Henrywood and Agarwal, Equation (13)

Average Error: 59.6 → 17.2

Time: 50.4s

Precision: binary64

Cost: 30540

Math

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

↓

\[\begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_2 := t_0 \cdot \left(t_1 + \sqrt{t_1 \cdot t_1 - M \cdot M}\right)\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{-214}:\\ \;\;\;\;d \cdot \frac{\frac{\frac{c0}{w}}{\frac{D}{\frac{c0}{\frac{D}{d}}}}}{w \cdot h}\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;0.25 \cdot \frac{\frac{D}{d} \cdot \left(h \cdot M\right)}{\frac{\frac{d}{D}}{M}}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+142}:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \left(c0 \cdot \frac{\frac{d \cdot d}{D}}{w \cdot \left(h \cdot D\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{h \cdot \left(D \cdot M\right)}{d} \cdot \frac{D \cdot M}{d}\right)\\ \end{array} \]

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 (/ c0 (* 2.0 w)))
        (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_2 (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))))
   (if (<= t_2 -5e-214)
     (* d (/ (/ (/ c0 w) (/ D (/ c0 (/ D d)))) (* w h)))
     (if (<= t_2 0.0)
       (* 0.25 (/ (* (/ D d) (* h M)) (/ (/ d D) M)))
       (if (<= t_2 2e+142)
         (* t_0 (* 2.0 (* c0 (/ (/ (* d d) D) (* w (* h D))))))
         (* 0.25 (* (/ (* h (* D M)) d) (/ (* D M) d))))))))

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 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_2 = t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= -5e-214) {
		tmp = d * (((c0 / w) / (D / (c0 / (D / d)))) / (w * h));
	} else if (t_2 <= 0.0) {
		tmp = 0.25 * (((D / d) * (h * M)) / ((d / D) / M));
	} else if (t_2 <= 2e+142) {
		tmp = t_0 * (2.0 * (c0 * (((d * d) / D) / (w * (h * D)))));
	} else {
		tmp = 0.25 * (((h * (D * M)) / d) * ((D * M) / d));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    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) :: t_2
    real(8) :: tmp
    t_0 = c0 / (2.0d0 * w)
    t_1 = (c0 * (d_1 * d_1)) / ((w * h) * (d * d))
    t_2 = t_0 * (t_1 + sqrt(((t_1 * t_1) - (m * m))))
    if (t_2 <= (-5d-214)) then
        tmp = d_1 * (((c0 / w) / (d / (c0 / (d / d_1)))) / (w * h))
    else if (t_2 <= 0.0d0) then
        tmp = 0.25d0 * (((d / d_1) * (h * m)) / ((d_1 / d) / m))
    else if (t_2 <= 2d+142) then
        tmp = t_0 * (2.0d0 * (c0 * (((d_1 * d_1) / d) / (w * (h * d)))))
    else
        tmp = 0.25d0 * (((h * (d * m)) / d_1) * ((d * m) / d_1))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	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 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_2 = t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= -5e-214) {
		tmp = d * (((c0 / w) / (D / (c0 / (D / d)))) / (w * h));
	} else if (t_2 <= 0.0) {
		tmp = 0.25 * (((D / d) * (h * M)) / ((d / D) / M));
	} else if (t_2 <= 2e+142) {
		tmp = t_0 * (2.0 * (c0 * (((d * d) / D) / (w * (h * D)))));
	} else {
		tmp = 0.25 * (((h * (D * M)) / d) * ((D * M) / d));
	}
	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 = c0 / (2.0 * w)
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	t_2 = t_0 * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))
	tmp = 0
	if t_2 <= -5e-214:
		tmp = d * (((c0 / w) / (D / (c0 / (D / d)))) / (w * h))
	elif t_2 <= 0.0:
		tmp = 0.25 * (((D / d) * (h * M)) / ((d / D) / M))
	elif t_2 <= 2e+142:
		tmp = t_0 * (2.0 * (c0 * (((d * d) / D) / (w * (h * D)))))
	else:
		tmp = 0.25 * (((h * (D * M)) / d) * ((D * M) / d))
	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(c0 / Float64(2.0 * w))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	t_2 = Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M)))))
	tmp = 0.0
	if (t_2 <= -5e-214)
		tmp = Float64(d * Float64(Float64(Float64(c0 / w) / Float64(D / Float64(c0 / Float64(D / d)))) / Float64(w * h)));
	elseif (t_2 <= 0.0)
		tmp = Float64(0.25 * Float64(Float64(Float64(D / d) * Float64(h * M)) / Float64(Float64(d / D) / M)));
	elseif (t_2 <= 2e+142)
		tmp = Float64(t_0 * Float64(2.0 * Float64(c0 * Float64(Float64(Float64(d * d) / D) / Float64(w * Float64(h * D))))));
	else
		tmp = Float64(0.25 * Float64(Float64(Float64(h * Float64(D * M)) / d) * Float64(Float64(D * M) / d)));
	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 = c0 / (2.0 * w);
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	t_2 = t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	tmp = 0.0;
	if (t_2 <= -5e-214)
		tmp = d * (((c0 / w) / (D / (c0 / (D / d)))) / (w * h));
	elseif (t_2 <= 0.0)
		tmp = 0.25 * (((D / d) * (h * M)) / ((d / D) / M));
	elseif (t_2 <= 2e+142)
		tmp = t_0 * (2.0 * (c0 * (((d * d) / D) / (w * (h * D)))));
	else
		tmp = 0.25 * (((h * (D * M)) / d) * ((D * M) / d));
	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[(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]}, Block[{t$95$2 = N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-214], N[(d * N[(N[(N[(c0 / w), $MachinePrecision] / N[(D / N[(c0 / N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(0.25 * N[(N[(N[(D / d), $MachinePrecision] * N[(h * M), $MachinePrecision]), $MachinePrecision] / N[(N[(d / D), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+142], N[(t$95$0 * N[(2.0 * N[(c0 * N[(N[(N[(d * d), $MachinePrecision] / D), $MachinePrecision] / N[(w * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(N[(h * N[(D * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < -4.9999999999999998e-214

