{\color{blue}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]
   15. Step-by-step derivation

       1. unpow2 40.3%

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

       2. unpow2 40.3%

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

       3. associate-*r* 45.8%

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

       4. times-frac 55.6%

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

   16. Simplified 55.6%

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

   if 3.8000000000000002e75 < D

   1. Initial program 14.0%

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

      1. times-frac 13.9%

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

      2. fma-def 13.9%

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

      3. associate-/r* 13.9%

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

      4. difference-of-squares 18.5%

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

   3. Simplified 23.2%

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

      1. fma-udef 23.1%

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

      2. associate-/l/ 23.1%

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

      3. times-frac 18.5%

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

      4. fma-udef 18.5%

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

      5. associate-/l/ 18.5%

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

      6. times-frac 18.4%

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

      7. associate-/l/ 18.4%

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

      8. times-frac 18.6%

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

   5. Applied egg-rr 4.7%

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

      1. unpow2 4.7%

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

      2. associate--r- 9.0%

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

      3. +-inverses 18.5%

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

      4. unpow2 18.5%

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

   7. Simplified 18.5%

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

   8. Taylor expanded in c0 around -inf 9.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{0 + M \cdot M}{\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2} - \color{blue}{\left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left(h \cdot {M}^{2}\right)\right)}{{d}^{2} \cdot c0} + -1 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}} \]
   9. Step-by-step derivation

      1. mul-1-neg 9.2%

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

      2. unsub-neg 9.2%

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

      3. associate-*r/ 9.2%

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

      4. associate-*r* 4.8%

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

      5. unpow2 4.8%

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

      6. unpow2 4.8%

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

      7. unpow2 4.8%

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

      8. associate-*r* 4.8%

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

      9. times-frac 9.3%

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

      10. unpow2 9.3%

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

      11. unpow2 9.3%

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

      12. times-frac 9.3%

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

      13. unpow2 9.3%

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

      14. *-commutative 9.3%

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

   10. Simplified 9.3%

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

   11. Taylor expanded in c0 around inf 23.4%

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

       1. unpow2 23.4%

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

       2. *-commutative 23.4%

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

       3. unpow2 23.4%

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

       4. associate-/l* 19.1%

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

       5. unpow2 19.1%

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

       6. unpow2 19.1%

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

       7. unpow2 19.1%

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

       8. *-commutative 19.1%

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

   13. Simplified 19.1%

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

   14. Taylor expanded in D around 0 23.4%

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

       1. *-commutative 23.4%

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

       2. unpow2 23.4%

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

       3. associate-*r/ 23.3%

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

       4. unpow2 23.3%

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

       5. unpow2 23.3%

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

       6. times-frac 55.1%

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

   16. Simplified 55.1%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 7.2 \cdot 10^{-146}:\\ \;\;\;\;0.25 \cdot \frac{D}{\frac{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}{D}}\\ \mathbf{elif}\;D \leq 6.6 \cdot 10^{-137}:\\ \;\;\;\;\frac{d \cdot d}{\frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(w \cdot w\right)\right)}{c0 \cdot c0}}\\ \mathbf{elif}\;D \leq 3.8 \cdot 10^{+75}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot D}{\frac{d}{h \cdot M} \cdot \frac{d}{M}}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\\ \end{array} \]

