Henrywood and Agarwal, Equation (13)

Average Accuracy: 6.6% → 75.2%

Time: 34.2s

Precision: binary64

Cost: 30540

Math

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

↓

\[\begin{array}{l} t_0 := c0 \cdot \left(d \cdot d\right)\\ t_1 := \left(w \cdot h\right) \cdot \left(D \cdot D\right)\\ t_2 := \frac{c0}{2 \cdot w}\\ t_3 := \frac{t_0}{t_1}\\ t_4 := t_2 \cdot \left(t_3 + \sqrt{t_3 \cdot t_3 - M \cdot M}\right)\\ \mathbf{if}\;t_4 \leq -1 \cdot 10^{+74}:\\ \;\;\;\;t_2 \cdot \left(2 \cdot {\left(\frac{d}{D} \cdot \sqrt{\frac{c0}{w \cdot h}}\right)}^{2}\right)\\ \mathbf{elif}\;t_4 \leq 0:\\ \;\;\;\;h \cdot \left(\frac{D}{d} \cdot \left(\frac{M}{\frac{d}{M}} \cdot \left(D \cdot 0.25\right)\right)\right)\\ \mathbf{elif}\;t_4 \leq 10^{+228}:\\ \;\;\;\;t_2 \cdot \frac{2 \cdot t_0}{t_1}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{D}{\frac{d}{M}} \cdot \left(h \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\\ \end{array} \]

