Henrywood and Agarwal, Equation (13)

Average Error: 59.6 → 21.9

Time: 36.5s

Precision: binary64

Cost: 13968

Math

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

↓

\[\begin{array}{l} t_0 := \frac{d}{M \cdot D}\\ t_1 := 0.25 \cdot \left(\frac{M \cdot h}{\frac{d}{D}} \cdot \left(D \cdot \frac{M}{d}\right)\right)\\ \mathbf{if}\;M \leq -9.8 \cdot 10^{+51}:\\ \;\;\;\;h \cdot \frac{c0 \cdot \frac{0.25}{c0}}{{t_0}^{2}}\\ \mathbf{elif}\;M \leq -1.75 \cdot 10^{-102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;M \leq 1.4 \cdot 10^{-214}:\\ \;\;\;\;\frac{1}{w} \cdot \left(\frac{w}{t_0 \cdot \frac{t_0}{h}} \cdot \left(\frac{0.5}{c0} \cdot \left(c0 \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;M \leq 5.1 \cdot 10^{-169}:\\ \;\;\;\;{\left(\frac{d}{D} \cdot \frac{c0}{w \cdot \sqrt{h}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :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))))))

↓

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ d (* M D)))
        (t_1 (* 0.25 (* (/ (* M h) (/ d D)) (* D (/ M d))))))
   (if (<= M -9.8e+51)
     (* h (/ (* c0 (/ 0.25 c0)) (pow t_0 2.0)))
     (if (<= M -1.75e-102)
       t_1
       (if (<= M 1.4e-214)
         (* (/ 1.0 w) (* (/ w (* t_0 (/ t_0 h))) (* (/ 0.5 c0) (* c0 0.5))))
         (if (<= M 5.1e-169)
           (pow (* (/ d D) (/ c0 (* w (sqrt h)))) 2.0)
           t_1))))))

C

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

↓

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = d / (M * D);
	double t_1 = 0.25 * (((M * h) / (d / D)) * (D * (M / d)));
	double tmp;
	if (M <= -9.8e+51) {
		tmp = h * ((c0 * (0.25 / c0)) / pow(t_0, 2.0));
	} else if (M <= -1.75e-102) {
		tmp = t_1;
	} else if (M <= 1.4e-214) {
		tmp = (1.0 / w) * ((w / (t_0 * (t_0 / h))) * ((0.5 / c0) * (c0 * 0.5)));
	} else if (M <= 5.1e-169) {
		tmp = pow(((d / D) * (c0 / (w * sqrt(h)))), 2.0);
	} else {
		tmp = t_1;
	}
	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
    code = (c0 / (2.0d0 * w)) * (((c0 * (d_1 * d_1)) / ((w * h) * (d * d))) + sqrt(((((c0 * (d_1 * d_1)) / ((w * h) * (d * d))) * ((c0 * (d_1 * d_1)) / ((w * h) * (d * d)))) - (m * m))))
end function

↓

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = d_1 / (m * d)
    t_1 = 0.25d0 * (((m * h) / (d_1 / d)) * (d * (m / d_1)))
    if (m <= (-9.8d+51)) then
        tmp = h * ((c0 * (0.25d0 / c0)) / (t_0 ** 2.0d0))
    else if (m <= (-1.75d-102)) then
        tmp = t_1
    else if (m <= 1.4d-214) then
        tmp = (1.0d0 / w) * ((w / (t_0 * (t_0 / h))) * ((0.5d0 / c0) * (c0 * 0.5d0)))
    else if (m <= 5.1d-169) then
        tmp = ((d_1 / d) * (c0 / (w * sqrt(h)))) ** 2.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

