Henrywood and Agarwal, Equation (13)

Percentage Accurate: 25.2% → 72.0%

Time: 46.9s

Precision: binary64

Cost: 36237

• Unsound rule application detected in e-graph. Results may not be sound.

• could not determine a ground truth

  1. c0 = -3.4405165695783617e+248
  2. w = 1.2627104096679672e-303
  3. h = 7.51938790541639e-195
  4. D = 2.1855805455467244e-186
  5. d = 7.002294375962175e+290
  6. M = 0.00047445757330859907

Math

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

↓

\[\begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_1 := \frac{c0}{2 \cdot w} \cdot \left(t_0 + \sqrt{t_0 \cdot t_0 - M \cdot M}\right)\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{\left(c0 \cdot \left(2 \cdot \frac{0.5}{w}\right)\right) \cdot \left(d \cdot \left(\frac{d}{h} \cdot \frac{c0}{D}\right)\right)}{w \cdot D}\\ \mathbf{elif}\;t_1 \leq 0 \lor \neg \left(t_1 \leq \infty\right):\\ \;\;\;\;0.25 \cdot \frac{M \cdot \left(\frac{D}{d} \cdot \left(h \cdot M\right)\right)}{\frac{d}{D}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{d}{D} \cdot \frac{c0}{w}\right)}^{2}}{h}\\ \end{array} \]

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)) (* (* w h) (* D D))))
        (t_1 (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))))
   (if (<= t_1 (- INFINITY))
     (/ (* (* c0 (* 2.0 (/ 0.5 w))) (* d (* (/ d h) (/ c0 D)))) (* w D))
     (if (or (<= t_1 0.0) (not (<= t_1 INFINITY)))
       (* 0.25 (/ (* M (* (/ D d) (* h M))) (/ d D)))
       (/ (pow (* (/ d D) (/ c0 w)) 2.0) h)))))

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)) / ((w * h) * (D * D));
	double t_1 = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((c0 * (2.0 * (0.5 / w))) * (d * ((d / h) * (c0 / D)))) / (w * D);
	} else if ((t_1 <= 0.0) || !(t_1 <= ((double) INFINITY))) {
		tmp = 0.25 * ((M * ((D / d) * (h * M))) / (d / D));
	} else {
		tmp = pow(((d / D) * (c0 / w)), 2.0) / h;
	}
	return tmp;
}

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)) / ((w * h) * (D * D));
	double t_1 = (c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = ((c0 * (2.0 * (0.5 / w))) * (d * ((d / h) * (c0 / D)))) / (w * D);
	} else if ((t_1 <= 0.0) || !(t_1 <= Double.POSITIVE_INFINITY)) {
		tmp = 0.25 * ((M * ((D / d) * (h * M))) / (d / D));
	} else {
		tmp = Math.pow(((d / D) * (c0 / w)), 2.0) / h;
	}
	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)) / ((w * h) * (D * D))
	t_1 = (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = ((c0 * (2.0 * (0.5 / w))) * (d * ((d / h) * (c0 / D)))) / (w * D)
	elif (t_1 <= 0.0) or not (t_1 <= math.inf):
		tmp = 0.25 * ((M * ((D / d) * (h * M))) / (d / D))
	else:
		tmp = math.pow(((d / D) * (c0 / w)), 2.0) / h
	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(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	t_1 = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(c0 * Float64(2.0 * Float64(0.5 / w))) * Float64(d * Float64(Float64(d / h) * Float64(c0 / D)))) / Float64(w * D));
	elseif ((t_1 <= 0.0) || !(t_1 <= Inf))
		tmp = Float64(0.25 * Float64(Float64(M * Float64(Float64(D / d) * Float64(h * M))) / Float64(d / D)));
	else
		tmp = Float64((Float64(Float64(d / D) * Float64(c0 / w)) ^ 2.0) / h);
	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)) / ((w * h) * (D * D));
	t_1 = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = ((c0 * (2.0 * (0.5 / w))) * (d * ((d / h) * (c0 / D)))) / (w * D);
	elseif ((t_1 <= 0.0) || ~((t_1 <= Inf)))
		tmp = 0.25 * ((M * ((D / d) * (h * M))) / (d / D));
	else
		tmp = (((d / D) * (c0 / w)) ^ 2.0) / h;
	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[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(c0 * N[(2.0 * N[(0.5 / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d * N[(N[(d / h), $MachinePrecision] * N[(c0 / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(w * D), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$1, 0.0], N[Not[LessEqual[t$95$1, Infinity]], $MachinePrecision]], N[(0.25 * N[(N[(M * N[(N[(D / d), $MachinePrecision] * N[(h * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(d / D), $MachinePrecision] * N[(c0 / w), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / h), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 87.5%

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

   2. Simplified 87.1%

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

      [Start] 87.5	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      times-frac [=>] 87.1	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      fma-def [=>] 87.1	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d \cdot d}{D \cdot D}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)} \]
      associate-/r* [=>] 87.1	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \color{blue}{\frac{\frac{d \cdot d}{D}}{D}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      difference-of-squares [=>] 87.1	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{\frac{d \cdot d}{D}}{D}, \sqrt{\color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + M\right) \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M\right)}}\right) \]