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 (* (* w h) (* D D)))
        (t_2 (/ c0 (* 2.0 w)))
        (t_3 (/ t_0 t_1))
        (t_4 (* t_2 (+ t_3 (sqrt (- (* t_3 t_3) (* M M)))))))
   (if (<= t_4 -1e+74)
     (* t_2 (* 2.0 (pow (* (/ d D) (sqrt (/ c0 (* w h)))) 2.0)))
     (if (<= t_4 0.0)
       (* h (* (/ D d) (* (/ M (/ d M)) (* D 0.25))))
       (if (<= t_4 1e+228)
         (* t_2 (/ (* 2.0 t_0) t_1))
         (* 0.25 (* (/ D (/ d M)) (* h (* 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 * (d * d);
	double t_1 = (w * h) * (D * D);
	double t_2 = c0 / (2.0 * w);
	double t_3 = t_0 / t_1;
	double t_4 = t_2 * (t_3 + sqrt(((t_3 * t_3) - (M * M))));
	double tmp;
	if (t_4 <= -1e+74) {
		tmp = t_2 * (2.0 * pow(((d / D) * sqrt((c0 / (w * h)))), 2.0));
	} else if (t_4 <= 0.0) {
		tmp = h * ((D / d) * ((M / (d / M)) * (D * 0.25)));
	} else if (t_4 <= 1e+228) {
		tmp = t_2 * ((2.0 * t_0) / t_1);
	} else {
		tmp = 0.25 * ((D / (d / M)) * (h * (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) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = c0 * (d_1 * d_1)
    t_1 = (w * h) * (d * d)
    t_2 = c0 / (2.0d0 * w)
    t_3 = t_0 / t_1
    t_4 = t_2 * (t_3 + sqrt(((t_3 * t_3) - (m * m))))
    if (t_4 <= (-1d+74)) then
        tmp = t_2 * (2.0d0 * (((d_1 / d) * sqrt((c0 / (w * h)))) ** 2.0d0))
    else if (t_4 <= 0.0d0) then
        tmp = h * ((d / d_1) * ((m / (d_1 / m)) * (d * 0.25d0)))
    else if (t_4 <= 1d+228) then
        tmp = t_2 * ((2.0d0 * t_0) / t_1)
    else
        tmp = 0.25d0 * ((d / (d_1 / m)) * (h * (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 * (d * d);
	double t_1 = (w * h) * (D * D);
	double t_2 = c0 / (2.0 * w);
	double t_3 = t_0 / t_1;
	double t_4 = t_2 * (t_3 + Math.sqrt(((t_3 * t_3) - (M * M))));
	double tmp;
	if (t_4 <= -1e+74) {
		tmp = t_2 * (2.0 * Math.pow(((d / D) * Math.sqrt((c0 / (w * h)))), 2.0));
	} else if (t_4 <= 0.0) {
		tmp = h * ((D / d) * ((M / (d / M)) * (D * 0.25)));
	} else if (t_4 <= 1e+228) {
		tmp = t_2 * ((2.0 * t_0) / t_1);
	} else {
		tmp = 0.25 * ((D / (d / M)) * (h * (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 * (d * d)
	t_1 = (w * h) * (D * D)
	t_2 = c0 / (2.0 * w)
	t_3 = t_0 / t_1
	t_4 = t_2 * (t_3 + math.sqrt(((t_3 * t_3) - (M * M))))
	tmp = 0
	if t_4 <= -1e+74:
		tmp = t_2 * (2.0 * math.pow(((d / D) * math.sqrt((c0 / (w * h)))), 2.0))
	elif t_4 <= 0.0:
		tmp = h * ((D / d) * ((M / (d / M)) * (D * 0.25)))
	elif t_4 <= 1e+228:
		tmp = t_2 * ((2.0 * t_0) / t_1)
	else:
		tmp = 0.25 * ((D / (d / M)) * (h * (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(d * d))
	t_1 = Float64(Float64(w * h) * Float64(D * D))
	t_2 = Float64(c0 / Float64(2.0 * w))
	t_3 = Float64(t_0 / t_1)
	t_4 = Float64(t_2 * Float64(t_3 + sqrt(Float64(Float64(t_3 * t_3) - Float64(M * M)))))
	tmp = 0.0
	if (t_4 <= -1e+74)
		tmp = Float64(t_2 * Float64(2.0 * (Float64(Float64(d / D) * sqrt(Float64(c0 / Float64(w * h)))) ^ 2.0)));
	elseif (t_4 <= 0.0)
		tmp = Float64(h * Float64(Float64(D / d) * Float64(Float64(M / Float64(d / M)) * Float64(D * 0.25))));
	elseif (t_4 <= 1e+228)
		tmp = Float64(t_2 * Float64(Float64(2.0 * t_0) / t_1));
	else
		tmp = Float64(0.25 * Float64(Float64(D / Float64(d / M)) * Float64(h * Float64(D * Float64(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 * (d * d);
	t_1 = (w * h) * (D * D);
	t_2 = c0 / (2.0 * w);
	t_3 = t_0 / t_1;
	t_4 = t_2 * (t_3 + sqrt(((t_3 * t_3) - (M * M))));
	tmp = 0.0;
	if (t_4 <= -1e+74)
		tmp = t_2 * (2.0 * (((d / D) * sqrt((c0 / (w * h)))) ^ 2.0));
	elseif (t_4 <= 0.0)
		tmp = h * ((D / d) * ((M / (d / M)) * (D * 0.25)));
	elseif (t_4 <= 1e+228)
		tmp = t_2 * ((2.0 * t_0) / t_1);
	else
		tmp = 0.25 * ((D / (d / M)) * (h * (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[(d * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 / t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 * N[(t$95$3 + N[Sqrt[N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -1e+74], N[(t$95$2 * N[(2.0 * N[Power[N[(N[(d / D), $MachinePrecision] * N[Sqrt[N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[(h * N[(N[(D / d), $MachinePrecision] * N[(N[(M / N[(d / M), $MachinePrecision]), $MachinePrecision] * N[(D * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 1e+228], N[(t$95$2 * N[(N[(2.0 * t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(D / N[(d / M), $MachinePrecision]), $MachinePrecision] * N[(h * N[(D * N[(M / d), $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))))) < -9.99999999999999952e73

   1. Initial program 10.3%

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

   2. Simplified 14.4%

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

      [Start] 10.3	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      times-frac [=>] 8.2	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      fma-def [=>] 8.1	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d \cdot d}{D \cdot D}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)} \]
      associate-/r* [=>] 8.2	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \color{blue}{\frac{\frac{d \cdot d}{D}}{D}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      difference-of-squares [=>] 8.2	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{\frac{d \cdot d}{D}}{D}, \sqrt{\color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + M\right) \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M\right)}}\right) \]