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

↓

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = d / (M * D);
	double t_1 = 0.25 * (((M * h) / (d / D)) * (D * (M / d)));
	double tmp;
	if (M <= -9.8e+51) {
		tmp = h * ((c0 * (0.25 / c0)) / Math.pow(t_0, 2.0));
	} else if (M <= -1.75e-102) {
		tmp = t_1;
	} else if (M <= 1.4e-214) {
		tmp = (1.0 / w) * ((w / (t_0 * (t_0 / h))) * ((0.5 / c0) * (c0 * 0.5)));
	} else if (M <= 5.1e-169) {
		tmp = Math.pow(((d / D) * (c0 / (w * Math.sqrt(h)))), 2.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

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

↓

def code(c0, w, h, D, d, M):
	t_0 = d / (M * D)
	t_1 = 0.25 * (((M * h) / (d / D)) * (D * (M / d)))
	tmp = 0
	if M <= -9.8e+51:
		tmp = h * ((c0 * (0.25 / c0)) / math.pow(t_0, 2.0))
	elif M <= -1.75e-102:
		tmp = t_1
	elif M <= 1.4e-214:
		tmp = (1.0 / w) * ((w / (t_0 * (t_0 / h))) * ((0.5 / c0) * (c0 * 0.5)))
	elif M <= 5.1e-169:
		tmp = math.pow(((d / D) * (c0 / (w * math.sqrt(h)))), 2.0)
	else:
		tmp = t_1
	return tmp

Julia

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

↓

function code(c0, w, h, D, d, M)
	t_0 = Float64(d / Float64(M * D))
	t_1 = Float64(0.25 * Float64(Float64(Float64(M * h) / Float64(d / D)) * Float64(D * Float64(M / d))))
	tmp = 0.0
	if (M <= -9.8e+51)
		tmp = Float64(h * Float64(Float64(c0 * Float64(0.25 / c0)) / (t_0 ^ 2.0)));
	elseif (M <= -1.75e-102)
		tmp = t_1;
	elseif (M <= 1.4e-214)
		tmp = Float64(Float64(1.0 / w) * Float64(Float64(w / Float64(t_0 * Float64(t_0 / h))) * Float64(Float64(0.5 / c0) * Float64(c0 * 0.5))));
	elseif (M <= 5.1e-169)
		tmp = Float64(Float64(d / D) * Float64(c0 / Float64(w * sqrt(h)))) ^ 2.0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = d / (M * D);
	t_1 = 0.25 * (((M * h) / (d / D)) * (D * (M / d)));
	tmp = 0.0;
	if (M <= -9.8e+51)
		tmp = h * ((c0 * (0.25 / c0)) / (t_0 ^ 2.0));
	elseif (M <= -1.75e-102)
		tmp = t_1;
	elseif (M <= 1.4e-214)
		tmp = (1.0 / w) * ((w / (t_0 * (t_0 / h))) * ((0.5 / c0) * (c0 * 0.5)));
	elseif (M <= 5.1e-169)
		tmp = ((d / D) * (c0 / (w * sqrt(h)))) ^ 2.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(d / N[(M * D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.25 * N[(N[(N[(M * h), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision] * N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, -9.8e+51], N[(h * N[(N[(c0 * N[(0.25 / c0), $MachinePrecision]), $MachinePrecision] / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[M, -1.75e-102], t$95$1, If[LessEqual[M, 1.4e-214], N[(N[(1.0 / w), $MachinePrecision] * N[(N[(w / N[(t$95$0 * N[(t$95$0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 / c0), $MachinePrecision] * N[(c0 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 5.1e-169], N[Power[N[(N[(d / D), $MachinePrecision] * N[(c0 / N[(w * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if M < -9.79999999999999967e51

   1. Initial program 63.8

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

   2. Simplified 63.8

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

      [Start] 63.8	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      times-frac [=>] 63.9	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      fma-neg [=>] 63.9	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\color{blue}{\mathsf{fma}\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, -M \cdot M\right)}}\right) \]
      times-frac [=>] 63.9	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\mathsf{fma}\left(\color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}}, \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, -M \cdot M\right)}\right) \]
      times-frac [=>] 63.8	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}, \color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}}, -M \cdot M\right)}\right) \]