   3. Taylor expanded in c0 around inf 87.5%

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

   4. Simplified 80.9%

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

      [Start] 87.5	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]
      times-frac [=>] 87.2	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{{d}^{2}}{{D}^{2}} \cdot \frac{c0}{w \cdot h}\right)}\right) \]
      associate-*l/ [=>] 87.2	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{{d}^{2} \cdot \frac{c0}{w \cdot h}}{{D}^{2}}}\right) \]
      unpow2 [=>] 87.2	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2} \cdot \frac{c0}{w \cdot h}}{\color{blue}{D \cdot D}}\right) \]
      times-frac [=>] 87.2	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{{d}^{2}}{D} \cdot \frac{\frac{c0}{w \cdot h}}{D}\right)}\right) \]
      unpow2 [=>] 87.2	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\color{blue}{d \cdot d}}{D} \cdot \frac{\frac{c0}{w \cdot h}}{D}\right)\right) \]
      associate-*r/ [=>] 87.2	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{\frac{d \cdot d}{D} \cdot \frac{c0}{w \cdot h}}{D}}\right) \]
      associate-*l/ [<=] 87.2	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{\frac{d \cdot d}{D}}{D} \cdot \frac{c0}{w \cdot h}\right)}\right) \]
      associate-/r* [=>] 87.2	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\frac{d \cdot d}{D}}{D} \cdot \color{blue}{\frac{\frac{c0}{w}}{h}}\right)\right) \]
      times-frac [<=] 81.0	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{\frac{d \cdot d}{D} \cdot \frac{c0}{w}}{D \cdot h}}\right) \]
      associate-/l* [=>] 80.9	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{\frac{d \cdot d}{D}}{\frac{D \cdot h}{\frac{c0}{w}}}}\right) \]
      associate-/l* [=>] 80.9	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\color{blue}{\frac{d}{\frac{D}{d}}}}{\frac{D \cdot h}{\frac{c0}{w}}}\right) \]

   5. Applied egg-rr 84.1%

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

      [Start] 80.9	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{d}{\frac{D}{d}}}{\frac{D \cdot h}{\frac{c0}{w}}}\right) \]
      div-inv [=>] 80.9	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\color{blue}{d \cdot \frac{1}{\frac{D}{d}}}}{\frac{D \cdot h}{\frac{c0}{w}}}\right) \]
      associate-/r/ [=>] 80.9	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{d \cdot \frac{1}{\frac{D}{d}}}{\color{blue}{\frac{D \cdot h}{c0} \cdot w}}\right) \]
      times-frac [=>] 80.9	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{d}{\frac{D \cdot h}{c0}} \cdot \frac{\frac{1}{\frac{D}{d}}}{w}\right)}\right) \]
      *-commutative [=>] 80.9	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{d}{\frac{\color{blue}{h \cdot D}}{c0}} \cdot \frac{\frac{1}{\frac{D}{d}}}{w}\right)\right) \]
      associate-/l* [=>] 84.1	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{d}{\color{blue}{\frac{h}{\frac{c0}{D}}}} \cdot \frac{\frac{1}{\frac{D}{d}}}{w}\right)\right) \]
      clear-num [<=] 84.1	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{d}{\frac{h}{\frac{c0}{D}}} \cdot \frac{\color{blue}{\frac{d}{D}}}{w}\right)\right) \]

   6. Simplified 87.5%

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

      [Start] 84.1	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{d}{\frac{h}{\frac{c0}{D}}} \cdot \frac{\frac{d}{D}}{w}\right)\right) \]
      associate-/r/ [=>] 87.5	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\color{blue}{\left(\frac{d}{h} \cdot \frac{c0}{D}\right)} \cdot \frac{\frac{d}{D}}{w}\right)\right) \]
      associate-/l/ [=>] 87.5	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\left(\frac{d}{h} \cdot \frac{c0}{D}\right) \cdot \color{blue}{\frac{d}{w \cdot D}}\right)\right) \]

   7. Applied egg-rr 93.6%

      \[\leadsto \color{blue}{\frac{\left(c0 \cdot \left(\frac{0.5}{w} \cdot 2\right)\right) \cdot \left(d \cdot \left(\frac{d}{h} \cdot \frac{c0}{D}\right)\right)}{D \cdot w}} \]
      Step-by-step derivation

      [Start] 87.5	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\left(\frac{d}{h} \cdot \frac{c0}{D}\right) \cdot \frac{d}{w \cdot D}\right)\right) \]
      associate-*r* [=>] 87.5	\[ \color{blue}{\left(\frac{c0}{2 \cdot w} \cdot 2\right) \cdot \left(\left(\frac{d}{h} \cdot \frac{c0}{D}\right) \cdot \frac{d}{w \cdot D}\right)} \]
      associate-*r/ [=>] 90.5	\[ \left(\frac{c0}{2 \cdot w} \cdot 2\right) \cdot \color{blue}{\frac{\left(\frac{d}{h} \cdot \frac{c0}{D}\right) \cdot d}{w \cdot D}} \]
      associate-*r/ [=>] 93.6	\[ \color{blue}{\frac{\left(\frac{c0}{2 \cdot w} \cdot 2\right) \cdot \left(\left(\frac{d}{h} \cdot \frac{c0}{D}\right) \cdot d\right)}{w \cdot D}} \]
      div-inv [=>] 93.6	\[ \frac{\left(\color{blue}{\left(c0 \cdot \frac{1}{2 \cdot w}\right)} \cdot 2\right) \cdot \left(\left(\frac{d}{h} \cdot \frac{c0}{D}\right) \cdot d\right)}{w \cdot D} \]
      associate-*l* [=>] 93.6	\[ \frac{\color{blue}{\left(c0 \cdot \left(\frac{1}{2 \cdot w} \cdot 2\right)\right)} \cdot \left(\left(\frac{d}{h} \cdot \frac{c0}{D}\right) \cdot d\right)}{w \cdot D} \]
      associate-/r* [=>] 93.6	\[ \frac{\left(c0 \cdot \left(\color{blue}{\frac{\frac{1}{2}}{w}} \cdot 2\right)\right) \cdot \left(\left(\frac{d}{h} \cdot \frac{c0}{D}\right) \cdot d\right)}{w \cdot D} \]
      metadata-eval [=>] 93.6	\[ \frac{\left(c0 \cdot \left(\frac{\color{blue}{0.5}}{w} \cdot 2\right)\right) \cdot \left(\left(\frac{d}{h} \cdot \frac{c0}{D}\right) \cdot d\right)}{w \cdot D} \]
      *-commutative [=>] 93.6	\[ \frac{\left(c0 \cdot \left(\frac{0.5}{w} \cdot 2\right)\right) \cdot \color{blue}{\left(d \cdot \left(\frac{d}{h} \cdot \frac{c0}{D}\right)\right)}}{w \cdot D} \]
      *-commutative [=>] 93.6	\[ \frac{\left(c0 \cdot \left(\frac{0.5}{w} \cdot 2\right)\right) \cdot \left(d \cdot \left(\frac{d}{h} \cdot \frac{c0}{D}\right)\right)}{\color{blue}{D \cdot w}} \]

   if -inf.0 < (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < -0.0 or +inf.0 < (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 4.8%