   1. Initial program 49.4

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

   2. Taylor expanded in c0 around inf 45.2

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

   3. Simplified 45.6

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

      [Start] 45.2	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]
      associate-*r/ [=>] 45.2	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left({d}^{2} \cdot c0\right)}{{D}^{2} \cdot \left(w \cdot h\right)}} \]
      *-commutative [=>] 45.2	\[ \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left({d}^{2} \cdot c0\right)}{\color{blue}{\left(w \cdot h\right) \cdot {D}^{2}}} \]
      unpow2 [=>] 45.2	\[ \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left({d}^{2} \cdot c0\right)}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} \]
      *-commutative [=>] 45.2	\[ \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \color{blue}{\left(c0 \cdot {d}^{2}\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \]
      unpow2 [=>] 45.2	\[ \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \color{blue}{\left(d \cdot d\right)}\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \]
      associate-*r* [=>] 40.7	\[ \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} \]
      associate-*r* [<=] 41.1	\[ \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{\left(w \cdot \left(h \cdot D\right)\right)} \cdot D} \]
      *-commutative [<=] 41.1	\[ \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}} \]
      associate-*r/ [<=] 41.1	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right)} \]
      associate-*r/ [<=] 43.0	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(c0 \cdot \frac{d \cdot d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right)}\right) \]
      *-commutative [=>] 43.0	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(c0 \cdot \frac{d \cdot d}{\color{blue}{\left(w \cdot \left(h \cdot D\right)\right) \cdot D}}\right)\right) \]
      associate-*r* [=>] 42.5	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(c0 \cdot \frac{d \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right)} \cdot D}\right)\right) \]
      associate-*r* [<=] 46.8	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(c0 \cdot \frac{d \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}}\right)\right) \]
      associate-/l/ [<=] 48.0	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(c0 \cdot \color{blue}{\frac{\frac{d \cdot d}{D \cdot D}}{w \cdot h}}\right)\right) \]
      associate-/r* [<=] 46.8	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(c0 \cdot \color{blue}{\frac{d \cdot d}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}\right)\right) \]
      times-frac [=>] 45.0	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(c0 \cdot \color{blue}{\left(\frac{d}{D \cdot D} \cdot \frac{d}{w \cdot h}\right)}\right)\right) \]
      associate-/r* [=>] 45.6	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(c0 \cdot \left(\frac{d}{D \cdot D} \cdot \color{blue}{\frac{\frac{d}{w}}{h}}\right)\right)\right) \]