Alternative 4: 43.2% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 2.4 \cdot 10^{-184} \lor \neg \left(D \leq 6.8 \cdot 10^{+101}\right):\\ \;\;\;\;0.25 \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot D}{\frac{d}{h} \cdot \frac{d}{M \cdot M}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (or (<= D 2.4e-184) (not (<= D 6.8e+101)))
   (* 0.25 (* (* h (* M M)) (* (/ D d) (/ D d))))
   (* 0.25 (/ (* D D) (* (/ d h) (/ d (* M M)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((D <= 2.4e-184) || !(D <= 6.8e+101)) {
		tmp = 0.25 * ((h * (M * M)) * ((D / d) * (D / d)));
	} else {
		tmp = 0.25 * ((D * D) / ((d / h) * (d / (M * M))));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((d <= 2.4d-184) .or. (.not. (d <= 6.8d+101))) then
        tmp = 0.25d0 * ((h * (m * m)) * ((d / d_1) * (d / d_1)))
    else
        tmp = 0.25d0 * ((d * d) / ((d_1 / h) * (d_1 / (m * m))))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((D <= 2.4e-184) || !(D <= 6.8e+101)) {
		tmp = 0.25 * ((h * (M * M)) * ((D / d) * (D / d)));
	} else {
		tmp = 0.25 * ((D * D) / ((d / h) * (d / (M * M))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (D <= 2.4e-184) or not (D <= 6.8e+101):
		tmp = 0.25 * ((h * (M * M)) * ((D / d) * (D / d)))
	else:
		tmp = 0.25 * ((D * D) / ((d / h) * (d / (M * M))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if ((D <= 2.4e-184) || !(D <= 6.8e+101))
		tmp = Float64(0.25 * Float64(Float64(h * Float64(M * M)) * Float64(Float64(D / d) * Float64(D / d))));
	else
		tmp = Float64(0.25 * Float64(Float64(D * D) / Float64(Float64(d / h) * Float64(d / Float64(M * M)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((D <= 2.4e-184) || ~((D <= 6.8e+101)))
		tmp = 0.25 * ((h * (M * M)) * ((D / d) * (D / d)));
	else
		tmp = 0.25 * ((D * D) / ((d / h) * (d / (M * M))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[Or[LessEqual[D, 2.4e-184], N[Not[LessEqual[D, 6.8e+101]], $MachinePrecision]], N[(0.25 * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(D * D), $MachinePrecision] / N[(N[(d / h), $MachinePrecision] * N[(d / N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 2.40000000000000024e-184 or 6.80000000000000034e101 < D

   1. Initial program 23.7%

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

      1. times-frac 22.0%

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

      2. fma-def 22.0%

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

      3. associate-/r* 22.1%

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

      4. difference-of-squares 30.7%

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

   3. Simplified 34.0%

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

      1. fma-udef 35.1%

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

      2. associate-/l/ 32.9%

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

      3. times-frac 31.2%

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

      4. fma-udef 31.2%

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

      5. associate-/l/ 31.1%

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

      6. times-frac 31.1%

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

      7. associate-/l/ 31.1%

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

      8. times-frac 32.3%

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

   5. Applied egg-rr 3.4%

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

      1. unpow2 3.4%

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

      2. associate--r- 6.6%

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

      3. +-inverses 26.4%

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

      4. unpow2 26.4%

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

   7. Simplified 26.4%

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

   8. Taylor expanded in c0 around -inf 18.7%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{0 + M \cdot M}{\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2} - \color{blue}{\left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left(h \cdot {M}^{2}\right)\right)}{{d}^{2} \cdot c0} + -1 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}} \]
   9. Step-by-step derivation