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

   2. Simplified 14.5%

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

      [Start] 4.8	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      times-frac [=>] 1.9	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      fma-def [=>] 1.9	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d \cdot d}{D \cdot D}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)} \]
      associate-/r* [=>] 1.9	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \color{blue}{\frac{\frac{d \cdot d}{D}}{D}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      difference-of-squares [=>] 7.4	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{\frac{d \cdot d}{D}}{D}, \sqrt{\color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + M\right) \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M\right)}}\right) \]

   3. Taylor expanded in c0 around -inf 4.3%

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

   4. Simplified 28.7%

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

      [Start] 4.3

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

      associate-*r* [=>] 4.3

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

   5. Taylor expanded in c0 around 0 43.1%

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

   6. Simplified 59.7%

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

      [Start] 43.1	\[ 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \]
      unpow2 [=>] 43.1	\[ 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{d \cdot d}} \]
      unpow2 [=>] 43.1	\[ 0.25 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)}{d \cdot d} \]
      *-commutative [<=] 43.1	\[ 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{d \cdot d} \]
      associate-/l* [=>] 40.6	\[ 0.25 \cdot \color{blue}{\frac{D \cdot D}{\frac{d \cdot d}{h \cdot {M}^{2}}}} \]
      associate-/r/ [=>] 42.1	\[ 0.25 \cdot \color{blue}{\left(\frac{D \cdot D}{d \cdot d} \cdot \left(h \cdot {M}^{2}\right)\right)} \]
      times-frac [=>] 57.6	\[ 0.25 \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \left(h \cdot {M}^{2}\right)\right) \]
      unpow2 [<=] 57.6	\[ 0.25 \cdot \left(\color{blue}{{\left(\frac{D}{d}\right)}^{2}} \cdot \left(h \cdot {M}^{2}\right)\right) \]
      unpow2 [=>] 57.6	\[ 0.25 \cdot \left({\left(\frac{D}{d}\right)}^{2} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right) \]
      associate-*r* [=>] 59.7	\[ 0.25 \cdot \left({\left(\frac{D}{d}\right)}^{2} \cdot \color{blue}{\left(\left(h \cdot M\right) \cdot M\right)}\right) \]

   7. Applied egg-rr 59.7%

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

      [Start] 59.7	\[ 0.25 \cdot \left({\left(\frac{D}{d}\right)}^{2} \cdot \left(\left(h \cdot M\right) \cdot M\right)\right) \]
      unpow2 [=>] 59.7	\[ 0.25 \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \left(\left(h \cdot M\right) \cdot M\right)\right) \]

   8. Applied egg-rr 68.6%

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

      [Start] 59.7	\[ 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\left(h \cdot M\right) \cdot M\right)\right) \]
      associate-*l* [=>] 62.5	\[ 0.25 \cdot \color{blue}{\left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \left(\left(h \cdot M\right) \cdot M\right)\right)\right)} \]
      clear-num [=>] 62.5	\[ 0.25 \cdot \left(\color{blue}{\frac{1}{\frac{d}{D}}} \cdot \left(\frac{D}{d} \cdot \left(\left(h \cdot M\right) \cdot M\right)\right)\right) \]
      associate-*l/ [=>] 62.6	\[ 0.25 \cdot \color{blue}{\frac{1 \cdot \left(\frac{D}{d} \cdot \left(\left(h \cdot M\right) \cdot M\right)\right)}{\frac{d}{D}}} \]
      *-un-lft-identity [<=] 62.6	\[ 0.25 \cdot \frac{\color{blue}{\frac{D}{d} \cdot \left(\left(h \cdot M\right) \cdot M\right)}}{\frac{d}{D}} \]
      associate-*r* [=>] 68.6	\[ 0.25 \cdot \frac{\color{blue}{\left(\frac{D}{d} \cdot \left(h \cdot M\right)\right) \cdot M}}{\frac{d}{D}} \]

   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))))) < +inf.0

   1. Initial program 75.0%

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

   2. Simplified 76.9%

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

      [Start] 75.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      times-frac [=>] 74.5	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      fma-def [=>] 74.5	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d \cdot d}{D \cdot D}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)} \]
      associate-/r* [=>] 74.6	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \color{blue}{\frac{\frac{d \cdot d}{D}}{D}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      difference-of-squares [=>] 74.6	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{\frac{d \cdot d}{D}}{D}, \sqrt{\color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + M\right) \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M\right)}}\right) \]