   3. Taylor expanded in c0 around inf 18.6%

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

   4. Simplified 31.5%

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

      [Start] 18.6	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]
      times-frac [=>] 19.9	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{{d}^{2}}{{D}^{2}} \cdot \frac{c0}{w \cdot h}\right)}\right) \]
      associate-/r* [=>] 22.6	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{{d}^{2}}{{D}^{2}} \cdot \color{blue}{\frac{\frac{c0}{w}}{h}}\right)\right) \]
      times-frac [<=] 23.1	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{{d}^{2} \cdot \frac{c0}{w}}{{D}^{2} \cdot h}}\right) \]
      *-commutative [<=] 23.1	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\color{blue}{\frac{c0}{w} \cdot {d}^{2}}}{{D}^{2} \cdot h}\right) \]
      *-commutative [<=] 23.1	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{w} \cdot {d}^{2}}{\color{blue}{h \cdot {D}^{2}}}\right) \]
      unpow2 [=>] 23.1	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{w} \cdot {d}^{2}}{h \cdot \color{blue}{\left(D \cdot D\right)}}\right) \]
      associate-/l* [=>] 21.3	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{\frac{c0}{w}}{\frac{h \cdot \left(D \cdot D\right)}{{d}^{2}}}}\right) \]
      associate-*r/ [=>] 21.3	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \frac{c0}{w}}{\frac{h \cdot \left(D \cdot D\right)}{{d}^{2}}}} \]
      associate-*r* [=>] 24.8	\[ \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \frac{c0}{w}}{\frac{\color{blue}{\left(h \cdot D\right) \cdot D}}{{d}^{2}}} \]
      associate-*r/ [<=] 29.4	\[ \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \frac{c0}{w}}{\color{blue}{\left(h \cdot D\right) \cdot \frac{D}{{d}^{2}}}} \]
      unpow2 [=>] 29.4	\[ \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \frac{c0}{w}}{\left(h \cdot D\right) \cdot \frac{D}{\color{blue}{d \cdot d}}} \]
      associate-*r* [<=] 25.4	\[ \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \frac{c0}{w}}{\color{blue}{h \cdot \left(D \cdot \frac{D}{d \cdot d}\right)}} \]
      associate-*r/ [<=] 25.4	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{\frac{c0}{w}}{h \cdot \left(D \cdot \frac{D}{d \cdot d}\right)}\right)} \]
      associate-/r* [=>] 28.5	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{\frac{\frac{c0}{w}}{h}}{D \cdot \frac{D}{d \cdot d}}}\right) \]
      associate-/r* [<=] 25.8	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\color{blue}{\frac{c0}{w \cdot h}}}{D \cdot \frac{D}{d \cdot d}}\right) \]

   5. Applied egg-rr 44.4%

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

      [Start] 31.5	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\right)\right) \]
      add-sqr-sqrt [=>] 31.3	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\sqrt{\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}} \cdot \sqrt{\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}}\right)}\right) \]
      pow2 [=>] 31.3	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{{\left(\sqrt{\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}}\right)}^{2}}\right) \]
      *-commutative [=>] 31.3	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot {\left(\sqrt{\color{blue}{{\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h}}}\right)}^{2}\right) \]
      sqrt-prod [=>] 31.3	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{d}{D}\right)}^{2}} \cdot \sqrt{\frac{c0}{w \cdot h}}\right)}}^{2}\right) \]
      unpow2 [=>] 31.3	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot {\left(\sqrt{\color{blue}{\frac{d}{D} \cdot \frac{d}{D}}} \cdot \sqrt{\frac{c0}{w \cdot h}}\right)}^{2}\right) \]
      sqrt-prod [=>] 23.3	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot {\left(\color{blue}{\left(\sqrt{\frac{d}{D}} \cdot \sqrt{\frac{d}{D}}\right)} \cdot \sqrt{\frac{c0}{w \cdot h}}\right)}^{2}\right) \]
      add-sqr-sqrt [<=] 44.4	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot {\left(\color{blue}{\frac{d}{D}} \cdot \sqrt{\frac{c0}{w \cdot h}}\right)}^{2}\right) \]

   if -9.99999999999999952e73 < (*.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 59.7%

