Henrywood and Agarwal, Equation (13)

Average Accuracy: 7.3% → 73.0%

Time: 35.5s

Precision: binary64

Cost: 30540

Math

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

↓

\[\begin{array}{l} t_0 := c0 \cdot \left(d \cdot d\right)\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{t_0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_3 := t_1 \cdot \left(t_2 + \sqrt{t_2 \cdot t_2 - M \cdot M}\right)\\ \mathbf{if}\;t_3 \leq -2 \cdot 10^{-251}:\\ \;\;\;\;\frac{\frac{2 \cdot d}{h \cdot \frac{w}{c0}}}{\left(2 \cdot \frac{w}{c0}\right) \cdot \left(D \cdot \frac{D}{d}\right)}\\ \mathbf{elif}\;t_3 \leq 10^{-290}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \frac{D \cdot \left(h \cdot \frac{M}{d}\right)}{\frac{d}{M}}\right)\\ \mathbf{elif}\;t_3 \leq 4 \cdot 10^{+272}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \frac{\frac{\frac{t_0}{w \cdot h}}{D}}{D}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{D}{d} \cdot \left(M \cdot \left(\frac{D}{d} \cdot \left(h \cdot M\right)\right)\right)\right)\\ \end{array} \]

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 (* d d)))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (/ t_0 (* (* w h) (* D D))))
        (t_3 (* t_1 (+ t_2 (sqrt (- (* t_2 t_2) (* M M)))))))
   (if (<= t_3 -2e-251)
     (/ (/ (* 2.0 d) (* h (/ w c0))) (* (* 2.0 (/ w c0)) (* D (/ D d))))
     (if (<= t_3 1e-290)
       (* 0.25 (* D (/ (* D (* h (/ M d))) (/ d M))))
       (if (<= t_3 4e+272)
         (* t_1 (* 2.0 (/ (/ (/ t_0 (* w h)) D) D)))
         (* 0.25 (* (/ D d) (* M (* (/ D d) (* h M))))))))))

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 * (d * d);
	double t_1 = c0 / (2.0 * w);
	double t_2 = t_0 / ((w * h) * (D * D));
	double t_3 = t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))));
	double tmp;
	if (t_3 <= -2e-251) {
		tmp = ((2.0 * d) / (h * (w / c0))) / ((2.0 * (w / c0)) * (D * (D / d)));
	} else if (t_3 <= 1e-290) {
		tmp = 0.25 * (D * ((D * (h * (M / d))) / (d / M)));
	} else if (t_3 <= 4e+272) {
		tmp = t_1 * (2.0 * (((t_0 / (w * h)) / D) / D));
	} else {
		tmp = 0.25 * ((D / d) * (M * ((D / d) * (h * 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
    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) :: t_3
    real(8) :: tmp
    t_0 = c0 * (d_1 * d_1)
    t_1 = c0 / (2.0d0 * w)
    t_2 = t_0 / ((w * h) * (d * d))
    t_3 = t_1 * (t_2 + sqrt(((t_2 * t_2) - (m * m))))
    if (t_3 <= (-2d-251)) then
        tmp = ((2.0d0 * d_1) / (h * (w / c0))) / ((2.0d0 * (w / c0)) * (d * (d / d_1)))
    else if (t_3 <= 1d-290) then
        tmp = 0.25d0 * (d * ((d * (h * (m / d_1))) / (d_1 / m)))
    else if (t_3 <= 4d+272) then
        tmp = t_1 * (2.0d0 * (((t_0 / (w * h)) / d) / d))
    else
        tmp = 0.25d0 * ((d / d_1) * (m * ((d / d_1) * (h * m))))
    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 * (d * d);
	double t_1 = c0 / (2.0 * w);
	double t_2 = t_0 / ((w * h) * (D * D));
	double t_3 = t_1 * (t_2 + Math.sqrt(((t_2 * t_2) - (M * M))));
	double tmp;
	if (t_3 <= -2e-251) {
		tmp = ((2.0 * d) / (h * (w / c0))) / ((2.0 * (w / c0)) * (D * (D / d)));
	} else if (t_3 <= 1e-290) {
		tmp = 0.25 * (D * ((D * (h * (M / d))) / (d / M)));
	} else if (t_3 <= 4e+272) {
		tmp = t_1 * (2.0 * (((t_0 / (w * h)) / D) / D));
	} else {
		tmp = 0.25 * ((D / d) * (M * ((D / d) * (h * M))));
	}
	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 * (d * d)
	t_1 = c0 / (2.0 * w)
	t_2 = t_0 / ((w * h) * (D * D))
	t_3 = t_1 * (t_2 + math.sqrt(((t_2 * t_2) - (M * M))))
	tmp = 0
	if t_3 <= -2e-251:
		tmp = ((2.0 * d) / (h * (w / c0))) / ((2.0 * (w / c0)) * (D * (D / d)))
	elif t_3 <= 1e-290:
		tmp = 0.25 * (D * ((D * (h * (M / d))) / (d / M)))
	elif t_3 <= 4e+272:
		tmp = t_1 * (2.0 * (((t_0 / (w * h)) / D) / D))
	else:
		tmp = 0.25 * ((D / d) * (M * ((D / d) * (h * M))))
	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(d * d))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(t_0 / Float64(Float64(w * h) * Float64(D * D)))
	t_3 = Float64(t_1 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M)))))
	tmp = 0.0
	if (t_3 <= -2e-251)
		tmp = Float64(Float64(Float64(2.0 * d) / Float64(h * Float64(w / c0))) / Float64(Float64(2.0 * Float64(w / c0)) * Float64(D * Float64(D / d))));
	elseif (t_3 <= 1e-290)
		tmp = Float64(0.25 * Float64(D * Float64(Float64(D * Float64(h * Float64(M / d))) / Float64(d / M))));
	elseif (t_3 <= 4e+272)
		tmp = Float64(t_1 * Float64(2.0 * Float64(Float64(Float64(t_0 / Float64(w * h)) / D) / D)));
	else
		tmp = Float64(0.25 * Float64(Float64(D / d) * Float64(M * Float64(Float64(D / d) * Float64(h * M)))));
	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 * (d * d);
	t_1 = c0 / (2.0 * w);
	t_2 = t_0 / ((w * h) * (D * D));
	t_3 = t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))));
	tmp = 0.0;
	if (t_3 <= -2e-251)
		tmp = ((2.0 * d) / (h * (w / c0))) / ((2.0 * (w / c0)) * (D * (D / d)));
	elseif (t_3 <= 1e-290)
		tmp = 0.25 * (D * ((D * (h * (M / d))) / (d / M)));
	elseif (t_3 <= 4e+272)
		tmp = t_1 * (2.0 * (((t_0 / (w * h)) / D) / D));
	else
		tmp = 0.25 * ((D / d) * (M * ((D / d) * (h * M))));
	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[(d * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -2e-251], N[(N[(N[(2.0 * d), $MachinePrecision] / N[(h * N[(w / c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * N[(w / c0), $MachinePrecision]), $MachinePrecision] * N[(D * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e-290], N[(0.25 * N[(D * N[(N[(D * N[(h * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 4e+272], N[(t$95$1 * N[(2.0 * N[(N[(N[(t$95$0 / N[(w * h), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(D / d), $MachinePrecision] * N[(M * N[(N[(D / d), $MachinePrecision] * N[(h * M), $MachinePrecision]), $MachinePrecision]), $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))))) < -2.00000000000000003e-251