      1. mul-1-neg 18.7%

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

      2. unsub-neg 18.7%

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

      3. associate-*r/ 18.7%

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

      4. associate-*r* 18.8%

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

      5. unpow2 18.8%

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

      6. unpow2 18.8%

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

      7. unpow2 18.8%

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

      8. associate-*r* 18.8%

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

      9. times-frac 18.8%

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

      10. unpow2 18.8%

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

      11. unpow2 18.8%

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

      12. times-frac 19.3%

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

      13. unpow2 19.3%

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

      14. *-commutative 19.3%

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

   10. Simplified 19.4%

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

   11. Taylor expanded in c0 around inf 33.0%

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

       1. unpow2 33.0%

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

       2. *-commutative 33.0%

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

       3. unpow2 33.0%

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

       4. associate-/l* 32.7%

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

       5. unpow2 32.7%

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

       6. unpow2 32.7%

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

       7. unpow2 32.7%

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

       8. *-commutative 32.7%

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

   13. Simplified 32.7%

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

   14. Taylor expanded in D around 0 33.0%

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

       1. *-commutative 33.0%

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

       2. unpow2 33.0%

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

       3. associate-*r/ 32.4%

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

       4. unpow2 32.4%

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

       5. unpow2 32.4%

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

       6. times-frac 45.1%

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

   16. Simplified 45.1%

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

   if 2.40000000000000024e-184 < D < 6.80000000000000034e101

   1. Initial program 34.4%

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

      1. times-frac 34.5%

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

      2. fma-def 33.1%

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

      3. associate-/r* 33.1%

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

      4. difference-of-squares 41.8%

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

   3. Simplified 40.3%

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

      1. fma-udef 41.7%

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

      2. associate-/l/ 41.7%

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

      3. times-frac 41.7%

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

      4. fma-udef 41.7%

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

      5. associate-/l/ 41.7%

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

      6. times-frac 41.7%

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

      7. associate-/l/ 41.7%

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

      8. times-frac 43.0%

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

   5. Applied egg-rr 4.3%

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

      1. unpow2 4.3%

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

      2. associate--r- 5.8%

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

      3. +-inverses 16.1%

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

      4. unpow2 16.1%

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

   7. Simplified 16.1%

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

   8. Taylor expanded in c0 around -inf 16.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{0 + M \cdot M}{\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2} - \color{blue}{\left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left(h \cdot {M}^{2}\right)\right)}{{d}^{2} \cdot c0} + -1 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}} \]
   9. Step-by-step derivation

      1. mul-1-neg 16.4%

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

      2. unsub-neg 16.4%

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

      3. associate-*r/ 16.4%

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

      4. associate-*r* 16.4%

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

      5. unpow2 16.4%

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

      6. unpow2 16.4%

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

      7. unpow2 16.4%

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

      8. associate-*r* 16.4%

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

      9. times-frac 15.0%

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

      10. unpow2 15.0%

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

      11. unpow2 15.0%

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

      12. times-frac 15.0%

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

      13. unpow2 15.0%

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

      14. *-commutative 15.0%

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

   10. Simplified 16.4%

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

   11. Taylor expanded in c0 around inf 41.1%

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

       1. unpow2 41.1%

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

       2. *-commutative 41.1%

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

       3. unpow2 41.1%

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

       4. associate-/l* 41.3%

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

       5. unpow2 41.3%

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

       6. unpow2 41.3%

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

       7. unpow2 41.3%

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

       8. *-commutative 41.3%

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

   13. Simplified 41.3%

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

       1. times-frac 48.1%

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

   15. Applied egg-rr 48.1%

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

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 2.4 \cdot 10^{-184} \lor \neg \left(D \leq 6.8 \cdot 10^{+101}\right):\\ \;\;\;\;0.25 \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot D}{\frac{d}{h} \cdot \frac{d}{M \cdot M}}\\ \end{array} \]

Alternative 5: 44.2% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 2.15 \cdot 10^{-184} \lor \neg \left(D \leq 3.9 \cdot 10^{+75}\right):\\ \;\;\;\;0.25 \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot D}{\frac{d}{h \cdot M} \cdot \frac{d}{M}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (or (<= D 2.15e-184) (not (<= D 3.9e+75)))
   (* 0.25 (* (* h (* M M)) (* (/ D d) (/ D d))))
   (* 0.25 (/ (* D D) (* (/ d (* h M)) (/ d M))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((D <= 2.15e-184) || !(D <= 3.9e+75)) {
		tmp = 0.25 * ((h * (M * M)) * ((D / d) * (D / d)));
	} else {
		tmp = 0.25 * ((D * D) / ((d / (h * M)) * (d / M)));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((d <= 2.15d-184) .or. (.not. (d <= 3.9d+75))) then
        tmp = 0.25d0 * ((h * (m * m)) * ((d / d_1) * (d / d_1)))
    else
        tmp = 0.25d0 * ((d * d) / ((d_1 / (h * m)) * (d_1 / m)))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((D <= 2.15e-184) || !(D <= 3.9e+75)) {
		tmp = 0.25 * ((h * (M * M)) * ((D / d) * (D / d)));
	} else {
		tmp = 0.25 * ((D * D) / ((d / (h * M)) * (d / M)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (D <= 2.15e-184) or not (D <= 3.9e+75):
		tmp = 0.25 * ((h * (M * M)) * ((D / d) * (D / d)))
	else:
		tmp = 0.25 * ((D * D) / ((d / (h * M)) * (d / M)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if ((D <= 2.15e-184) || !(D <= 3.9e+75))
		tmp = Float64(0.25 * Float64(Float64(h * Float64(M * M)) * Float64(Float64(D / d) * Float64(D / d))));
	else
		tmp = Float64(0.25 * Float64(Float64(D * D) / Float64(Float64(d / Float64(h * M)) * Float64(d / M))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((D <= 2.15e-184) || ~((D <= 3.9e+75)))
		tmp = 0.25 * ((h * (M * M)) * ((D / d) * (D / d)));
	else
		tmp = 0.25 * ((D * D) / ((d / (h * M)) * (d / M)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[Or[LessEqual[D, 2.15e-184], N[Not[LessEqual[D, 3.9e+75]], $MachinePrecision]], N[(0.25 * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(D * D), $MachinePrecision] / N[(N[(d / N[(h * M), $MachinePrecision]), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 2.15000000000000003e-184 or 3.90000000000000038e75 < D