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

   2. Applied egg-rr 55.1%

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

      [Start] 59.7	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      distribute-rgt-in [=>] 59.1	\[ \color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0}{2 \cdot w} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M} \cdot \frac{c0}{2 \cdot w}} \]
      associate-*r* [=>] 54.2	\[ \frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0}{2 \cdot w} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M} \cdot \frac{c0}{2 \cdot w} \]
      associate-*l* [=>] 43.5	\[ \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)}} \cdot \frac{c0}{2 \cdot w} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M} \cdot \frac{c0}{2 \cdot w} \]
      times-frac [=>] 40.3	\[ \color{blue}{\left(\frac{c0 \cdot d}{w} \cdot \frac{d}{h \cdot \left(D \cdot D\right)}\right)} \cdot \frac{c0}{2 \cdot w} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M} \cdot \frac{c0}{2 \cdot w} \]
      associate-*r* [=>] 38.1	\[ \left(\frac{c0 \cdot d}{w} \cdot \frac{d}{\color{blue}{\left(h \cdot D\right) \cdot D}}\right) \cdot \frac{c0}{2 \cdot w} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M} \cdot \frac{c0}{2 \cdot w} \]
      *-un-lft-identity [=>] 38.1	\[ \left(\frac{c0 \cdot d}{w} \cdot \frac{d}{\left(h \cdot D\right) \cdot D}\right) \cdot \frac{\color{blue}{1 \cdot c0}}{2 \cdot w} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M} \cdot \frac{c0}{2 \cdot w} \]
      times-frac [=>] 37.9	\[ \left(\frac{c0 \cdot d}{w} \cdot \frac{d}{\left(h \cdot D\right) \cdot D}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{c0}{w}\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M} \cdot \frac{c0}{2 \cdot w} \]
      metadata-eval [=>] 37.9	\[ \left(\frac{c0 \cdot d}{w} \cdot \frac{d}{\left(h \cdot D\right) \cdot D}\right) \cdot \left(\color{blue}{0.5} \cdot \frac{c0}{w}\right) + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M} \cdot \frac{c0}{2 \cdot w} \]

   3. Taylor expanded in c0 around -inf 35.7%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}} + \left(0.5 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left({w}^{2} \cdot h\right)} + -0.5 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left({w}^{2} \cdot h\right)}\right) \cdot {c0}^{2}} \]

   4. Simplified 46.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c0 \cdot c0, 0, 0.25 \cdot \frac{D \cdot D}{\frac{d}{h} \cdot \frac{d}{M \cdot M}}\right)} \]
      Proof

      [Start] 35.7	\[ 0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}} + \left(0.5 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left({w}^{2} \cdot h\right)} + -0.5 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left({w}^{2} \cdot h\right)}\right) \cdot {c0}^{2} \]
      +-commutative [=>] 35.7	\[ \color{blue}{\left(0.5 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left({w}^{2} \cdot h\right)} + -0.5 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left({w}^{2} \cdot h\right)}\right) \cdot {c0}^{2} + 0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]
      *-commutative [=>] 35.7	\[ \color{blue}{{c0}^{2} \cdot \left(0.5 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left({w}^{2} \cdot h\right)} + -0.5 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left({w}^{2} \cdot h\right)}\right)} + 0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}} \]
      fma-def [=>] 35.7	\[ \color{blue}{\mathsf{fma}\left({c0}^{2}, 0.5 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left({w}^{2} \cdot h\right)} + -0.5 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left({w}^{2} \cdot h\right)}, 0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}\right)} \]
      unpow2 [=>] 35.7	\[ \mathsf{fma}\left(\color{blue}{c0 \cdot c0}, 0.5 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left({w}^{2} \cdot h\right)} + -0.5 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left({w}^{2} \cdot h\right)}, 0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}\right) \]
      distribute-rgt-out [=>] 35.7	\[ \mathsf{fma}\left(c0 \cdot c0, \color{blue}{\frac{{d}^{2}}{{D}^{2} \cdot \left({w}^{2} \cdot h\right)} \cdot \left(0.5 + -0.5\right)}, 0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}\right) \]
      metadata-eval [=>] 35.7	\[ \mathsf{fma}\left(c0 \cdot c0, \frac{{d}^{2}}{{D}^{2} \cdot \left({w}^{2} \cdot h\right)} \cdot \color{blue}{0}, 0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}\right) \]
      mul0-rgt [=>] 44.6	\[ \mathsf{fma}\left(c0 \cdot c0, \color{blue}{0}, 0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}\right) \]
      associate-/l* [=>] 44.1	\[ \mathsf{fma}\left(c0 \cdot c0, 0, 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}}\right) \]
      *-commutative [<=] 44.1	\[ \mathsf{fma}\left(c0 \cdot c0, 0, 0.25 \cdot \frac{{D}^{2}}{\frac{{d}^{2}}{\color{blue}{{M}^{2} \cdot h}}}\right) \]