   1. Initial program 24.3%

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

   2. Simplified 29.2%

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

      [Start] 24.3	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      times-frac [=>] 21.5	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      fma-def [=>] 21.5	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d \cdot d}{D \cdot D}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)} \]
      associate-/r* [=>] 21.5	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\color{blue}{\frac{\frac{c0}{w}}{h}}, \frac{d \cdot d}{D \cdot D}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      times-frac [=>] 21.4	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \color{blue}{\frac{d}{D} \cdot \frac{d}{D}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      difference-of-squares [=>] 21.4	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \sqrt{\color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + M\right) \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M\right)}}\right) \]
      sub-neg [=>] 21.4	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \sqrt{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + M\right) \cdot \color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \left(-M\right)\right)}}\right) \]

   3. Taylor expanded in c0 around inf 32.6%

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

   4. Simplified 35.1%

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

      [Start] 32.6	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]
      times-frac [=>] 31.4	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{{d}^{2}}{{D}^{2}} \cdot \frac{c0}{w \cdot h}\right)}\right) \]
      associate-*l/ [=>] 34.6	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{{d}^{2} \cdot \frac{c0}{w \cdot h}}{{D}^{2}}}\right) \]
      unpow2 [=>] 34.6	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\color{blue}{\left(d \cdot d\right)} \cdot \frac{c0}{w \cdot h}}{{D}^{2}}\right) \]
      unpow2 [=>] 34.6	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\left(d \cdot d\right) \cdot \frac{c0}{w \cdot h}}{\color{blue}{D \cdot D}}\right) \]
      times-frac [=>] 41.4	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{d \cdot d}{D} \cdot \frac{\frac{c0}{w \cdot h}}{D}\right)}\right) \]
      associate-*r/ [=>] 41.4	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{\frac{d \cdot d}{D} \cdot \frac{c0}{w \cdot h}}{D}}\right) \]
      associate-*l/ [<=] 34.9	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{\frac{d \cdot d}{D}}{D} \cdot \frac{c0}{w \cdot h}\right)}\right) \]
      associate-*l/ [<=] 39.0	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\color{blue}{\frac{d}{D} \cdot d}}{D} \cdot \frac{c0}{w \cdot h}\right)\right) \]
      associate-*r/ [<=] 41.5	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D}\right)} \cdot \frac{c0}{w \cdot h}\right)\right) \]
      unpow2 [<=] 41.5	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\color{blue}{{\left(\frac{d}{D}\right)}^{2}} \cdot \frac{c0}{w \cdot h}\right)\right) \]
      associate-*r/ [=>] 39.0	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{{\left(\frac{d}{D}\right)}^{2} \cdot c0}{w \cdot h}}\right) \]
      associate-/l* [=>] 41.3	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{{\left(\frac{d}{D}\right)}^{2}}{\frac{w \cdot h}{c0}}}\right) \]
      associate-/r/ [=>] 35.1	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h} \cdot c0\right)}\right) \]