   4. Applied egg-rr 39.7

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

   5. Applied egg-rr 61.5

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

   6. Simplified 35.8

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

      [Start] 61.5	\[ e^{\mathsf{log1p}\left(\left(\frac{c0}{D} \cdot \left(\frac{\frac{d}{D}}{w \cdot h} \cdot d\right)\right) \cdot \left(1 \cdot \frac{c0}{w}\right)\right)} - 1 \]
      expm1-def [=>] 54.0	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{c0}{D} \cdot \left(\frac{\frac{d}{D}}{w \cdot h} \cdot d\right)\right) \cdot \left(1 \cdot \frac{c0}{w}\right)\right)\right)} \]
      expm1-log1p [=>] 36.1	\[ \color{blue}{\left(\frac{c0}{D} \cdot \left(\frac{\frac{d}{D}}{w \cdot h} \cdot d\right)\right) \cdot \left(1 \cdot \frac{c0}{w}\right)} \]
      associate-/r/ [<=] 37.8	\[ \color{blue}{\frac{c0}{\frac{D}{\frac{\frac{d}{D}}{w \cdot h} \cdot d}}} \cdot \left(1 \cdot \frac{c0}{w}\right) \]
      associate-*l/ [=>] 40.3	\[ \color{blue}{\frac{c0 \cdot \left(1 \cdot \frac{c0}{w}\right)}{\frac{D}{\frac{\frac{d}{D}}{w \cdot h} \cdot d}}} \]
      *-commutative [<=] 40.3	\[ \frac{\color{blue}{\left(1 \cdot \frac{c0}{w}\right) \cdot c0}}{\frac{D}{\frac{\frac{d}{D}}{w \cdot h} \cdot d}} \]
      associate-/r/ [<=] 41.0	\[ \frac{\left(1 \cdot \frac{c0}{w}\right) \cdot c0}{\frac{D}{\color{blue}{\frac{\frac{d}{D}}{\frac{w \cdot h}{d}}}}} \]
      associate-/l* [<=] 40.4	\[ \frac{\left(1 \cdot \frac{c0}{w}\right) \cdot c0}{\color{blue}{\frac{D \cdot \frac{w \cdot h}{d}}{\frac{d}{D}}}} \]
      associate-/l* [<=] 34.9	\[ \color{blue}{\frac{\left(\left(1 \cdot \frac{c0}{w}\right) \cdot c0\right) \cdot \frac{d}{D}}{D \cdot \frac{w \cdot h}{d}}} \]
      associate-*r* [<=] 31.1	\[ \frac{\color{blue}{\left(1 \cdot \frac{c0}{w}\right) \cdot \left(c0 \cdot \frac{d}{D}\right)}}{D \cdot \frac{w \cdot h}{d}} \]
      associate-/r* [=>] 34.1	\[ \color{blue}{\frac{\frac{\left(1 \cdot \frac{c0}{w}\right) \cdot \left(c0 \cdot \frac{d}{D}\right)}{D}}{\frac{w \cdot h}{d}}} \]
      associate-/r/ [=>] 33.3	\[ \color{blue}{\frac{\frac{\left(1 \cdot \frac{c0}{w}\right) \cdot \left(c0 \cdot \frac{d}{D}\right)}{D}}{w \cdot h} \cdot d} \]

   if -4.9999999999999998e-214 < (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < 0.0

   1. Initial program 28.9

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

   2. Taylor expanded in c0 around -inf 30.9

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

   3. Simplified 27.7

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

      [Start] 30.9	\[ \frac{c0}{2 \cdot w} \cdot \left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0} + -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right) \]
      fma-def [=>] 30.9	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0}, -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right)} \]