   1. Initial program 23.6%

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

      1. times-frac 21.9%

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

      2. fma-def 21.9%

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

      3. associate-/r* 22.0%

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

      4. difference-of-squares 30.4%

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

   3. Simplified 33.6%

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

      1. fma-udef 34.7%

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

      2. associate-/l/ 32.6%

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

      3. times-frac 30.9%

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

      4. fma-udef 30.9%

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

      5. associate-/l/ 30.8%

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

      6. times-frac 30.8%

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

      7. associate-/l/ 30.8%

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

      8. times-frac 31.9%

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

   5. Applied egg-rr 3.9%

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

      1. unpow2 3.9%

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

      2. associate--r- 6.9%

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

      3. +-inverses 26.8%

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

      4. unpow2 26.8%

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

   7. Simplified 26.8%

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

   8. Taylor expanded in c0 around -inf 18.7%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{0 + M \cdot M}{\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2} - \color{blue}{\left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left(h \cdot {M}^{2}\right)\right)}{{d}^{2} \cdot c0} + -1 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}} \]
   9. Step-by-step derivation

      1. mul-1-neg 18.7%

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

      2. unsub-neg 18.7%

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

      3. associate-*r/ 18.7%

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

      4. associate-*r* 18.8%

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

      5. unpow2 18.8%

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

      6. unpow2 18.8%

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

      7. unpow2 18.8%

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

      8. associate-*r* 18.8%

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

      9. times-frac 19.3%

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

      10. unpow2 19.3%

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

      11. unpow2 19.3%

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

      12. times-frac 19.8%

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

      13. unpow2 19.8%

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

      14. *-commutative 19.8%

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

   10. Simplified 19.9%

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

   11. Taylor expanded in c0 around inf 33.7%

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

       1. unpow2 33.7%

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

       2. *-commutative 33.7%

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

       3. unpow2 33.7%

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

       4. associate-/l* 33.5%

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

       5. unpow2 33.5%

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

       6. unpow2 33.5%

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

       7. unpow2 33.5%

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

       8. *-commutative 33.5%

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

   13. Simplified 33.5%

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

   14. Taylor expanded in D around 0 33.7%

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

       1. *-commutative 33.7%

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

       2. unpow2 33.7%

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

       3. associate-*r/ 33.2%

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

       4. unpow2 33.2%

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

       5. unpow2 33.2%

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

       6. times-frac 45.5%

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

   16. Simplified 45.5%

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

   if 2.15000000000000003e-184 < D < 3.90000000000000038e75

   1. Initial program 35.5%

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

      1. times-frac 35.6%

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

      2. fma-def 34.1%

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

      3. associate-/r* 34.1%

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

      4. difference-of-squares 43.4%

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

   3. Simplified 41.9%

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

      1. fma-udef 43.3%

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

      2. associate-/l/ 43.3%

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

      3. times-frac 43.4%

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

      4. fma-udef 43.4%

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

      5. associate-/l/ 43.4%

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

      6. times-frac 43.4%

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

      7. associate-/l/ 43.4%

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

      8. times-frac 44.8%

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

   5. Applied egg-rr 3.1%

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

      1. unpow2 3.1%

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

      2. associate--r- 4.7%

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

      3. +-inverses 14.2%

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

      4. unpow2 14.2%

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

   7. Simplified 14.2%

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

   8. Taylor expanded in c0 around -inf 16.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{0 + M \cdot M}{\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2} - \color{blue}{\left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left(h \cdot {M}^{2}\right)\right)}{{d}^{2} \cdot c0} + -1 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}} \]
   9. Step-by-step derivation