   3. Taylor expanded in c0 around -inf 63.4

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

   4. Simplified 52.4

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

      [Start] 63.4	\[ \frac{c0}{2 \cdot w} \cdot \left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0} + -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right) \]
      fma-def [=>] 63.4	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0}, -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right)} \]

   5. Applied egg-rr 46.4

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

   6. Simplified 46.5

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

      [Start] 46.4	\[ \frac{\left(\frac{0.5}{c0} \cdot {\left(\frac{\sqrt{w \cdot h} \cdot M}{\frac{d}{D}}\right)}^{2}\right) \cdot \left(-c0\right)}{w \cdot -2} \]
      times-frac [=>] 47.2	\[ \color{blue}{\frac{\frac{0.5}{c0} \cdot {\left(\frac{\sqrt{w \cdot h} \cdot M}{\frac{d}{D}}\right)}^{2}}{w} \cdot \frac{-c0}{-2}} \]
      *-commutative [=>] 47.2	\[ \frac{\color{blue}{{\left(\frac{\sqrt{w \cdot h} \cdot M}{\frac{d}{D}}\right)}^{2} \cdot \frac{0.5}{c0}}}{w} \cdot \frac{-c0}{-2} \]
      associate-/l* [=>] 46.5	\[ \frac{{\color{blue}{\left(\frac{\sqrt{w \cdot h}}{\frac{\frac{d}{D}}{M}}\right)}}^{2} \cdot \frac{0.5}{c0}}{w} \cdot \frac{-c0}{-2} \]

   7. Applied egg-rr 28.6

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

   8. Simplified 24.9

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

      [Start] 28.6	\[ \frac{1}{\frac{w}{\frac{w \cdot h}{{\left(\frac{d}{D \cdot M}\right)}^{2}} \cdot \left(\frac{0.5}{c0} \cdot \left(c0 \cdot 0.5\right)\right)}} \]
      associate-/r/ [=>] 28.5	\[ \color{blue}{\frac{1}{w} \cdot \left(\frac{w \cdot h}{{\left(\frac{d}{D \cdot M}\right)}^{2}} \cdot \left(\frac{0.5}{c0} \cdot \left(c0 \cdot 0.5\right)\right)\right)} \]
      associate-/l* [=>] 24.9	\[ \frac{1}{w} \cdot \left(\color{blue}{\frac{w}{\frac{{\left(\frac{d}{D \cdot M}\right)}^{2}}{h}}} \cdot \left(\frac{0.5}{c0} \cdot \left(c0 \cdot 0.5\right)\right)\right) \]
      *-commutative [=>] 24.9	\[ \frac{1}{w} \cdot \left(\frac{w}{\frac{{\left(\frac{d}{D \cdot M}\right)}^{2}}{h}} \cdot \left(\frac{0.5}{c0} \cdot \color{blue}{\left(0.5 \cdot c0\right)}\right)\right) \]

   9. Applied egg-rr 39.6

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{w \cdot \left(c0 \cdot \frac{0.25}{c0}\right)}{w \cdot \frac{{\left(\frac{d}{D \cdot M}\right)}^{2}}{h}}\right)} - 1} \]

   10. Simplified 20.7

       \[\leadsto \color{blue}{h \cdot \frac{c0 \cdot \frac{0.25}{c0}}{{\left(\frac{d}{D \cdot M}\right)}^{2}}} \]
       Proof