   4. Taylor expanded in c0 around 0 24.8

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

   5. Simplified 23.0

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

      [Start] 24.8	\[ 0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}} \]
      *-commutative [=>] 24.8	\[ 0.25 \cdot \frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{{d}^{2}} \]
      associate-/l* [=>] 24.2	\[ 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{{M}^{2} \cdot h}}} \]
      unpow2 [=>] 24.2	\[ 0.25 \cdot \frac{\color{blue}{D \cdot D}}{\frac{{d}^{2}}{{M}^{2} \cdot h}} \]
      unpow2 [=>] 24.2	\[ 0.25 \cdot \frac{D \cdot D}{\frac{\color{blue}{d \cdot d}}{{M}^{2} \cdot h}} \]
      unpow2 [=>] 24.2	\[ 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{\color{blue}{\left(M \cdot M\right)} \cdot h}} \]
      associate-*r* [<=] 21.4	\[ 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{\color{blue}{M \cdot \left(M \cdot h\right)}}} \]
      associate-/r/ [=>] 22.0	\[ 0.25 \cdot \color{blue}{\left(\frac{D \cdot D}{d \cdot d} \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)} \]
      times-frac [=>] 19.2	\[ 0.25 \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \left(M \cdot \left(M \cdot h\right)\right)\right) \]
      *-commutative [=>] 19.2	\[ 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \color{blue}{\left(\left(M \cdot h\right) \cdot M\right)}\right) \]
      *-commutative [=>] 19.2	\[ 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\color{blue}{\left(h \cdot M\right)} \cdot M\right)\right) \]
      associate-*r* [<=] 23.0	\[ 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \color{blue}{\left(h \cdot \left(M \cdot M\right)\right)}\right) \]

   6. Applied egg-rr 13.0

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

   if 0.0 < (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < 2.0000000000000001e142

   1. Initial program 11.9

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

   2. Applied egg-rr 26.7

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \color{blue}{\sqrt[3]{{\left({\left(\frac{c0}{w \cdot \left(h \cdot D\right)} \cdot \frac{d \cdot d}{D}\right)}^{2} - M \cdot M\right)}^{1.5}}}\right) \]

   3. Taylor expanded in c0 around inf 6.5

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

   4. Simplified 14.8

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

      [Start] 6.5	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]
      associate-/l* [=>] 10.0	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{{d}^{2}}{\frac{{D}^{2} \cdot \left(w \cdot h\right)}{c0}}}\right) \]
      associate-/r/ [=>] 13.3	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} \cdot c0\right)}\right) \]
      associate-/r* [=>] 17.0	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\color{blue}{\frac{\frac{{d}^{2}}{{D}^{2}}}{w \cdot h}} \cdot c0\right)\right) \]
      associate-/l/ [<=] 21.3	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\color{blue}{\frac{\frac{\frac{{d}^{2}}{{D}^{2}}}{h}}{w}} \cdot c0\right)\right) \]
      unpow2 [=>] 21.3	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\frac{\frac{{d}^{2}}{\color{blue}{D \cdot D}}}{h}}{w} \cdot c0\right)\right) \]
      associate-/r* [=>] 21.0	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\frac{\color{blue}{\frac{\frac{{d}^{2}}{D}}{D}}}{h}}{w} \cdot c0\right)\right) \]
      unpow2 [=>] 21.0	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\frac{\frac{\frac{\color{blue}{d \cdot d}}{D}}{D}}{h}}{w} \cdot c0\right)\right) \]
      associate-*l/ [<=] 20.9	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\frac{\frac{\color{blue}{\frac{d}{D} \cdot d}}{D}}{h}}{w} \cdot c0\right)\right) \]
      associate-/r/ [<=] 20.8	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\frac{\frac{\color{blue}{\frac{d}{\frac{D}{d}}}}{D}}{h}}{w} \cdot c0\right)\right) \]
      associate-/l/ [=>] 19.7	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\color{blue}{\frac{\frac{d}{\frac{D}{d}}}{h \cdot D}}}{w} \cdot c0\right)\right) \]
      associate-/l/ [=>] 14.7	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\color{blue}{\frac{\frac{d}{\frac{D}{d}}}{w \cdot \left(h \cdot D\right)}} \cdot c0\right)\right) \]
      associate-/r/ [=>] 14.6	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\color{blue}{\frac{d}{D} \cdot d}}{w \cdot \left(h \cdot D\right)} \cdot c0\right)\right) \]
      associate-*l/ [=>] 14.8	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\color{blue}{\frac{d \cdot d}{D}}}{w \cdot \left(h \cdot D\right)} \cdot c0\right)\right) \]
      *-commutative [=>] 14.8	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\frac{d \cdot d}{D}}{w \cdot \color{blue}{\left(D \cdot h\right)}} \cdot c0\right)\right) \]

   if 2.0000000000000001e142 < (*.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 63.7