   3. Taylor expanded in c0 around inf 75.2%

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

   4. Simplified 77.6%

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

      [Start] 75.2	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]
      times-frac [=>] 72.9	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{{d}^{2}}{{D}^{2}} \cdot \frac{c0}{w \cdot h}\right)}\right) \]
      associate-*l/ [=>] 72.9	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{{d}^{2} \cdot \frac{c0}{w \cdot h}}{{D}^{2}}}\right) \]
      unpow2 [=>] 72.9	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2} \cdot \frac{c0}{w \cdot h}}{\color{blue}{D \cdot D}}\right) \]
      times-frac [=>] 75.3	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{{d}^{2}}{D} \cdot \frac{\frac{c0}{w \cdot h}}{D}\right)}\right) \]
      unpow2 [=>] 75.3	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\color{blue}{d \cdot d}}{D} \cdot \frac{\frac{c0}{w \cdot h}}{D}\right)\right) \]
      associate-*r/ [=>] 75.3	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{\frac{d \cdot d}{D} \cdot \frac{c0}{w \cdot h}}{D}}\right) \]
      associate-*l/ [<=] 75.3	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{\frac{d \cdot d}{D}}{D} \cdot \frac{c0}{w \cdot h}\right)}\right) \]
      associate-/r* [=>] 77.6	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\frac{d \cdot d}{D}}{D} \cdot \color{blue}{\frac{\frac{c0}{w}}{h}}\right)\right) \]
      times-frac [<=] 77.4	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{\frac{d \cdot d}{D} \cdot \frac{c0}{w}}{D \cdot h}}\right) \]
      associate-/l* [=>] 75.2	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{\frac{d \cdot d}{D}}{\frac{D \cdot h}{\frac{c0}{w}}}}\right) \]
      associate-/l* [=>] 77.6	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\color{blue}{\frac{d}{\frac{D}{d}}}}{\frac{D \cdot h}{\frac{c0}{w}}}\right) \]

   5. Applied egg-rr 71.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{\frac{d}{D}}{h}\right)\right) \cdot \left(c0 \cdot \left(\frac{0.5}{w} \cdot 2\right)\right)\right)} - 1} \]
      Step-by-step derivation

      [Start] 77.6	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{d}{\frac{D}{d}}}{\frac{D \cdot h}{\frac{c0}{w}}}\right) \]
      expm1-log1p-u [=>] 77.2	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{d}{\frac{D}{d}}}{\frac{D \cdot h}{\frac{c0}{w}}}\right)\right)\right)} \]
      expm1-udef [=>] 71.9	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{d}{\frac{D}{d}}}{\frac{D \cdot h}{\frac{c0}{w}}}\right)\right)} - 1} \]

   6. Simplified 73.7%

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

      [Start] 71.4	\[ e^{\mathsf{log1p}\left(\left(\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{\frac{d}{D}}{h}\right)\right) \cdot \left(c0 \cdot \left(\frac{0.5}{w} \cdot 2\right)\right)\right)} - 1 \]
      expm1-def [=>] 73.6	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{\frac{d}{D}}{h}\right)\right) \cdot \left(c0 \cdot \left(\frac{0.5}{w} \cdot 2\right)\right)\right)\right)} \]
      expm1-log1p [=>] 74.0	\[ \color{blue}{\left(\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{\frac{d}{D}}{h}\right)\right) \cdot \left(c0 \cdot \left(\frac{0.5}{w} \cdot 2\right)\right)} \]
      *-commutative [=>] 74.0	\[ \color{blue}{\left(c0 \cdot \left(\frac{0.5}{w} \cdot 2\right)\right) \cdot \left(\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{\frac{d}{D}}{h}\right)\right)} \]
      associate-*r* [=>] 73.7	\[ \color{blue}{\left(\left(c0 \cdot \left(\frac{0.5}{w} \cdot 2\right)\right) \cdot \frac{c0}{w}\right) \cdot \left(\frac{d}{D} \cdot \frac{\frac{d}{D}}{h}\right)} \]
      associate-*l/ [=>] 73.7	\[ \left(\left(c0 \cdot \color{blue}{\frac{0.5 \cdot 2}{w}}\right) \cdot \frac{c0}{w}\right) \cdot \left(\frac{d}{D} \cdot \frac{\frac{d}{D}}{h}\right) \]
      metadata-eval [=>] 73.7	\[ \left(\left(c0 \cdot \frac{\color{blue}{1}}{w}\right) \cdot \frac{c0}{w}\right) \cdot \left(\frac{d}{D} \cdot \frac{\frac{d}{D}}{h}\right) \]
      *-commutative [=>] 73.7	\[ \left(\color{blue}{\left(\frac{1}{w} \cdot c0\right)} \cdot \frac{c0}{w}\right) \cdot \left(\frac{d}{D} \cdot \frac{\frac{d}{D}}{h}\right) \]
      associate-*l/ [=>] 73.7	\[ \left(\color{blue}{\frac{1 \cdot c0}{w}} \cdot \frac{c0}{w}\right) \cdot \left(\frac{d}{D} \cdot \frac{\frac{d}{D}}{h}\right) \]
      associate-*r/ [<=] 73.7	\[ \left(\color{blue}{\left(1 \cdot \frac{c0}{w}\right)} \cdot \frac{c0}{w}\right) \cdot \left(\frac{d}{D} \cdot \frac{\frac{d}{D}}{h}\right) \]
      *-lft-identity [=>] 73.7	\[ \left(\color{blue}{\frac{c0}{w}} \cdot \frac{c0}{w}\right) \cdot \left(\frac{d}{D} \cdot \frac{\frac{d}{D}}{h}\right) \]
      associate-*r/ [=>] 73.7	\[ \left(\frac{c0}{w} \cdot \frac{c0}{w}\right) \cdot \color{blue}{\frac{\frac{d}{D} \cdot \frac{d}{D}}{h}} \]