      1. mul-1-neg 16.1%

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

      2. unsub-neg 16.1%

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

      3. associate-*r/ 16.1%

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

      4. associate-*r* 16.1%

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

      5. unpow2 16.1%

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

      6. unpow2 16.1%

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

      7. unpow2 16.1%

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

      8. associate-*r* 16.1%

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

      9. times-frac 13.0%

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

      10. unpow2 13.0%

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

      11. unpow2 13.0%

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

      12. times-frac 13.1%

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

      13. unpow2 13.1%

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

      14. *-commutative 13.1%

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

   10. Simplified 14.6%

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

   11. Taylor expanded in c0 around inf 39.5%

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

       1. unpow2 39.5%

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

       2. *-commutative 39.5%

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

       3. unpow2 39.5%

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

       4. associate-/l* 39.6%

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

       5. unpow2 39.6%

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

       6. unpow2 39.6%

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

       7. unpow2 39.6%

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

       8. *-commutative 39.6%

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

   13. Simplified 39.6%

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

   14. Taylor expanded in d around 0 39.6%

       \[\leadsto 0.25 \cdot \frac{D \cdot D}{\color{blue}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]
   15. Step-by-step derivation

       1. unpow2 39.6%

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

       2. unpow2 39.6%

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

       3. associate-*r* 45.6%

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

       4. times-frac 53.2%

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

   16. Simplified 53.2%

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

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 2.15 \cdot 10^{-184} \lor \neg \left(D \leq 3.9 \cdot 10^{+75}\right):\\ \;\;\;\;0.25 \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot D}{\frac{d}{h \cdot M} \cdot \frac{d}{M}}\\ \end{array} \]

Alternative 6: 43.3% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := h \cdot \left(M \cdot M\right)\\ \mathbf{if}\;D \leq 2.3 \cdot 10^{-184}:\\ \;\;\;\;0.25 \cdot \frac{D}{\frac{\frac{d \cdot d}{t_0}}{D}}\\ \mathbf{elif}\;D \leq 4 \cdot 10^{+75}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot D}{\frac{d}{h \cdot M} \cdot \frac{d}{M}}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(t_0 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* h (* M M))))
   (if (<= D 2.3e-184)
     (* 0.25 (/ D (/ (/ (* d d) t_0) D)))
     (if (<= D 4e+75)
       (* 0.25 (/ (* D D) (* (/ d (* h M)) (/ d M))))
       (* 0.25 (* t_0 (* (/ D d) (/ D d))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = h * (M * M);
	double tmp;
	if (D <= 2.3e-184) {
		tmp = 0.25 * (D / (((d * d) / t_0) / D));
	} else if (D <= 4e+75) {
		tmp = 0.25 * ((D * D) / ((d / (h * M)) * (d / M)));
	} else {
		tmp = 0.25 * (t_0 * ((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) :: tmp
    t_0 = h * (m * m)
    if (d <= 2.3d-184) then
        tmp = 0.25d0 * (d / (((d_1 * d_1) / t_0) / d))
    else if (d <= 4d+75) then
        tmp = 0.25d0 * ((d * d) / ((d_1 / (h * m)) * (d_1 / m)))
    else
        tmp = 0.25d0 * (t_0 * ((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 t_0 = h * (M * M);
	double tmp;
	if (D <= 2.3e-184) {
		tmp = 0.25 * (D / (((d * d) / t_0) / D));
	} else if (D <= 4e+75) {
		tmp = 0.25 * ((D * D) / ((d / (h * M)) * (d / M)));
	} else {
		tmp = 0.25 * (t_0 * ((D / d) * (D / d)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = h * (M * M)
	tmp = 0
	if D <= 2.3e-184:
		tmp = 0.25 * (D / (((d * d) / t_0) / D))
	elif D <= 4e+75:
		tmp = 0.25 * ((D * D) / ((d / (h * M)) * (d / M)))
	else:
		tmp = 0.25 * (t_0 * ((D / d) * (D / d)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(h * Float64(M * M))
	tmp = 0.0
	if (D <= 2.3e-184)
		tmp = Float64(0.25 * Float64(D / Float64(Float64(Float64(d * d) / t_0) / D)));
	elseif (D <= 4e+75)
		tmp = Float64(0.25 * Float64(Float64(D * D) / Float64(Float64(d / Float64(h * M)) * Float64(d / M))));
	else
		tmp = Float64(0.25 * Float64(t_0 * Float64(Float64(D / d) * Float64(D / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = h * (M * M);
	tmp = 0.0;
	if (D <= 2.3e-184)
		tmp = 0.25 * (D / (((d * d) / t_0) / D));
	elseif (D <= 4e+75)
		tmp = 0.25 * ((D * D) / ((d / (h * M)) * (d / M)));
	else
		tmp = 0.25 * (t_0 * ((D / d) * (D / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[D, 2.3e-184], N[(0.25 * N[(D / N[(N[(N[(d * d), $MachinePrecision] / t$95$0), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[D, 4e+75], N[(0.25 * N[(N[(D * D), $MachinePrecision] / N[(N[(d / N[(h * M), $MachinePrecision]), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(t$95$0 * N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if D < 2.2999999999999999e-184