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

   2. Taylor expanded in c0 around -inf 62.6

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

   3. Simplified 41.2

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

      [Start] 62.6	\[ \frac{c0}{2 \cdot w} \cdot \left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0} + -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right) \]
      fma-def [=>] 62.6	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0}, -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right)} \]

   4. Taylor expanded in c0 around 0 34.9

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

   5. Simplified 25.9

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

      [Start] 34.9	\[ 0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}} \]
      *-commutative [=>] 34.9	\[ 0.25 \cdot \frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{{d}^{2}} \]
      associate-*r* [=>] 34.3	\[ 0.25 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2}} \]
      *-commutative [=>] 34.3	\[ 0.25 \cdot \frac{\color{blue}{h \cdot \left({D}^{2} \cdot {M}^{2}\right)}}{{d}^{2}} \]
      unpow2 [=>] 34.3	\[ 0.25 \cdot \frac{h \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right)}{{d}^{2}} \]
      unpow2 [=>] 34.3	\[ 0.25 \cdot \frac{h \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right)}{{d}^{2}} \]
      unswap-sqr [=>] 25.9	\[ 0.25 \cdot \frac{h \cdot \color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)}}{{d}^{2}} \]
      unpow2 [=>] 25.9	\[ 0.25 \cdot \frac{h \cdot \left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)}{\color{blue}{d \cdot d}} \]

   6. Applied egg-rr 16.2

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

4. Final simplification 17.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq -5 \cdot 10^{-214}:\\ \;\;\;\;d \cdot \frac{\frac{\frac{c0}{w}}{\frac{D}{\frac{c0}{\frac{D}{d}}}}}{w \cdot h}\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 0:\\ \;\;\;\;0.25 \cdot \frac{\frac{D}{d} \cdot \left(h \cdot M\right)}{\frac{\frac{d}{D}}{M}}\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 2 \cdot 10^{+142}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(c0 \cdot \frac{\frac{d \cdot d}{D}}{w \cdot \left(h \cdot D\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{h \cdot \left(D \cdot M\right)}{d} \cdot \frac{D \cdot M}{d}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	18.4
Cost	1864

\[\begin{array}{l} \mathbf{if}\;D \cdot D \leq 0.05:\\ \;\;\;\;0.25 \cdot \left(\frac{h \cdot \left(D \cdot M\right)}{d} \cdot \frac{D \cdot M}{d}\right)\\ \mathbf{elif}\;D \cdot D \leq 2 \cdot 10^{+16}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(c0 \cdot \left(\frac{d}{D \cdot D} \cdot \frac{\frac{d}{w}}{h}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{M \cdot \left(h \cdot \frac{D}{d}\right)}{\frac{\frac{d}{D}}{M}}\\ \end{array} \]

Alternative 2
Error	18.4
Cost	1608

\[\begin{array}{l} \mathbf{if}\;D \cdot D \leq 0.05:\\ \;\;\;\;0.25 \cdot \left(\frac{h \cdot \left(D \cdot M\right)}{d} \cdot \frac{D \cdot M}{d}\right)\\ \mathbf{elif}\;D \cdot D \leq 2 \cdot 10^{+16}:\\ \;\;\;\;c0 \cdot \frac{\left(\frac{d}{D} \cdot \frac{d}{w}\right) \cdot \frac{\frac{c0}{D}}{h}}{w}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{M \cdot \left(h \cdot \frac{D}{d}\right)}{\frac{\frac{d}{D}}{M}}\\ \end{array} \]