   5. Applied egg-rr 47.9%

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

      [Start] 35.1	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h} \cdot c0\right)\right) \]
      associate-/r/ [<=] 41.3	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{{\left(\frac{d}{D}\right)}^{2}}{\frac{w \cdot h}{c0}}}\right) \]
      unpow2 [=>] 41.3	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\color{blue}{\frac{d}{D} \cdot \frac{d}{D}}}{\frac{w \cdot h}{c0}}\right) \]
      clear-num [=>] 41.4	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{d}{D} \cdot \color{blue}{\frac{1}{\frac{D}{d}}}}{\frac{w \cdot h}{c0}}\right) \]
      un-div-inv [=>] 41.4	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\color{blue}{\frac{\frac{d}{D}}{\frac{D}{d}}}}{\frac{w \cdot h}{c0}}\right) \]
      associate-/l/ [=>] 41.0	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\color{blue}{\frac{d}{\frac{D}{d} \cdot D}}}{\frac{w \cdot h}{c0}}\right) \]
      associate-/l/ [=>] 47.9	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{d}{\frac{w \cdot h}{c0} \cdot \left(\frac{D}{d} \cdot D\right)}}\right) \]

   6. Applied egg-rr 54.0%

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

      [Start] 47.9	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{d}{\frac{w \cdot h}{c0} \cdot \left(\frac{D}{d} \cdot D\right)}\right) \]
      *-commutative [=>] 47.9	\[ \color{blue}{\left(2 \cdot \frac{d}{\frac{w \cdot h}{c0} \cdot \left(\frac{D}{d} \cdot D\right)}\right) \cdot \frac{c0}{2 \cdot w}} \]
      clear-num [=>] 47.7	\[ \left(2 \cdot \frac{d}{\frac{w \cdot h}{c0} \cdot \left(\frac{D}{d} \cdot D\right)}\right) \cdot \color{blue}{\frac{1}{\frac{2 \cdot w}{c0}}} \]
      un-div-inv [=>] 47.7	\[ \color{blue}{\frac{2 \cdot \frac{d}{\frac{w \cdot h}{c0} \cdot \left(\frac{D}{d} \cdot D\right)}}{\frac{2 \cdot w}{c0}}} \]
      associate-*r/ [=>] 47.7	\[ \frac{\color{blue}{\frac{2 \cdot d}{\frac{w \cdot h}{c0} \cdot \left(\frac{D}{d} \cdot D\right)}}}{\frac{2 \cdot w}{c0}} \]
      associate-/r* [=>] 47.1	\[ \frac{\color{blue}{\frac{\frac{2 \cdot d}{\frac{w \cdot h}{c0}}}{\frac{D}{d} \cdot D}}}{\frac{2 \cdot w}{c0}} \]
      associate-/l/ [=>] 52.5	\[ \color{blue}{\frac{\frac{2 \cdot d}{\frac{w \cdot h}{c0}}}{\frac{2 \cdot w}{c0} \cdot \left(\frac{D}{d} \cdot D\right)}} \]
      *-commutative [=>] 52.5	\[ \frac{\frac{\color{blue}{d \cdot 2}}{\frac{w \cdot h}{c0}}}{\frac{2 \cdot w}{c0} \cdot \left(\frac{D}{d} \cdot D\right)} \]
      associate-/l* [=>] 45.6	\[ \frac{\frac{d \cdot 2}{\color{blue}{\frac{w}{\frac{c0}{h}}}}}{\frac{2 \cdot w}{c0} \cdot \left(\frac{D}{d} \cdot D\right)} \]
      associate-/r/ [=>] 54.0	\[ \frac{\frac{d \cdot 2}{\color{blue}{\frac{w}{c0} \cdot h}}}{\frac{2 \cdot w}{c0} \cdot \left(\frac{D}{d} \cdot D\right)} \]
      *-un-lft-identity [=>] 54.0	\[ \frac{\frac{d \cdot 2}{\frac{w}{c0} \cdot h}}{\frac{2 \cdot w}{\color{blue}{1 \cdot c0}} \cdot \left(\frac{D}{d} \cdot D\right)} \]
      times-frac [=>] 54.0	\[ \frac{\frac{d \cdot 2}{\frac{w}{c0} \cdot h}}{\color{blue}{\left(\frac{2}{1} \cdot \frac{w}{c0}\right)} \cdot \left(\frac{D}{d} \cdot D\right)} \]
      metadata-eval [=>] 54.0	\[ \frac{\frac{d \cdot 2}{\frac{w}{c0} \cdot h}}{\left(\color{blue}{2} \cdot \frac{w}{c0}\right) \cdot \left(\frac{D}{d} \cdot D\right)} \]
      *-commutative [=>] 54.0	\[ \frac{\frac{d \cdot 2}{\frac{w}{c0} \cdot h}}{\left(2 \cdot \frac{w}{c0}\right) \cdot \color{blue}{\left(D \cdot \frac{D}{d}\right)}} \]