       [Start] 39.6	\[ e^{\mathsf{log1p}\left(\frac{w \cdot \left(c0 \cdot \frac{0.25}{c0}\right)}{w \cdot \frac{{\left(\frac{d}{D \cdot M}\right)}^{2}}{h}}\right)} - 1 \]
       expm1-def [=>] 29.7	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{w \cdot \left(c0 \cdot \frac{0.25}{c0}\right)}{w \cdot \frac{{\left(\frac{d}{D \cdot M}\right)}^{2}}{h}}\right)\right)} \]
       expm1-log1p [=>] 26.0	\[ \color{blue}{\frac{w \cdot \left(c0 \cdot \frac{0.25}{c0}\right)}{w \cdot \frac{{\left(\frac{d}{D \cdot M}\right)}^{2}}{h}}} \]
       associate-/r* [=>] 20.7	\[ \color{blue}{\frac{\frac{w \cdot \left(c0 \cdot \frac{0.25}{c0}\right)}{w}}{\frac{{\left(\frac{d}{D \cdot M}\right)}^{2}}{h}}} \]
       *-rgt-identity [<=] 20.7	\[ \frac{\frac{\color{blue}{\left(w \cdot \left(c0 \cdot \frac{0.25}{c0}\right)\right) \cdot 1}}{w}}{\frac{{\left(\frac{d}{D \cdot M}\right)}^{2}}{h}} \]
       associate-*r/ [<=] 20.7	\[ \frac{\color{blue}{\left(w \cdot \left(c0 \cdot \frac{0.25}{c0}\right)\right) \cdot \frac{1}{w}}}{\frac{{\left(\frac{d}{D \cdot M}\right)}^{2}}{h}} \]
       associate-/r/ [=>] 20.7	\[ \color{blue}{\frac{\left(w \cdot \left(c0 \cdot \frac{0.25}{c0}\right)\right) \cdot \frac{1}{w}}{{\left(\frac{d}{D \cdot M}\right)}^{2}} \cdot h} \]
       *-commutative [=>] 20.7	\[ \color{blue}{h \cdot \frac{\left(w \cdot \left(c0 \cdot \frac{0.25}{c0}\right)\right) \cdot \frac{1}{w}}{{\left(\frac{d}{D \cdot M}\right)}^{2}}} \]
       *-commutative [=>] 20.7	\[ h \cdot \frac{\color{blue}{\frac{1}{w} \cdot \left(w \cdot \left(c0 \cdot \frac{0.25}{c0}\right)\right)}}{{\left(\frac{d}{D \cdot M}\right)}^{2}} \]
       associate-*r* [=>] 20.7	\[ h \cdot \frac{\color{blue}{\left(\frac{1}{w} \cdot w\right) \cdot \left(c0 \cdot \frac{0.25}{c0}\right)}}{{\left(\frac{d}{D \cdot M}\right)}^{2}} \]
       *-commutative [=>] 20.7	\[ h \cdot \frac{\color{blue}{\left(c0 \cdot \frac{0.25}{c0}\right) \cdot \left(\frac{1}{w} \cdot w\right)}}{{\left(\frac{d}{D \cdot M}\right)}^{2}} \]
       lft-mult-inverse [=>] 20.7	\[ h \cdot \frac{\left(c0 \cdot \frac{0.25}{c0}\right) \cdot \color{blue}{1}}{{\left(\frac{d}{D \cdot M}\right)}^{2}} \]
       *-rgt-identity [=>] 20.7	\[ h \cdot \frac{\color{blue}{c0 \cdot \frac{0.25}{c0}}}{{\left(\frac{d}{D \cdot M}\right)}^{2}} \]

   if -9.79999999999999967e51 < M < -1.74999999999999993e-102 or 5.09999999999999997e-169 < M

   1. Initial program 61.5

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

   2. Simplified 61.4

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

      [Start] 61.5	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      times-frac [=>] 61.9	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      fma-neg [=>] 61.9	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\color{blue}{\mathsf{fma}\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, -M \cdot M\right)}}\right) \]
      times-frac [=>] 61.9	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\mathsf{fma}\left(\color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}}, \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, -M \cdot M\right)}\right) \]
      times-frac [=>] 61.4	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}, \color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}}, -M \cdot M\right)}\right) \]