   1. Initial program 24.8%

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

      1. times-frac 23.0%

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

      2. fma-def 23.0%

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

      3. associate-/r* 23.1%

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

      4. difference-of-squares 32.0%

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

   3. Simplified 35.0%

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

      1. fma-udef 36.2%

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

      2. associate-/l/ 33.8%

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

      3. times-frac 32.5%

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

      4. fma-udef 32.5%

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

      5. associate-/l/ 32.4%

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

      6. times-frac 32.4%

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

      7. associate-/l/ 32.4%

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

      8. times-frac 33.7%

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

   5. Applied egg-rr 3.8%

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

      1. unpow2 3.8%

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

      2. associate--r- 6.7%

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

      3. +-inverses 27.9%

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

      4. unpow2 27.9%

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

   7. Simplified 27.9%

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

   8. Taylor expanded in c0 around -inf 20.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{0 + M \cdot M}{\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2} - \color{blue}{\left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left(h \cdot {M}^{2}\right)\right)}{{d}^{2} \cdot c0} + -1 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}} \]
   9. Step-by-step derivation

      1. mul-1-neg 20.0%

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

      2. unsub-neg 20.0%

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

      3. associate-*r/ 20.0%

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

      4. associate-*r* 20.6%

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

      5. unpow2 20.6%

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

      6. unpow2 20.6%

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

      7. unpow2 20.6%

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

      8. associate-*r* 20.6%

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

      9. times-frac 20.6%

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

      10. unpow2 20.6%

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

      11. unpow2 20.6%

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

      12. times-frac 21.2%

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

      13. unpow2 21.2%

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

      14. *-commutative 21.2%

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

   10. Simplified 21.3%

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

   11. Taylor expanded in c0 around inf 35.1%

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

       1. unpow2 35.1%

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

       2. *-commutative 35.1%

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

       3. unpow2 35.1%

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

       4. associate-/l* 35.3%

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

       5. unpow2 35.3%

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

       6. unpow2 35.3%

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

       7. unpow2 35.3%

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

       8. *-commutative 35.3%

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

   13. Simplified 35.3%

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

       1. *-un-lft-identity 35.3%

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

       2. associate-/l* 44.6%

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

   15. Applied egg-rr 44.6%

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

   if 2.2999999999999999e-184 < D < 3.99999999999999971e75

   1. Initial program 35.5%

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

      1. times-frac 35.6%

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

      2. fma-def 34.1%

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

      3. associate-/r* 34.1%

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

      4. difference-of-squares 43.4%

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

   3. Simplified 41.9%

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

      1. fma-udef 43.3%

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

      2. associate-/l/ 43.3%

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

      3. times-frac 43.4%

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

      4. fma-udef 43.4%

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

      5. associate-/l/ 43.4%

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

      6. times-frac 43.4%

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

      7. associate-/l/ 43.4%

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

      8. times-frac 44.8%

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

   5. Applied egg-rr 3.1%

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

      1. unpow2 3.1%

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

      2. associate--r- 4.7%

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

      3. +-inverses 14.2%

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

      4. unpow2 14.2%

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

   7. Simplified 14.2%

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

   8. Taylor expanded in c0 around -inf 16.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{0 + M \cdot M}{\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2} - \color{blue}{\left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left(h \cdot {M}^{2}\right)\right)}{{d}^{2} \cdot c0} + -1 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}} \]
   9. Step-by-step derivation