   if -2.00000000000000003e-251 < (*.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))))) < 1.0000000000000001e-290

   1. Initial program 55.7%

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

   2. Simplified 15.1%

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

      [Start] 55.7	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      associate-*l/ [<=] 46.3	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \left(d \cdot d\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      *-commutative [=>] 46.3	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(d \cdot d\right) \cdot \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      fma-def [=>] 32.5	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(d \cdot d, \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)} \]
      associate-*l* [=>] 28.6	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{\color{blue}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      associate-/r* [=>] 27.9	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \color{blue}{\frac{\frac{c0}{w}}{h \cdot \left(D \cdot D\right)}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      associate-*r* [=>] 26.9	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{\color{blue}{\left(h \cdot D\right) \cdot D}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      *-commutative [=>] 26.9	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{\color{blue}{D \cdot \left(h \cdot D\right)}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Taylor expanded in c0 around -inf 54.9%

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

   4. Simplified 63.6%

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

      [Start] 54.9	\[ \frac{c0}{2 \cdot w} \cdot \left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0} + -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right) \]
      fma-def [=>] 54.9	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0}, -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right)} \]

   5. Taylor expanded in c0 around 0 60.9%

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

   6. Simplified 64.6%

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

      [Start] 60.9	\[ 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \]
      unpow2 [=>] 60.9	\[ 0.25 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \]
      unpow2 [=>] 60.9	\[ 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot h\right)}{{d}^{2}} \]
      associate-*r* [<=] 64.6	\[ 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot \left(M \cdot h\right)\right)}}{{d}^{2}} \]
      *-commutative [<=] 64.6	\[ 0.25 \cdot \frac{\color{blue}{\left(M \cdot \left(M \cdot h\right)\right) \cdot \left(D \cdot D\right)}}{{d}^{2}} \]
      unpow2 [=>] 64.6	\[ 0.25 \cdot \frac{\left(M \cdot \left(M \cdot h\right)\right) \cdot \left(D \cdot D\right)}{\color{blue}{d \cdot d}} \]

   7. Applied egg-rr 64.3%

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

      [Start] 64.6	\[ 0.25 \cdot \frac{\left(M \cdot \left(M \cdot h\right)\right) \cdot \left(D \cdot D\right)}{d \cdot d} \]
      associate-/l* [=>] 65.2	\[ 0.25 \cdot \color{blue}{\frac{M \cdot \left(M \cdot h\right)}{\frac{d \cdot d}{D \cdot D}}} \]
      frac-2neg [=>] 65.2	\[ 0.25 \cdot \frac{M \cdot \left(M \cdot h\right)}{\color{blue}{\frac{-d \cdot d}{-D \cdot D}}} \]
      associate-/r/ [=>] 64.3	\[ 0.25 \cdot \color{blue}{\left(\frac{M \cdot \left(M \cdot h\right)}{-d \cdot d} \cdot \left(-D \cdot D\right)\right)} \]
      distribute-rgt-neg-in [=>] 64.3	\[ 0.25 \cdot \left(\frac{M \cdot \left(M \cdot h\right)}{\color{blue}{d \cdot \left(-d\right)}} \cdot \left(-D \cdot D\right)\right) \]