   7. Taylor expanded in c0 around 0 61.7%

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

   8. Simplified 83.8%

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

      [Start] 61.7	\[ \frac{{d}^{2} \cdot {c0}^{2}}{{D}^{2} \cdot \left({w}^{2} \cdot h\right)} \]
      unpow2 [=>] 61.7	\[ \frac{{d}^{2} \cdot \color{blue}{\left(c0 \cdot c0\right)}}{{D}^{2} \cdot \left({w}^{2} \cdot h\right)} \]
      unpow2 [=>] 61.7	\[ \frac{{d}^{2} \cdot \left(c0 \cdot c0\right)}{\color{blue}{\left(D \cdot D\right)} \cdot \left({w}^{2} \cdot h\right)} \]
      times-frac [=>] 61.7	\[ \color{blue}{\frac{{d}^{2}}{D \cdot D} \cdot \frac{c0 \cdot c0}{{w}^{2} \cdot h}} \]
      associate-/r* [=>] 61.8	\[ \color{blue}{\frac{\frac{{d}^{2}}{D}}{D}} \cdot \frac{c0 \cdot c0}{{w}^{2} \cdot h} \]
      unpow2 [=>] 61.8	\[ \frac{\frac{\color{blue}{d \cdot d}}{D}}{D} \cdot \frac{c0 \cdot c0}{{w}^{2} \cdot h} \]
      associate-*r/ [<=] 61.9	\[ \frac{\color{blue}{d \cdot \frac{d}{D}}}{D} \cdot \frac{c0 \cdot c0}{{w}^{2} \cdot h} \]
      associate-*l/ [<=] 61.9	\[ \color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot c0}{{w}^{2} \cdot h} \]
      unpow2 [<=] 61.9	\[ \color{blue}{{\left(\frac{d}{D}\right)}^{2}} \cdot \frac{c0 \cdot c0}{{w}^{2} \cdot h} \]
      unpow2 [=>] 61.9	\[ {\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0 \cdot c0}{\color{blue}{\left(w \cdot w\right)} \cdot h} \]
      associate-/r* [=>] 62.0	\[ {\left(\frac{d}{D}\right)}^{2} \cdot \color{blue}{\frac{\frac{c0 \cdot c0}{w \cdot w}}{h}} \]
      times-frac [=>] 76.8	\[ {\left(\frac{d}{D}\right)}^{2} \cdot \frac{\color{blue}{\frac{c0}{w} \cdot \frac{c0}{w}}}{h} \]
      unpow2 [<=] 76.8	\[ {\left(\frac{d}{D}\right)}^{2} \cdot \frac{\color{blue}{{\left(\frac{c0}{w}\right)}^{2}}}{h} \]
      *-commutative [=>] 76.8	\[ \color{blue}{\frac{{\left(\frac{c0}{w}\right)}^{2}}{h} \cdot {\left(\frac{d}{D}\right)}^{2}} \]
      associate-*l/ [=>] 79.1	\[ \color{blue}{\frac{{\left(\frac{c0}{w}\right)}^{2} \cdot {\left(\frac{d}{D}\right)}^{2}}{h}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq -\infty:\\ \;\;\;\;\frac{\left(c0 \cdot \left(2 \cdot \frac{0.5}{w}\right)\right) \cdot \left(d \cdot \left(\frac{d}{h} \cdot \frac{c0}{D}\right)\right)}{w \cdot D}\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 0 \lor \neg \left(\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty\right):\\ \;\;\;\;0.25 \cdot \frac{M \cdot \left(\frac{D}{d} \cdot \left(h \cdot M\right)\right)}{\frac{d}{D}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{d}{D} \cdot \frac{c0}{w}\right)}^{2}}{h}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	48.4%
Cost	7304

\[\begin{array}{l} t_0 := \frac{d}{\frac{D}{d}}\\ t_1 := M \cdot \left(\frac{D}{d} \cdot \left(h \cdot M\right)\right)\\ t_2 := 0.25 \cdot \frac{t_1}{\frac{d}{D}}\\ t_3 := \frac{c0}{2 \cdot w}\\ \mathbf{if}\;c0 \leq -1.65 \cdot 10^{+210}:\\ \;\;\;\;t_3 \cdot \left(2 \cdot \frac{t_0}{\frac{h \cdot D}{\frac{c0}{w}}}\right)\\ \mathbf{elif}\;c0 \leq -3.4 \cdot 10^{+193}:\\ \;\;\;\;0.25 \cdot \left(h \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\\ \mathbf{elif}\;c0 \leq -2.8 \cdot 10^{+163}:\\ \;\;\;\;\frac{2 \cdot \left(d \cdot \frac{d}{D}\right)}{\frac{w}{c0 \cdot 0.5} \cdot \left(D \cdot \left(h \cdot \frac{w}{c0}\right)\right)}\\ \mathbf{elif}\;c0 \leq -4.5 \cdot 10^{-7}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c0 \leq -2.6 \cdot 10^{-47}:\\ \;\;\;\;\frac{c0}{\frac{w \cdot w}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{h}\\ \mathbf{elif}\;c0 \leq 2.75 \cdot 10^{-80}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c0 \leq 860000000:\\ \;\;\;\;t_3 \cdot \left(2 \cdot \frac{t_0}{\frac{\left(w \cdot h\right) \cdot D}{c0}}\right)\\ \mathbf{elif}\;c0 \leq 4.5 \cdot 10^{+67}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot t_1}{d}\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \left(2 \cdot \frac{\frac{d}{w \cdot D} \cdot \left(c0 \cdot \frac{d}{h}\right)}{D}\right)\\ \end{array} \]