      1. mul-1-neg 16.1%

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

      2. unsub-neg 16.1%

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

      3. associate-*r/ 16.1%

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

      4. associate-*r* 16.1%

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

      5. unpow2 16.1%

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

      6. unpow2 16.1%

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

      7. unpow2 16.1%

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

      8. associate-*r* 16.1%

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

      9. times-frac 13.0%

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

      10. unpow2 13.0%

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

      11. unpow2 13.0%

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

      12. times-frac 13.1%

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

      13. unpow2 13.1%

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

      14. *-commutative 13.1%

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

   10. Simplified 14.6%

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

   11. Taylor expanded in c0 around inf 39.5%

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

       1. unpow2 39.5%

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

       2. *-commutative 39.5%

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

       3. unpow2 39.5%

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

       4. associate-/l* 39.6%

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

       5. unpow2 39.6%

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

       6. unpow2 39.6%

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

       7. unpow2 39.6%

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

       8. *-commutative 39.6%

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

   13. Simplified 39.6%

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

   14. Taylor expanded in d around 0 39.6%

       \[\leadsto 0.25 \cdot \frac{D \cdot D}{\color{blue}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]
   15. Step-by-step derivation

       1. unpow2 39.6%

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

       2. unpow2 39.6%

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

       3. associate-*r* 45.6%

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

       4. times-frac 53.2%

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

   16. Simplified 53.2%

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

   if 3.99999999999999971e75 < D

   1. Initial program 14.0%

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

      1. times-frac 13.9%

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

      2. fma-def 13.9%

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

      3. associate-/r* 13.9%

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

      4. difference-of-squares 18.5%

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

   3. Simplified 23.2%

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

      1. fma-udef 23.1%

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

      2. associate-/l/ 23.1%

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

      3. times-frac 18.5%

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

      4. fma-udef 18.5%

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

      5. associate-/l/ 18.5%

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

      6. times-frac 18.4%

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

      7. associate-/l/ 18.4%

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

      8. times-frac 18.6%

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

   5. Applied egg-rr 4.7%

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

      1. unpow2 4.7%

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

      2. associate--r- 9.0%

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

      3. +-inverses 18.5%

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

      4. unpow2 18.5%

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

   7. Simplified 18.5%

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

   8. Taylor expanded in c0 around -inf 9.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{0 + M \cdot M}{\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2} - \color{blue}{\left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left(h \cdot {M}^{2}\right)\right)}{{d}^{2} \cdot c0} + -1 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}} \]
   9. Step-by-step derivation

      1. mul-1-neg 9.2%

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

      2. unsub-neg 9.2%

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

      3. associate-*r/ 9.2%

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

      4. associate-*r* 4.8%

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

      5. unpow2 4.8%

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

      6. unpow2 4.8%

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

      7. unpow2 4.8%

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

      8. associate-*r* 4.8%

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

      9. times-frac 9.3%

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

      10. unpow2 9.3%

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

      11. unpow2 9.3%

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

      12. times-frac 9.3%

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

      13. unpow2 9.3%

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

      14. *-commutative 9.3%

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

   10. Simplified 9.3%

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

   11. Taylor expanded in c0 around inf 23.4%

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

       1. unpow2 23.4%

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

       2. *-commutative 23.4%

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

       3. unpow2 23.4%

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

       4. associate-/l* 19.1%

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

       5. unpow2 19.1%

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

       6. unpow2 19.1%

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

       7. unpow2 19.1%

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

       8. *-commutative 19.1%

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

   13. Simplified 19.1%

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

   14. Taylor expanded in D around 0 23.4%

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

       1. *-commutative 23.4%

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

       2. unpow2 23.4%

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

       3. associate-*r/ 23.3%

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

       4. unpow2 23.3%

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

       5. unpow2 23.3%

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

       6. times-frac 55.1%

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

   16. Simplified 55.1%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 2.3 \cdot 10^{-184}:\\ \;\;\;\;0.25 \cdot \frac{D}{\frac{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}{D}}\\ \mathbf{elif}\;D \leq 4 \cdot 10^{+75}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot D}{\frac{d}{h \cdot M} \cdot \frac{d}{M}}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\\ \end{array} \]