   8. Simplified 76.0%

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

      [Start] 64.3	\[ 0.25 \cdot \left(\frac{M \cdot \left(M \cdot h\right)}{d \cdot \left(-d\right)} \cdot \left(-D \cdot D\right)\right) \]
      *-commutative [=>] 64.3	\[ 0.25 \cdot \color{blue}{\left(\left(-D \cdot D\right) \cdot \frac{M \cdot \left(M \cdot h\right)}{d \cdot \left(-d\right)}\right)} \]
      distribute-rgt-neg-in [=>] 64.3	\[ 0.25 \cdot \left(\color{blue}{\left(D \cdot \left(-D\right)\right)} \cdot \frac{M \cdot \left(M \cdot h\right)}{d \cdot \left(-d\right)}\right) \]
      associate-*l* [=>] 67.9	\[ 0.25 \cdot \color{blue}{\left(D \cdot \left(\left(-D\right) \cdot \frac{M \cdot \left(M \cdot h\right)}{d \cdot \left(-d\right)}\right)\right)} \]
      times-frac [=>] 74.1	\[ 0.25 \cdot \left(D \cdot \left(\left(-D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{M \cdot h}{-d}\right)}\right)\right) \]
      *-commutative [=>] 74.1	\[ 0.25 \cdot \left(D \cdot \left(\left(-D\right) \cdot \left(\frac{M}{d} \cdot \frac{\color{blue}{h \cdot M}}{-d}\right)\right)\right) \]
      associate-/l* [=>] 76.0	\[ 0.25 \cdot \left(D \cdot \left(\left(-D\right) \cdot \left(\frac{M}{d} \cdot \color{blue}{\frac{h}{\frac{-d}{M}}}\right)\right)\right) \]

   9. Applied egg-rr 78.6%

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

      [Start] 76.0	\[ 0.25 \cdot \left(D \cdot \left(\left(-D\right) \cdot \left(\frac{M}{d} \cdot \frac{h}{\frac{-d}{M}}\right)\right)\right) \]
      *-commutative [=>] 76.0	\[ 0.25 \cdot \left(D \cdot \color{blue}{\left(\left(\frac{M}{d} \cdot \frac{h}{\frac{-d}{M}}\right) \cdot \left(-D\right)\right)}\right) \]
      associate-*r/ [=>] 76.0	\[ 0.25 \cdot \left(D \cdot \left(\color{blue}{\frac{\frac{M}{d} \cdot h}{\frac{-d}{M}}} \cdot \left(-D\right)\right)\right) \]
      associate-*l/ [=>] 78.6	\[ 0.25 \cdot \left(D \cdot \color{blue}{\frac{\left(\frac{M}{d} \cdot h\right) \cdot \left(-D\right)}{\frac{-d}{M}}}\right) \]
      *-commutative [=>] 78.6	\[ 0.25 \cdot \left(D \cdot \frac{\color{blue}{\left(h \cdot \frac{M}{d}\right)} \cdot \left(-D\right)}{\frac{-d}{M}}\right) \]
      add-sqr-sqrt [=>] 38.3	\[ 0.25 \cdot \left(D \cdot \frac{\left(h \cdot \frac{M}{d}\right) \cdot \color{blue}{\left(\sqrt{-D} \cdot \sqrt{-D}\right)}}{\frac{-d}{M}}\right) \]
      sqrt-unprod [=>] 61.9	\[ 0.25 \cdot \left(D \cdot \frac{\left(h \cdot \frac{M}{d}\right) \cdot \color{blue}{\sqrt{\left(-D\right) \cdot \left(-D\right)}}}{\frac{-d}{M}}\right) \]
      sqr-neg [=>] 61.9	\[ 0.25 \cdot \left(D \cdot \frac{\left(h \cdot \frac{M}{d}\right) \cdot \sqrt{\color{blue}{D \cdot D}}}{\frac{-d}{M}}\right) \]
      sqrt-unprod [<=] 27.2	\[ 0.25 \cdot \left(D \cdot \frac{\left(h \cdot \frac{M}{d}\right) \cdot \color{blue}{\left(\sqrt{D} \cdot \sqrt{D}\right)}}{\frac{-d}{M}}\right) \]
      add-sqr-sqrt [<=] 54.0	\[ 0.25 \cdot \left(D \cdot \frac{\left(h \cdot \frac{M}{d}\right) \cdot \color{blue}{D}}{\frac{-d}{M}}\right) \]
      add-sqr-sqrt [=>] 27.1	\[ 0.25 \cdot \left(D \cdot \frac{\left(h \cdot \frac{M}{d}\right) \cdot D}{\frac{\color{blue}{\sqrt{-d} \cdot \sqrt{-d}}}{M}}\right) \]
      sqrt-unprod [=>] 64.2	\[ 0.25 \cdot \left(D \cdot \frac{\left(h \cdot \frac{M}{d}\right) \cdot D}{\frac{\color{blue}{\sqrt{\left(-d\right) \cdot \left(-d\right)}}}{M}}\right) \]
      sqr-neg [=>] 64.2	\[ 0.25 \cdot \left(D \cdot \frac{\left(h \cdot \frac{M}{d}\right) \cdot D}{\frac{\sqrt{\color{blue}{d \cdot d}}}{M}}\right) \]
      sqrt-unprod [<=] 40.6	\[ 0.25 \cdot \left(D \cdot \frac{\left(h \cdot \frac{M}{d}\right) \cdot D}{\frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{M}}\right) \]
      add-sqr-sqrt [<=] 78.6	\[ 0.25 \cdot \left(D \cdot \frac{\left(h \cdot \frac{M}{d}\right) \cdot D}{\frac{\color{blue}{d}}{M}}\right) \]

   if 1.0000000000000001e-290 < (*.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))))) < 4.0000000000000003e272