   3. Taylor expanded in c0 around -inf 61.1

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

   4. Simplified 38.0

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

      [Start] 61.1	\[ \frac{c0}{2 \cdot w} \cdot \left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0} + -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right) \]
      fma-def [=>] 61.1	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0}, -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right)} \]

   5. Applied egg-rr 33.1

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

   6. Taylor expanded in c0 around 0 35.8

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

   7. Simplified 26.4

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

      [Start] 35.8	\[ 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \]
      *-commutative [=>] 35.8	\[ 0.25 \cdot \frac{\color{blue}{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}}{{d}^{2}} \]
      associate-/l* [=>] 36.0	\[ 0.25 \cdot \color{blue}{\frac{{M}^{2} \cdot h}{\frac{{d}^{2}}{{D}^{2}}}} \]
      unpow2 [=>] 36.0	\[ 0.25 \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{\frac{{d}^{2}}{{D}^{2}}} \]
      associate-*l* [=>] 34.5	\[ 0.25 \cdot \frac{\color{blue}{M \cdot \left(M \cdot h\right)}}{\frac{{d}^{2}}{{D}^{2}}} \]
      unpow2 [=>] 34.5	\[ 0.25 \cdot \frac{M \cdot \left(M \cdot h\right)}{\frac{\color{blue}{d \cdot d}}{{D}^{2}}} \]
      unpow2 [=>] 34.5	\[ 0.25 \cdot \frac{M \cdot \left(M \cdot h\right)}{\frac{d \cdot d}{\color{blue}{D \cdot D}}} \]
      times-frac [=>] 26.4	\[ 0.25 \cdot \frac{M \cdot \left(M \cdot h\right)}{\color{blue}{\frac{d}{D} \cdot \frac{d}{D}}} \]
      unpow2 [<=] 26.4	\[ 0.25 \cdot \frac{M \cdot \left(M \cdot h\right)}{\color{blue}{{\left(\frac{d}{D}\right)}^{2}}} \]

   8. Applied egg-rr 20.5

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

   if -1.74999999999999993e-102 < M < 1.4000000000000001e-214

   1. Initial program 56.0

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

   2. Simplified 55.5

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

      [Start] 56.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      times-frac [=>] 58.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      fma-neg [=>] 58.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\color{blue}{\mathsf{fma}\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, -M \cdot M\right)}}\right) \]
      times-frac [=>] 57.7	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\mathsf{fma}\left(\color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}}, \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, -M \cdot M\right)}\right) \]
      times-frac [=>] 55.5	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}, \color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}}, -M \cdot M\right)}\right) \]

   3. Taylor expanded in c0 around -inf 57.9

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

   4. Simplified 27.5

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

      [Start] 57.9	\[ \frac{c0}{2 \cdot w} \cdot \left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0} + -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right) \]
      fma-def [=>] 57.9	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0}, -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right)} \]

   5. Applied egg-rr 41.9

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

   6. Simplified 42.0

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

      [Start] 41.9	\[ \frac{\left(\frac{0.5}{c0} \cdot {\left(\frac{\sqrt{w \cdot h} \cdot M}{\frac{d}{D}}\right)}^{2}\right) \cdot \left(-c0\right)}{w \cdot -2} \]
      times-frac [=>] 42.0	\[ \color{blue}{\frac{\frac{0.5}{c0} \cdot {\left(\frac{\sqrt{w \cdot h} \cdot M}{\frac{d}{D}}\right)}^{2}}{w} \cdot \frac{-c0}{-2}} \]
      *-commutative [=>] 42.0	\[ \frac{\color{blue}{{\left(\frac{\sqrt{w \cdot h} \cdot M}{\frac{d}{D}}\right)}^{2} \cdot \frac{0.5}{c0}}}{w} \cdot \frac{-c0}{-2} \]
      associate-/l* [=>] 42.0	\[ \frac{{\color{blue}{\left(\frac{\sqrt{w \cdot h}}{\frac{\frac{d}{D}}{M}}\right)}}^{2} \cdot \frac{0.5}{c0}}{w} \cdot \frac{-c0}{-2} \]