   5. Taylor expanded in c0 around 0 50.9%

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

   6. Simplified 59.5%

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

      [Start] 50.9	\[ 0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}} \]
      unpow2 [=>] 50.9	\[ 0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)}{{d}^{2}} \]
      *-commutative [=>] 50.9	\[ 0.25 \cdot \frac{{D}^{2} \cdot \color{blue}{\left(\left(M \cdot M\right) \cdot h\right)}}{{d}^{2}} \]
      unpow2 [<=] 50.9	\[ 0.25 \cdot \frac{{D}^{2} \cdot \left(\color{blue}{{M}^{2}} \cdot h\right)}{{d}^{2}} \]
      unpow2 [=>] 50.9	\[ 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{d \cdot d}} \]
      associate-*r/ [=>] 50.9	\[ \color{blue}{\frac{0.25 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{d \cdot d}} \]
      unpow2 [=>] 50.9	\[ \frac{0.25 \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{d \cdot d} \]
      unpow2 [=>] 50.9	\[ \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot h\right)\right)}{d \cdot d} \]
      *-commutative [<=] 50.9	\[ \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot \left(M \cdot M\right)\right)}\right)}{d \cdot d} \]
      associate-*r* [=>] 50.9	\[ \frac{\color{blue}{\left(0.25 \cdot \left(D \cdot D\right)\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}}{d \cdot d} \]
      *-commutative [<=] 50.9	\[ \frac{\color{blue}{\left(\left(D \cdot D\right) \cdot 0.25\right)} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d} \]
      times-frac [=>] 54.8	\[ \color{blue}{\frac{\left(D \cdot D\right) \cdot 0.25}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}} \]
      associate-/l* [=>] 55.1	\[ \frac{\left(D \cdot D\right) \cdot 0.25}{d} \cdot \color{blue}{\frac{h}{\frac{d}{M \cdot M}}} \]
      associate-/l/ [<=] 57.4	\[ \frac{\left(D \cdot D\right) \cdot 0.25}{d} \cdot \frac{h}{\color{blue}{\frac{\frac{d}{M}}{M}}} \]
      times-frac [<=] 56.3	\[ \color{blue}{\frac{\left(\left(D \cdot D\right) \cdot 0.25\right) \cdot h}{d \cdot \frac{\frac{d}{M}}{M}}} \]
      associate-*l/ [<=] 56.5	\[ \color{blue}{\frac{\left(D \cdot D\right) \cdot 0.25}{d \cdot \frac{\frac{d}{M}}{M}} \cdot h} \]

   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))))) < 9.9999999999999992e227

   1. Initial program 83.9%

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

   2. Taylor expanded in c0 around inf 87.5%

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

   3. Simplified 87.5%

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

      [Start] 87.5	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]
      associate-*r/ [=>] 87.5	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left({d}^{2} \cdot c0\right)}{{D}^{2} \cdot \left(w \cdot h\right)}} \]
      unpow2 [=>] 87.5	\[ \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\color{blue}{\left(d \cdot d\right)} \cdot c0\right)}{{D}^{2} \cdot \left(w \cdot h\right)} \]
      *-commutative [<=] 87.5	\[ \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \color{blue}{\left(c0 \cdot \left(d \cdot d\right)\right)}}{{D}^{2} \cdot \left(w \cdot h\right)} \]
      unpow2 [=>] 87.5	\[ \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{\left(D \cdot D\right)} \cdot \left(w \cdot h\right)} \]
      *-commutative [=>] 87.5	\[ \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} \]

   if 9.9999999999999992e227 < (*.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 0.1%

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

   2. Simplified 0.4%

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

      [Start] 0.1	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      associate-*l/ [<=] 0.1	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \left(d \cdot d\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      *-commutative [=>] 0.1	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(d \cdot d\right) \cdot \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      fma-def [=>] 0.1	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(d \cdot d, \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)} \]

   3. Taylor expanded in c0 around -inf 1.8%

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

   4. Simplified 46.7%

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

      [Start] 1.8	\[ \frac{c0}{2 \cdot w} \cdot \left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0} + -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right) \]
      fma-def [=>] 1.8	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0}, -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right)} \]