   1. Initial program 88.9%

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

   2. Simplified 79.0%

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

      [Start] 88.9	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      times-frac [=>] 79.2	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      fma-neg [=>] 79.2	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\color{blue}{\mathsf{fma}\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, -M \cdot M\right)}}\right) \]
      times-frac [=>] 79.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\mathsf{fma}\left(\color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}}, \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, -M \cdot M\right)}\right) \]
      times-frac [=>] 79.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}, \color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}}, -M \cdot M\right)}\right) \]

   3. Taylor expanded in c0 around inf 88.8%

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

   4. Simplified 86.7%

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

      [Start] 88.8	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]
      *-commutative [=>] 88.8	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2} \cdot c0}{\color{blue}{\left(w \cdot h\right) \cdot {D}^{2}}}\right) \]
      unpow2 [=>] 88.8	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2} \cdot c0}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}}\right) \]
      *-commutative [=>] 88.8	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\color{blue}{c0 \cdot {d}^{2}}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right) \]
      unpow2 [=>] 88.8	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right) \]
      associate-/r* [=>] 86.7	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{\frac{c0 \cdot \left(d \cdot d\right)}{w \cdot h}}{D \cdot D}}\right) \]
      *-commutative [=>] 86.7	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{\color{blue}{\left(d \cdot d\right) \cdot c0}}{w \cdot h}}{D \cdot D}\right) \]
      associate-/l* [=>] 80.7	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\color{blue}{\frac{d \cdot d}{\frac{w \cdot h}{c0}}}}{D \cdot D}\right) \]
      *-commutative [=>] 80.7	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{d \cdot d}{\frac{\color{blue}{h \cdot w}}{c0}}}{D \cdot D}\right) \]
      associate-*l/ [<=] 66.7	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{d \cdot d}{\color{blue}{\frac{h}{c0} \cdot w}}}{D \cdot D}\right) \]
      associate-/r* [=>] 66.8	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{\frac{\frac{d \cdot d}{\frac{h}{c0} \cdot w}}{D}}{D}}\right) \]
      associate-*l/ [=>] 80.7	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{\frac{d \cdot d}{\color{blue}{\frac{h \cdot w}{c0}}}}{D}}{D}\right) \]
      *-commutative [<=] 80.7	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{\frac{d \cdot d}{\frac{\color{blue}{w \cdot h}}{c0}}}{D}}{D}\right) \]
      associate-/l* [<=] 86.7	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{\color{blue}{\frac{\left(d \cdot d\right) \cdot c0}{w \cdot h}}}{D}}{D}\right) \]
      *-commutative [<=] 86.7	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{\frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{w \cdot h}}{D}}{D}\right) \]

   if 4.0000000000000003e272 < (*.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. Simplified 0.7%

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

      [Start] 0.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      associate-*l/ [<=] 0.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \left(d \cdot d\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      *-commutative [=>] 0.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(d \cdot d\right) \cdot \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      fma-def [=>] 0.0	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(d \cdot d, \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)} \]
      associate-*l* [=>] 0.0	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{\color{blue}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      associate-/r* [=>] 0.0	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \color{blue}{\frac{\frac{c0}{w}}{h \cdot \left(D \cdot D\right)}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      associate-*r* [=>] 0.0	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{\color{blue}{\left(h \cdot D\right) \cdot D}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      *-commutative [=>] 0.0	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{\color{blue}{D \cdot \left(h \cdot D\right)}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Taylor expanded in c0 around -inf 2.0%

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

   4. Simplified 47.8%

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

      [Start] 2.0	\[ \frac{c0}{2 \cdot w} \cdot \left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0} + -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right) \]
      fma-def [=>] 2.0	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0}, -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right)} \]

   5. Taylor expanded in c0 around 0 46.3%

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

   6. Simplified 61.5%

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

      [Start] 46.3	\[ 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \]
      *-commutative [<=] 46.3	\[ 0.25 \cdot \frac{{D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{{d}^{2}} \]
      associate-/l* [=>] 46.2	\[ 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]
      unpow2 [=>] 46.2	\[ 0.25 \cdot \frac{\color{blue}{D \cdot D}}{\frac{{d}^{2}}{h \cdot {M}^{2}}} \]
      unpow2 [=>] 46.2	\[ 0.25 \cdot \frac{D \cdot D}{\frac{\color{blue}{d \cdot d}}{h \cdot {M}^{2}}} \]
      *-commutative [=>] 46.2	\[ 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{\color{blue}{{M}^{2} \cdot h}}} \]
      unpow2 [=>] 46.2	\[ 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{\color{blue}{\left(M \cdot M\right)} \cdot h}} \]
      associate-*r* [<=] 49.2	\[ 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{\color{blue}{M \cdot \left(M \cdot h\right)}}} \]
      associate-/r/ [=>] 49.0	\[ 0.25 \cdot \color{blue}{\left(\frac{D \cdot D}{d \cdot d} \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)} \]
      associate-/r* [=>] 53.8	\[ 0.25 \cdot \left(\color{blue}{\frac{\frac{D \cdot D}{d}}{d}} \cdot \left(M \cdot \left(M \cdot h\right)\right)\right) \]
      associate-*r/ [<=] 61.5	\[ 0.25 \cdot \left(\frac{\color{blue}{D \cdot \frac{D}{d}}}{d} \cdot \left(M \cdot \left(M \cdot h\right)\right)\right) \]