Alternative 2
Accuracy	47.9%
Cost	2136

\[\begin{array}{l} t_0 := M \cdot \left(\frac{D}{d} \cdot \left(h \cdot M\right)\right)\\ t_1 := \frac{c0}{2 \cdot w}\\ \mathbf{if}\;c0 \leq -6.2 \cdot 10^{+211}:\\ \;\;\;\;\frac{c0}{w} \cdot \left(\frac{d}{w \cdot \frac{D}{d}} \cdot \frac{c0}{h \cdot D}\right)\\ \mathbf{elif}\;c0 \leq -8.8 \cdot 10^{-8}:\\ \;\;\;\;0.25 \cdot \left(h \cdot \frac{\left(M \cdot \frac{D}{d}\right) \cdot \left(D \cdot M\right)}{d}\right)\\ \mathbf{elif}\;c0 \leq -3.4 \cdot 10^{-43}:\\ \;\;\;\;\frac{c0}{\frac{w \cdot w}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{h}\\ \mathbf{elif}\;c0 \leq 1.3 \cdot 10^{-86}:\\ \;\;\;\;0.25 \cdot \frac{t_0}{\frac{d}{D}}\\ \mathbf{elif}\;c0 \leq 35000000000:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \left(\frac{d}{h} \cdot \frac{\frac{d}{D}}{D}\right)\right)\right)\\ \mathbf{elif}\;c0 \leq 3.7 \cdot 10^{+65}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot t_0}{d}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \frac{\frac{d}{w \cdot D} \cdot \left(c0 \cdot \frac{d}{h}\right)}{D}\right)\\ \end{array} \]

Alternative 3
Accuracy	47.9%
Cost	2136

\[\begin{array}{l} t_0 := M \cdot \left(\frac{D}{d} \cdot \left(h \cdot M\right)\right)\\ t_1 := \frac{c0}{2 \cdot w}\\ \mathbf{if}\;c0 \leq -3.2 \cdot 10^{+208}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \frac{\frac{d}{\frac{D}{d}}}{\frac{h \cdot D}{\frac{c0}{w}}}\right)\\ \mathbf{elif}\;c0 \leq -1.25 \cdot 10^{-7}:\\ \;\;\;\;0.25 \cdot \left(h \cdot \frac{\left(M \cdot \frac{D}{d}\right) \cdot \left(D \cdot M\right)}{d}\right)\\ \mathbf{elif}\;c0 \leq -2.6 \cdot 10^{-47}:\\ \;\;\;\;\frac{c0}{\frac{w \cdot w}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{h}\\ \mathbf{elif}\;c0 \leq 5.2 \cdot 10^{-98}:\\ \;\;\;\;0.25 \cdot \frac{t_0}{\frac{d}{D}}\\ \mathbf{elif}\;c0 \leq 15200000000:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \left(\frac{d}{h} \cdot \frac{\frac{d}{D}}{D}\right)\right)\right)\\ \mathbf{elif}\;c0 \leq 6.2 \cdot 10^{+62}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot t_0}{d}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \frac{\frac{d}{w \cdot D} \cdot \left(c0 \cdot \frac{d}{h}\right)}{D}\right)\\ \end{array} \]

Alternative 4
Accuracy	48.1%
Cost	2136

\[\begin{array}{l} t_0 := \frac{d}{\frac{D}{d}}\\ t_1 := M \cdot \left(\frac{D}{d} \cdot \left(h \cdot M\right)\right)\\ t_2 := \frac{c0}{2 \cdot w}\\ \mathbf{if}\;c0 \leq -8.8 \cdot 10^{+207}:\\ \;\;\;\;t_2 \cdot \left(2 \cdot \frac{t_0}{\frac{h \cdot D}{\frac{c0}{w}}}\right)\\ \mathbf{elif}\;c0 \leq -1.04 \cdot 10^{-7}:\\ \;\;\;\;0.25 \cdot \left(h \cdot \frac{\left(M \cdot \frac{D}{d}\right) \cdot \left(D \cdot M\right)}{d}\right)\\ \mathbf{elif}\;c0 \leq -3.2 \cdot 10^{-43}:\\ \;\;\;\;\frac{c0}{\frac{w \cdot w}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{h}\\ \mathbf{elif}\;c0 \leq 6.3 \cdot 10^{-80}:\\ \;\;\;\;0.25 \cdot \frac{t_1}{\frac{d}{D}}\\ \mathbf{elif}\;c0 \leq 10000000000:\\ \;\;\;\;t_2 \cdot \left(2 \cdot \frac{t_0}{\frac{\left(w \cdot h\right) \cdot D}{c0}}\right)\\ \mathbf{elif}\;c0 \leq 1.95 \cdot 10^{+62}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot t_1}{d}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(2 \cdot \frac{\frac{d}{w \cdot D} \cdot \left(c0 \cdot \frac{d}{h}\right)}{D}\right)\\ \end{array} \]

Alternative 5
Accuracy	48.0%
Cost	2136

\[\begin{array}{l} t_0 := \frac{d}{\frac{D}{d}}\\ t_1 := M \cdot \left(\frac{D}{d} \cdot \left(h \cdot M\right)\right)\\ t_2 := \frac{c0}{2 \cdot w}\\ \mathbf{if}\;c0 \leq -1.7 \cdot 10^{+226}:\\ \;\;\;\;t_2 \cdot \left(2 \cdot \frac{t_0}{\frac{1}{c0} \cdot \frac{h \cdot D}{\frac{1}{w}}}\right)\\ \mathbf{elif}\;c0 \leq -9 \cdot 10^{-8}:\\ \;\;\;\;0.25 \cdot \left(h \cdot \frac{\left(M \cdot \frac{D}{d}\right) \cdot \left(D \cdot M\right)}{d}\right)\\ \mathbf{elif}\;c0 \leq -3.4 \cdot 10^{-43}:\\ \;\;\;\;\frac{c0}{\frac{w \cdot w}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{h}\\ \mathbf{elif}\;c0 \leq 7.6 \cdot 10^{-93}:\\ \;\;\;\;0.25 \cdot \frac{t_1}{\frac{d}{D}}\\ \mathbf{elif}\;c0 \leq 6500000000:\\ \;\;\;\;t_2 \cdot \left(2 \cdot \frac{t_0}{\frac{\left(w \cdot h\right) \cdot D}{c0}}\right)\\ \mathbf{elif}\;c0 \leq 2.2 \cdot 10^{+62}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot t_1}{d}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(2 \cdot \frac{\frac{d}{w \cdot D} \cdot \left(c0 \cdot \frac{d}{h}\right)}{D}\right)\\ \end{array} \]