   5. Taylor expanded in c0 around 0 45.2%

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

   6. Simplified 48.5%

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

      [Start] 45.2	\[ 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \]
      unpow2 [=>] 45.2	\[ 0.25 \cdot \frac{{D}^{2} \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot h\right)}{{d}^{2}} \]
      associate-*r* [<=] 48.3	\[ 0.25 \cdot \frac{{D}^{2} \cdot \color{blue}{\left(M \cdot \left(M \cdot h\right)\right)}}{{d}^{2}} \]
      associate-/l* [=>] 48.5	\[ 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{M \cdot \left(M \cdot h\right)}}} \]
      unpow2 [=>] 48.5	\[ 0.25 \cdot \frac{\color{blue}{D \cdot D}}{\frac{{d}^{2}}{M \cdot \left(M \cdot h\right)}} \]
      unpow2 [=>] 48.5	\[ 0.25 \cdot \frac{D \cdot D}{\frac{\color{blue}{d \cdot d}}{M \cdot \left(M \cdot h\right)}} \]

   7. Applied egg-rr 70.6%

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

      [Start] 48.5	\[ 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{M \cdot \left(M \cdot h\right)}} \]
      times-frac [=>] 58.6	\[ 0.25 \cdot \frac{D \cdot D}{\color{blue}{\frac{d}{M} \cdot \frac{d}{M \cdot h}}} \]
      times-frac [=>] 70.6	\[ 0.25 \cdot \color{blue}{\left(\frac{D}{\frac{d}{M}} \cdot \frac{D}{\frac{d}{M \cdot h}}\right)} \]

   8. Applied egg-rr 78.3%

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

      [Start] 70.6	\[ 0.25 \cdot \left(\frac{D}{\frac{d}{M}} \cdot \frac{D}{\frac{d}{M \cdot h}}\right) \]
      associate-/r* [=>] 75.0	\[ 0.25 \cdot \left(\frac{D}{\frac{d}{M}} \cdot \frac{D}{\color{blue}{\frac{\frac{d}{M}}{h}}}\right) \]
      associate-/r/ [=>] 78.4	\[ 0.25 \cdot \left(\frac{D}{\frac{d}{M}} \cdot \color{blue}{\left(\frac{D}{\frac{d}{M}} \cdot h\right)}\right) \]
      div-inv [=>] 78.3	\[ 0.25 \cdot \left(\frac{D}{\frac{d}{M}} \cdot \left(\color{blue}{\left(D \cdot \frac{1}{\frac{d}{M}}\right)} \cdot h\right)\right) \]
      clear-num [<=] 78.3	\[ 0.25 \cdot \left(\frac{D}{\frac{d}{M}} \cdot \left(\left(D \cdot \color{blue}{\frac{M}{d}}\right) \cdot h\right)\right) \]

3. Recombined 4 regimes into one program.

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq -1 \cdot 10^{+74}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot {\left(\frac{d}{D} \cdot \sqrt{\frac{c0}{w \cdot h}}\right)}^{2}\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 0:\\ \;\;\;\;h \cdot \left(\frac{D}{d} \cdot \left(\frac{M}{\frac{d}{M}} \cdot \left(D \cdot 0.25\right)\right)\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 10^{+228}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{D}{\frac{d}{M}} \cdot \left(h \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	66.0%
Cost	1225

\[\begin{array}{l} \mathbf{if}\;d \leq -3 \cdot 10^{+38} \lor \neg \left(d \leq -3 \cdot 10^{-172}\right):\\ \;\;\;\;0.25 \cdot \left(D \cdot \frac{M \cdot \left(D \cdot \frac{M}{d}\right)}{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{D}{\frac{d}{h \cdot M}}\right)\\ \end{array} \]

Alternative 2
Accuracy	64.9%
Cost	1225

\[\begin{array}{l} \mathbf{if}\;M \leq -6 \cdot 10^{-200} \lor \neg \left(M \leq 5.5 \cdot 10^{+71}\right):\\ \;\;\;\;0.25 \cdot \left(D \cdot \frac{M \cdot \left(D \cdot \frac{M}{d}\right)}{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(M \cdot \left(\frac{\frac{D}{d}}{d} \cdot \left(D \cdot \left(h \cdot M\right)\right)\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	62.2%
Cost	960

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

Alternative 4
Accuracy	61.8%
Cost	960

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

Alternative 5
Accuracy	65.0%
Cost	960

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

Alternative 6
Accuracy	68.5%
Cost	960

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

Alternative 7
Accuracy	74.0%
Cost	960

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

Alternative 8
Accuracy	50.4%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023152 
(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))))))