   7. Applied egg-rr 67.3%

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

      [Start] 61.5	\[ 0.25 \cdot \left(\frac{D \cdot \frac{D}{d}}{d} \cdot \left(M \cdot \left(M \cdot h\right)\right)\right) \]
      add-sqr-sqrt [=>] 61.5	\[ 0.25 \cdot \left(\color{blue}{\left(\sqrt{\frac{D \cdot \frac{D}{d}}{d}} \cdot \sqrt{\frac{D \cdot \frac{D}{d}}{d}}\right)} \cdot \left(M \cdot \left(M \cdot h\right)\right)\right) \]
      associate-*l* [=>] 61.5	\[ 0.25 \cdot \color{blue}{\left(\sqrt{\frac{D \cdot \frac{D}{d}}{d}} \cdot \left(\sqrt{\frac{D \cdot \frac{D}{d}}{d}} \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)\right)} \]
      associate-*l/ [<=] 61.5	\[ 0.25 \cdot \left(\sqrt{\color{blue}{\frac{D}{d} \cdot \frac{D}{d}}} \cdot \left(\sqrt{\frac{D \cdot \frac{D}{d}}{d}} \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)\right) \]
      sqrt-unprod [<=] 38.3	\[ 0.25 \cdot \left(\color{blue}{\left(\sqrt{\frac{D}{d}} \cdot \sqrt{\frac{D}{d}}\right)} \cdot \left(\sqrt{\frac{D \cdot \frac{D}{d}}{d}} \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)\right) \]
      add-sqr-sqrt [<=] 55.8	\[ 0.25 \cdot \left(\color{blue}{\frac{D}{d}} \cdot \left(\sqrt{\frac{D \cdot \frac{D}{d}}{d}} \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)\right) \]
      frac-2neg [=>] 55.8	\[ 0.25 \cdot \left(\color{blue}{\frac{-D}{-d}} \cdot \left(\sqrt{\frac{D \cdot \frac{D}{d}}{d}} \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)\right) \]
      distribute-frac-neg [=>] 55.8	\[ 0.25 \cdot \left(\color{blue}{\left(-\frac{D}{-d}\right)} \cdot \left(\sqrt{\frac{D \cdot \frac{D}{d}}{d}} \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)\right) \]
      associate-*l/ [<=] 57.4	\[ 0.25 \cdot \left(\left(-\frac{D}{-d}\right) \cdot \left(\sqrt{\color{blue}{\frac{D}{d} \cdot \frac{D}{d}}} \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)\right) \]
      sqrt-unprod [<=] 41.2	\[ 0.25 \cdot \left(\left(-\frac{D}{-d}\right) \cdot \left(\color{blue}{\left(\sqrt{\frac{D}{d}} \cdot \sqrt{\frac{D}{d}}\right)} \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)\right) \]
      add-sqr-sqrt [<=] 67.3	\[ 0.25 \cdot \left(\left(-\frac{D}{-d}\right) \cdot \left(\color{blue}{\frac{D}{d}} \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)\right) \]
      distribute-lft-neg-out [=>] 67.3	\[ 0.25 \cdot \color{blue}{\left(-\frac{D}{-d} \cdot \left(\frac{D}{d} \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)\right)} \]