Alternative 6
Accuracy	47.9%
Cost	2004

\[\begin{array}{l} t_0 := w \cdot \frac{D}{d}\\ t_1 := M \cdot \left(\frac{D}{d} \cdot \left(h \cdot M\right)\right)\\ \mathbf{if}\;c0 \leq -6 \cdot 10^{+211}:\\ \;\;\;\;\frac{c0}{w} \cdot \left(\frac{d}{t_0} \cdot \frac{c0}{h \cdot D}\right)\\ \mathbf{elif}\;c0 \leq -1.04 \cdot 10^{-7}:\\ \;\;\;\;0.25 \cdot \left(h \cdot \frac{\left(M \cdot \frac{D}{d}\right) \cdot \left(D \cdot M\right)}{d}\right)\\ \mathbf{elif}\;c0 \leq -3.4 \cdot 10^{-43}:\\ \;\;\;\;\frac{c0}{\frac{w \cdot w}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{h}\\ \mathbf{elif}\;c0 \leq 8.2 \cdot 10^{-79}:\\ \;\;\;\;0.25 \cdot \frac{t_1}{\frac{d}{D}}\\ \mathbf{elif}\;c0 \leq 1960000000:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \left(\frac{d}{h} \cdot \frac{\frac{d}{D}}{D}\right)\right)\right)\\ \mathbf{elif}\;c0 \leq 7.5 \cdot 10^{+52}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot t_1}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{w} \cdot \frac{d \cdot \frac{\frac{c0}{D}}{h}}{t_0}\\ \end{array} \]

Alternative 7
Accuracy	47.4%
Cost	1881

\[\begin{array}{l} t_0 := M \cdot \left(\frac{D}{d} \cdot \left(h \cdot M\right)\right)\\ \mathbf{if}\;c0 \leq -1.4 \cdot 10^{+211}:\\ \;\;\;\;\frac{c0}{w} \cdot \left(\frac{d}{w \cdot \frac{D}{d}} \cdot \frac{c0}{h \cdot D}\right)\\ \mathbf{elif}\;c0 \leq -8.5 \cdot 10^{-8}:\\ \;\;\;\;0.25 \cdot \left(h \cdot \frac{\left(M \cdot \frac{D}{d}\right) \cdot \left(D \cdot M\right)}{d}\right)\\ \mathbf{elif}\;c0 \leq -3.4 \cdot 10^{-43}:\\ \;\;\;\;\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot c0}{h \cdot \left(w \cdot w\right)}\\ \mathbf{elif}\;c0 \leq 1.35 \cdot 10^{-78}:\\ \;\;\;\;0.25 \cdot \frac{t_0}{\frac{d}{D}}\\ \mathbf{elif}\;c0 \leq 340000000 \lor \neg \left(c0 \leq 2.1 \cdot 10^{+54}\right):\\ \;\;\;\;\left(\frac{c0}{w} \cdot \frac{c0}{w}\right) \cdot \frac{\frac{\frac{d}{D}}{\frac{D}{d}}}{h}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot t_0}{d}\\ \end{array} \]

Alternative 8
Accuracy	48.3%
Cost	1881

\[\begin{array}{l} t_0 := w \cdot \frac{D}{d}\\ t_1 := M \cdot \left(\frac{D}{d} \cdot \left(h \cdot M\right)\right)\\ \mathbf{if}\;c0 \leq -2.8 \cdot 10^{+211}:\\ \;\;\;\;\frac{c0}{w} \cdot \left(\frac{d}{t_0} \cdot \frac{c0}{h \cdot D}\right)\\ \mathbf{elif}\;c0 \leq -1.95 \cdot 10^{-7}:\\ \;\;\;\;0.25 \cdot \left(h \cdot \frac{\left(M \cdot \frac{D}{d}\right) \cdot \left(D \cdot M\right)}{d}\right)\\ \mathbf{elif}\;c0 \leq -2.3 \cdot 10^{-49}:\\ \;\;\;\;\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot c0}{h \cdot \left(w \cdot w\right)}\\ \mathbf{elif}\;c0 \leq 1.9 \cdot 10^{-87}:\\ \;\;\;\;0.25 \cdot \frac{t_1}{\frac{d}{D}}\\ \mathbf{elif}\;c0 \leq 34000000000 \lor \neg \left(c0 \leq 1.8 \cdot 10^{+53}\right):\\ \;\;\;\;\frac{c0}{w} \cdot \frac{d \cdot \frac{\frac{c0}{D}}{h}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot t_1}{d}\\ \end{array} \]

Alternative 9
Accuracy	47.8%
Cost	1880

\[\begin{array}{l} t_0 := \frac{d}{D} \cdot \frac{d}{D}\\ t_1 := w \cdot \frac{D}{d}\\ t_2 := M \cdot \left(\frac{D}{d} \cdot \left(h \cdot M\right)\right)\\ \mathbf{if}\;c0 \leq -3.9 \cdot 10^{+208}:\\ \;\;\;\;\frac{c0}{w} \cdot \left(\frac{d}{t_1} \cdot \frac{c0}{h \cdot D}\right)\\ \mathbf{elif}\;c0 \leq -4.6 \cdot 10^{-7}:\\ \;\;\;\;0.25 \cdot \left(h \cdot \frac{\left(M \cdot \frac{D}{d}\right) \cdot \left(D \cdot M\right)}{d}\right)\\ \mathbf{elif}\;c0 \leq -3.4 \cdot 10^{-43}:\\ \;\;\;\;t_0 \cdot \frac{c0 \cdot c0}{h \cdot \left(w \cdot w\right)}\\ \mathbf{elif}\;c0 \leq 5.5 \cdot 10^{-79}:\\ \;\;\;\;0.25 \cdot \frac{t_2}{\frac{d}{D}}\\ \mathbf{elif}\;c0 \leq 76000000:\\ \;\;\;\;t_0 \cdot \frac{c0}{w \cdot \left(w \cdot \frac{h}{c0}\right)}\\ \mathbf{elif}\;c0 \leq 3.7 \cdot 10^{+53}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot t_2}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{w} \cdot \frac{d \cdot \frac{\frac{c0}{D}}{h}}{t_1}\\ \end{array} \]