Alternative 7: 42.4% accurate, 10.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = 0.25d0 * ((h * (m * m)) * ((d / d_1) * (d / d_1)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := 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. Initial program 26.6%

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

   1. times-frac 25.4%

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

   2. fma-def 25.0%

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

   3. associate-/r* 25.1%

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

   4. difference-of-squares 33.7%

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

3. Simplified 35.7%

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

   1. fma-udef 36.9%

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

   2. associate-/l/ 35.3%

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

   3. times-frac 34.1%

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

   4. fma-udef 34.1%

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

   5. associate-/l/ 34.0%

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

   6. times-frac 34.0%

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

   7. associate-/l/ 34.0%

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

   8. times-frac 35.2%

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

5. Applied egg-rr 3.7%

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

   1. unpow2 3.7%

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

   2. associate--r- 6.4%

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

   3. +-inverses 23.6%

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

   4. unpow2 23.6%

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

7. Simplified 23.6%

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

8. Taylor expanded in c0 around -inf 18.1%

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{0 + M \cdot M}{\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2} - \color{blue}{\left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left(h \cdot {M}^{2}\right)\right)}{{d}^{2} \cdot c0} + -1 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}} \]
9. Step-by-step derivation

   1. mul-1-neg 18.1%

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

   2. unsub-neg 18.1%

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

   3. associate-*r/ 18.1%

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

   4. associate-*r* 18.1%

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

   5. unpow2 18.1%

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

   6. unpow2 18.1%

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

   7. unpow2 18.1%

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

   8. associate-*r* 18.1%

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

   9. times-frac 17.7%

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

   10. unpow2 17.7%

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

   11. unpow2 17.7%

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

   12. times-frac 18.1%

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

   13. unpow2 18.1%

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

   14. *-commutative 18.1%

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

10. Simplified 18.6%

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

11. Taylor expanded in c0 around inf 35.2%

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

    1. unpow2 35.2%

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

    2. *-commutative 35.2%

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

    3. unpow2 35.2%

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

    4. associate-/l* 35.0%

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

    5. unpow2 35.0%

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

    6. unpow2 35.0%

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

    7. unpow2 35.0%

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

    8. *-commutative 35.0%

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

13. Simplified 35.0%

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

14. Taylor expanded in D around 0 35.2%

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

    1. *-commutative 35.2%

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

    2. unpow2 35.2%

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

    3. associate-*r/ 34.1%

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

    4. unpow2 34.1%

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

    5. unpow2 34.1%

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

    6. times-frac 43.3%

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

16. Simplified 43.3%

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

17. Final simplification 43.3%

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

Alternative 8: 33.9% accurate, 30.2× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{0}{w} \end{array} \]

FPCore

(FPCore (c0 w h D d M) :precision binary64 (* -0.5 (/ 0.0 w)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return -0.5 * (0.0 / w);
}

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 = (-0.5d0) * (0.0d0 / w)
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return -0.5 * (0.0 / w);
}

Python

def code(c0, w, h, D, d, M):
	return -0.5 * (0.0 / w)

Julia

function code(c0, w, h, D, d, M)
	return Float64(-0.5 * Float64(0.0 / w))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = -0.5 * (0.0 / w);
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(-0.5 * N[(0.0 / w), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 26.6%

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

   1. associate-*l* 25.8%

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

   2. difference-of-squares 34.4%

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

   3. associate-*l* 34.0%

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

   4. associate-*l* 35.5%

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

3. Simplified 35.5%

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

4. Taylor expanded in c0 around -inf 3.0%

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

   1. *-commutative 3.0%

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

   2. unpow2 3.0%

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

   3. distribute-rgt1-in 3.0%

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

   4. metadata-eval 3.0%

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

   5. mul0-lft 25.4%

      \[\leadsto -0.5 \cdot \frac{\left(c0 \cdot c0\right) \cdot \color{blue}{0}}{w} \]

6. Simplified 25.4%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{\left(c0 \cdot c0\right) \cdot 0}{w}} \]

7. Taylor expanded in c0 around 0 32.7%

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

8. Final simplification 32.7%

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

Reproduce

herbie shell --seed 2023250 
(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))))))