   8. Applied egg-rr 73.6%

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

      [Start] 67.3	\[ 0.25 \cdot \left(-\frac{-D}{d} \cdot \left(\frac{D}{d} \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)\right) \]
      associate-*l/ [=>] 64.7	\[ 0.25 \cdot \left(-\frac{-D}{d} \cdot \color{blue}{\frac{D \cdot \left(M \cdot \left(M \cdot h\right)\right)}{d}}\right) \]
      *-un-lft-identity [=>] 64.7	\[ 0.25 \cdot \left(-\frac{-D}{d} \cdot \frac{D \cdot \left(M \cdot \left(M \cdot h\right)\right)}{\color{blue}{1 \cdot d}}\right) \]
      *-commutative [=>] 64.7	\[ 0.25 \cdot \left(-\frac{-D}{d} \cdot \frac{D \cdot \left(M \cdot \left(M \cdot h\right)\right)}{\color{blue}{d \cdot 1}}\right) \]
      associate-/r* [=>] 64.7	\[ 0.25 \cdot \left(-\frac{-D}{d} \cdot \color{blue}{\frac{\frac{D \cdot \left(M \cdot \left(M \cdot h\right)\right)}{d}}{1}}\right) \]
      associate-*l/ [<=] 67.3	\[ 0.25 \cdot \left(-\frac{-D}{d} \cdot \frac{\color{blue}{\frac{D}{d} \cdot \left(M \cdot \left(M \cdot h\right)\right)}}{1}\right) \]
      *-commutative [=>] 67.3	\[ 0.25 \cdot \left(-\frac{-D}{d} \cdot \frac{\color{blue}{\left(M \cdot \left(M \cdot h\right)\right) \cdot \frac{D}{d}}}{1}\right) \]
      associate-*l* [=>] 73.6	\[ 0.25 \cdot \left(-\frac{-D}{d} \cdot \frac{\color{blue}{M \cdot \left(\left(M \cdot h\right) \cdot \frac{D}{d}\right)}}{1}\right) \]

3. Recombined 4 regimes into one program.

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq -2 \cdot 10^{-251}:\\ \;\;\;\;\frac{\frac{2 \cdot d}{h \cdot \frac{w}{c0}}}{\left(2 \cdot \frac{w}{c0}\right) \cdot \left(D \cdot \frac{D}{d}\right)}\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 10^{-290}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \frac{D \cdot \left(h \cdot \frac{M}{d}\right)}{\frac{d}{M}}\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 4 \cdot 10^{+272}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{\frac{c0 \cdot \left(d \cdot d\right)}{w \cdot h}}{D}}{D}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{D}{d} \cdot \left(M \cdot \left(\frac{D}{d} \cdot \left(h \cdot M\right)\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	68.4%
Cost	1608

\[\begin{array}{l} \mathbf{if}\;w \leq 1.45 \cdot 10^{-114}:\\ \;\;\;\;0.25 \cdot \left(\frac{D}{d} \cdot \left(M \cdot \left(\frac{D}{d} \cdot \left(h \cdot M\right)\right)\right)\right)\\ \mathbf{elif}\;w \leq 1.35 \cdot 10^{-98}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{d}{D} \cdot \frac{\frac{c0}{h}}{w \cdot \frac{D}{d}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \frac{D \cdot \left(h \cdot \frac{M}{d}\right)}{\frac{d}{M}}\right)\\ \end{array} \]

Alternative 2
Accuracy	71.7%
Cost	1481

\[\begin{array}{l} \mathbf{if}\;M \cdot M \leq 2 \cdot 10^{-268} \lor \neg \left(M \cdot M \leq 2 \cdot 10^{+245}\right):\\ \;\;\;\;0.25 \cdot \left(D \cdot \frac{D \cdot \left(h \cdot \frac{M}{d}\right)}{\frac{d}{M}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{D}{d} \cdot \left(h \cdot \left(M \cdot \left(M \cdot \frac{D}{d}\right)\right)\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	68.0%
Cost	1480

\[\begin{array}{l} \mathbf{if}\;w \leq 1.7 \cdot 10^{-121}:\\ \;\;\;\;0.25 \cdot \left(\frac{D}{d} \cdot \left(M \cdot \left(\frac{D}{d} \cdot \left(h \cdot M\right)\right)\right)\right)\\ \mathbf{elif}\;w \leq 1.35 \cdot 10^{-98}:\\ \;\;\;\;\frac{-d}{\frac{w \cdot \frac{-D}{\frac{\frac{c0 \cdot d}{h}}{D}}}{\frac{c0}{w}}}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \frac{D \cdot \left(h \cdot \frac{M}{d}\right)}{\frac{d}{M}}\right)\\ \end{array} \]

Alternative 4
Accuracy	70.2%
Cost	1225

\[\begin{array}{l} \mathbf{if}\;D \leq 1.2 \cdot 10^{-225} \lor \neg \left(D \leq 6.9 \cdot 10^{+59}\right):\\ \;\;\;\;0.25 \cdot \left(\frac{D}{d} \cdot \left(M \cdot \left(\frac{D}{d} \cdot \left(h \cdot M\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \frac{D \cdot \left(h \cdot \frac{M}{d}\right)}{\frac{d}{M}}\right)\\ \end{array} \]

Alternative 5
Accuracy	64.5%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;d \leq 62000000000000:\\ \;\;\;\;0.25 \cdot \left(D \cdot \frac{D \cdot M}{\frac{d}{M} \cdot \frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \frac{D \cdot \left(M \cdot \left(h \cdot \frac{M}{d}\right)\right)}{d}\right)\\ \end{array} \]

Alternative 6
Accuracy	64.5%
Cost	960

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

Alternative 7
Accuracy	69.9%
Cost	960

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

Alternative 8
Accuracy	50.5%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023138 
(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))))))