   7. Applied egg-rr 24.3

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

   8. Simplified 20.4

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

      [Start] 24.3	\[ \frac{1}{\frac{w}{\frac{w \cdot h}{{\left(\frac{d}{D \cdot M}\right)}^{2}} \cdot \left(\frac{0.5}{c0} \cdot \left(c0 \cdot 0.5\right)\right)}} \]
      associate-/r/ [=>] 24.3	\[ \color{blue}{\frac{1}{w} \cdot \left(\frac{w \cdot h}{{\left(\frac{d}{D \cdot M}\right)}^{2}} \cdot \left(\frac{0.5}{c0} \cdot \left(c0 \cdot 0.5\right)\right)\right)} \]
      associate-/l* [=>] 20.4	\[ \frac{1}{w} \cdot \left(\color{blue}{\frac{w}{\frac{{\left(\frac{d}{D \cdot M}\right)}^{2}}{h}}} \cdot \left(\frac{0.5}{c0} \cdot \left(c0 \cdot 0.5\right)\right)\right) \]
      *-commutative [=>] 20.4	\[ \frac{1}{w} \cdot \left(\frac{w}{\frac{{\left(\frac{d}{D \cdot M}\right)}^{2}}{h}} \cdot \left(\frac{0.5}{c0} \cdot \color{blue}{\left(0.5 \cdot c0\right)}\right)\right) \]

   9. Applied egg-rr 19.7

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

   if 1.4000000000000001e-214 < M < 5.09999999999999997e-169

   1. Initial program 56.9

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

   2. Simplified 56.2

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

      [Start] 56.9	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      times-frac [=>] 58.2	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      fma-neg [=>] 58.2	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\color{blue}{\mathsf{fma}\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, -M \cdot M\right)}}\right) \]
      times-frac [=>] 58.2	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\mathsf{fma}\left(\color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}}, \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, -M \cdot M\right)}\right) \]
      times-frac [=>] 56.2	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}, \color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}}, -M \cdot M\right)}\right) \]

   3. Taylor expanded in c0 around inf 58.2

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

   4. Simplified 55.4

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

      [Start] 58.2	\[ \frac{{d}^{2} \cdot {c0}^{2}}{{D}^{2} \cdot \left({w}^{2} \cdot h\right)} \]
      times-frac [=>] 59.1	\[ \color{blue}{\frac{{d}^{2}}{{D}^{2}} \cdot \frac{{c0}^{2}}{{w}^{2} \cdot h}} \]
      unpow2 [=>] 59.1	\[ \frac{\color{blue}{d \cdot d}}{{D}^{2}} \cdot \frac{{c0}^{2}}{{w}^{2} \cdot h} \]
      unpow2 [=>] 59.1	\[ \frac{d \cdot d}{\color{blue}{D \cdot D}} \cdot \frac{{c0}^{2}}{{w}^{2} \cdot h} \]
      times-frac [=>] 54.4	\[ \color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D}\right)} \cdot \frac{{c0}^{2}}{{w}^{2} \cdot h} \]
      unpow2 [<=] 54.4	\[ \color{blue}{{\left(\frac{d}{D}\right)}^{2}} \cdot \frac{{c0}^{2}}{{w}^{2} \cdot h} \]
      *-commutative [=>] 54.4	\[ {\left(\frac{d}{D}\right)}^{2} \cdot \frac{{c0}^{2}}{\color{blue}{h \cdot {w}^{2}}} \]
      associate-/r* [=>] 55.4	\[ {\left(\frac{d}{D}\right)}^{2} \cdot \color{blue}{\frac{\frac{{c0}^{2}}{h}}{{w}^{2}}} \]
      unpow2 [=>] 55.4	\[ {\left(\frac{d}{D}\right)}^{2} \cdot \frac{\frac{\color{blue}{c0 \cdot c0}}{h}}{{w}^{2}} \]
      unpow2 [=>] 55.4	\[ {\left(\frac{d}{D}\right)}^{2} \cdot \frac{\frac{c0 \cdot c0}{h}}{\color{blue}{w \cdot w}} \]