Alternative 3
Error	18.4
Cost	1608

\[\begin{array}{l} \mathbf{if}\;D \cdot D \leq 0.05:\\ \;\;\;\;0.25 \cdot \left(\frac{h \cdot \left(D \cdot M\right)}{d} \cdot \frac{D \cdot M}{d}\right)\\ \mathbf{elif}\;D \cdot D \leq 2 \cdot 10^{+16}:\\ \;\;\;\;\frac{c0}{w} \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{w}\right) \cdot \frac{c0}{h \cdot D}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{M \cdot \left(h \cdot \frac{D}{d}\right)}{\frac{\frac{d}{D}}{M}}\\ \end{array} \]

Alternative 4
Error	18.4
Cost	1608

\[\begin{array}{l} \mathbf{if}\;D \cdot D \leq 0.05:\\ \;\;\;\;0.25 \cdot \left(\frac{h \cdot \left(D \cdot M\right)}{d} \cdot \frac{D \cdot M}{d}\right)\\ \mathbf{elif}\;D \cdot D \leq 2 \cdot 10^{+16}:\\ \;\;\;\;\frac{c0}{D} \cdot \left(\frac{\frac{\frac{d}{w}}{h}}{D} \cdot \left(d \cdot \frac{c0}{w}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{M \cdot \left(h \cdot \frac{D}{d}\right)}{\frac{\frac{d}{D}}{M}}\\ \end{array} \]

Alternative 5
Error	18.4
Cost	1608

\[\begin{array}{l} \mathbf{if}\;D \cdot D \leq 0.05:\\ \;\;\;\;0.25 \cdot \left(\frac{h \cdot \left(D \cdot M\right)}{d} \cdot \frac{D \cdot M}{d}\right)\\ \mathbf{elif}\;D \cdot D \leq 2 \cdot 10^{+16}:\\ \;\;\;\;\frac{c0}{\frac{h \cdot \frac{w}{d}}{\frac{d}{D \cdot D}} \cdot \frac{w}{c0}}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{M \cdot \left(h \cdot \frac{D}{d}\right)}{\frac{\frac{d}{D}}{M}}\\ \end{array} \]

Alternative 6
Error	18.7
Cost	1481

\[\begin{array}{l} \mathbf{if}\;d \cdot d \leq 10^{-262} \lor \neg \left(d \cdot d \leq 5 \cdot 10^{+302}\right):\\ \;\;\;\;0.25 \cdot \left(\left(h \cdot M\right) \cdot \left(\frac{D}{d} \cdot \left(M \cdot \frac{D}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot M\right) \cdot \frac{h}{\frac{d \cdot d}{D \cdot M}}\right)\\ \end{array} \]

Alternative 7
Error	23.6
Cost	1225

\[\begin{array}{l} \mathbf{if}\;M \leq -1.2 \cdot 10^{+135} \lor \neg \left(M \leq 1.1 \cdot 10^{+199}\right):\\ \;\;\;\;0.25 \cdot \left(\left(h \cdot M\right) \cdot \left(\frac{D}{d} \cdot \left(M \cdot \frac{D}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(h \cdot \frac{\frac{D}{d}}{d}\right) \cdot \left(M \cdot \left(D \cdot M\right)\right)\right)\\ \end{array} \]

Alternative 8
Error	17.3
Cost	1220

\[\begin{array}{l} \mathbf{if}\;M \cdot M \leq 2 \cdot 10^{+72}:\\ \;\;\;\;0.25 \cdot \frac{M \cdot \frac{h}{\frac{d}{D}}}{\frac{\frac{d}{D}}{M}}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{h \cdot \left(D \cdot M\right)}{d} \cdot \frac{D \cdot M}{d}\right)\\ \end{array} \]

Alternative 9
Error	17.3
Cost	1220

\[\begin{array}{l} \mathbf{if}\;M \cdot M \leq 2 \cdot 10^{+72}:\\ \;\;\;\;0.25 \cdot \frac{M \cdot \left(h \cdot \frac{D}{d}\right)}{\frac{\frac{d}{D}}{M}}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{h \cdot \left(D \cdot M\right)}{d} \cdot \frac{D \cdot M}{d}\right)\\ \end{array} \]

Alternative 10
Error	20.6
Cost	960

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

Alternative 11
Error	18.6
Cost	960

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

Alternative 12
Error	31.7
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023077 
(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))))))