   5. Applied egg-rr 53.7

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

4. Final simplification 21.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -9.8 \cdot 10^{+51}:\\ \;\;\;\;h \cdot \frac{c0 \cdot \frac{0.25}{c0}}{{\left(\frac{d}{M \cdot D}\right)}^{2}}\\ \mathbf{elif}\;M \leq -1.75 \cdot 10^{-102}:\\ \;\;\;\;0.25 \cdot \left(\frac{M \cdot h}{\frac{d}{D}} \cdot \left(D \cdot \frac{M}{d}\right)\right)\\ \mathbf{elif}\;M \leq 1.4 \cdot 10^{-214}:\\ \;\;\;\;\frac{1}{w} \cdot \left(\frac{w}{\frac{d}{M \cdot D} \cdot \frac{\frac{d}{M \cdot D}}{h}} \cdot \left(\frac{0.5}{c0} \cdot \left(c0 \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;M \leq 5.1 \cdot 10^{-169}:\\ \;\;\;\;{\left(\frac{d}{D} \cdot \frac{c0}{w \cdot \sqrt{h}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{M \cdot h}{\frac{d}{D}} \cdot \left(D \cdot \frac{M}{d}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	20.4
Cost	1993

\[\begin{array}{l} t_0 := \frac{d}{M \cdot D}\\ \mathbf{if}\;c0 \leq -1.9 \cdot 10^{+209} \lor \neg \left(c0 \leq 7.2 \cdot 10^{+218}\right):\\ \;\;\;\;\frac{1}{w} \cdot \left(\frac{w}{t_0 \cdot \frac{t_0}{h}} \cdot \left(\frac{0.5}{c0} \cdot \left(c0 \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{M \cdot h}{\frac{d}{D}} \cdot \left(D \cdot \frac{M}{d}\right)\right)\\ \end{array} \]

Alternative 2
Error	28.2
Cost	1480

\[\begin{array}{l} \mathbf{if}\;M \cdot M \leq 2 \cdot 10^{-269}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \cdot M \leq 2 \cdot 10^{+298}:\\ \;\;\;\;0.25 \cdot \left(\left(\frac{h}{d} \cdot \left(M \cdot M\right)\right) \cdot \frac{D \cdot D}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 3
Error	28.3
Cost	1480

\[\begin{array}{l} \mathbf{if}\;M \cdot M \leq 2 \cdot 10^{-269}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \cdot M \leq 2 \cdot 10^{+298}:\\ \;\;\;\;0.25 \cdot \left(\left(\frac{h}{d} \cdot \frac{M \cdot M}{d}\right) \cdot \left(D \cdot D\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 4
Error	21.0
Cost	1225

\[\begin{array}{l} \mathbf{if}\;w \leq -2.55 \cdot 10^{-84} \lor \neg \left(w \leq -1.2 \cdot 10^{-195}\right):\\ \;\;\;\;0.25 \cdot \left(\frac{M \cdot h}{\frac{d}{D}} \cdot \left(D \cdot \frac{M}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d} \cdot \frac{h}{d}\right)\\ \end{array} \]

Alternative 5
Error	25.2
Cost	1220

\[\begin{array}{l} \mathbf{if}\;M \cdot M \leq 2 \cdot 10^{+298}:\\ \;\;\;\;0.25 \cdot \left(\frac{M \cdot M}{\frac{d}{D}} \cdot \left(D \cdot \frac{h}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6
Error	20.6
Cost	960

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

Alternative 7
Error	31.6
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023039 
(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))))))