Alternative 10
Accuracy	47.8%
Cost	1880

\[\begin{array}{l} t_0 := \frac{d}{D} \cdot \frac{d}{D}\\ t_1 := w \cdot \frac{D}{d}\\ t_2 := M \cdot \left(\frac{D}{d} \cdot \left(h \cdot M\right)\right)\\ \mathbf{if}\;c0 \leq -2 \cdot 10^{+209}:\\ \;\;\;\;\frac{c0}{w} \cdot \left(\frac{d}{t_1} \cdot \frac{c0}{h \cdot D}\right)\\ \mathbf{elif}\;c0 \leq -1.25 \cdot 10^{-7}:\\ \;\;\;\;0.25 \cdot \left(h \cdot \frac{\left(M \cdot \frac{D}{d}\right) \cdot \left(D \cdot M\right)}{d}\right)\\ \mathbf{elif}\;c0 \leq -3.3 \cdot 10^{-44}:\\ \;\;\;\;\frac{c0}{\frac{w \cdot w}{c0}} \cdot \frac{t_0}{h}\\ \mathbf{elif}\;c0 \leq 1.42 \cdot 10^{-79}:\\ \;\;\;\;0.25 \cdot \frac{t_2}{\frac{d}{D}}\\ \mathbf{elif}\;c0 \leq 260000000:\\ \;\;\;\;t_0 \cdot \frac{c0}{w \cdot \left(w \cdot \frac{h}{c0}\right)}\\ \mathbf{elif}\;c0 \leq 2.4 \cdot 10^{+53}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot t_2}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{w} \cdot \frac{d \cdot \frac{\frac{c0}{D}}{h}}{t_1}\\ \end{array} \]

Alternative 11
Accuracy	48.3%
Cost	1880

\[\begin{array}{l} t_0 := w \cdot \frac{D}{d}\\ t_1 := M \cdot \left(\frac{D}{d} \cdot \left(h \cdot M\right)\right)\\ \mathbf{if}\;c0 \leq -9.5 \cdot 10^{+208}:\\ \;\;\;\;\frac{c0}{w} \cdot \left(\frac{d}{t_0} \cdot \frac{c0}{h \cdot D}\right)\\ \mathbf{elif}\;c0 \leq -9.5 \cdot 10^{-8}:\\ \;\;\;\;0.25 \cdot \left(h \cdot \frac{\left(M \cdot \frac{D}{d}\right) \cdot \left(D \cdot M\right)}{d}\right)\\ \mathbf{elif}\;c0 \leq -3.1 \cdot 10^{-44}:\\ \;\;\;\;\frac{c0}{\frac{w \cdot w}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{h}\\ \mathbf{elif}\;c0 \leq 1.66 \cdot 10^{-80}:\\ \;\;\;\;0.25 \cdot \frac{t_1}{\frac{d}{D}}\\ \mathbf{elif}\;c0 \leq 112000000:\\ \;\;\;\;\frac{\frac{d}{D} \cdot \left(c0 \cdot \frac{c0}{w}\right)}{w \cdot \left(D \cdot \frac{h}{d}\right)}\\ \mathbf{elif}\;c0 \leq 6.2 \cdot 10^{+53}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot t_1}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{w} \cdot \frac{d \cdot \frac{\frac{c0}{D}}{h}}{t_0}\\ \end{array} \]

Alternative 12
Accuracy	48.0%
Cost	1749

\[\begin{array}{l} t_0 := \frac{\frac{d}{D} \cdot \frac{d}{D}}{h} \cdot \left(\frac{c0}{w} \cdot \frac{c0}{w}\right)\\ t_1 := M \cdot \left(\frac{D}{d} \cdot \left(h \cdot M\right)\right)\\ t_2 := 0.25 \cdot \frac{t_1}{\frac{d}{D}}\\ \mathbf{if}\;c0 \leq -1.04 \cdot 10^{-7}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c0 \leq -1.05 \cdot 10^{-43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c0 \leq 2 \cdot 10^{-91}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c0 \leq 265000000 \lor \neg \left(c0 \leq 4.4 \cdot 10^{+52}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot t_1}{d}\\ \end{array} \]

Alternative 13
Accuracy	48.0%
Cost	1749

\[\begin{array}{l} t_0 := \frac{c0}{w} \cdot \frac{c0}{w}\\ t_1 := M \cdot \left(\frac{D}{d} \cdot \left(h \cdot M\right)\right)\\ t_2 := 0.25 \cdot \frac{t_1}{\frac{d}{D}}\\ \mathbf{if}\;c0 \leq -8.5 \cdot 10^{-8}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c0 \leq -2.6 \cdot 10^{-45}:\\ \;\;\;\;\frac{\frac{d}{D} \cdot \frac{d}{D}}{h} \cdot t_0\\ \mathbf{elif}\;c0 \leq 4.9 \cdot 10^{-93}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c0 \leq 3300000000 \lor \neg \left(c0 \leq 2.1 \cdot 10^{+52}\right):\\ \;\;\;\;t_0 \cdot \frac{\frac{\frac{d}{D}}{\frac{D}{d}}}{h}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot t_1}{d}\\ \end{array} \]

Alternative 14
Accuracy	48.0%
Cost	1749

\[\begin{array}{l} t_0 := M \cdot \left(\frac{D}{d} \cdot \left(h \cdot M\right)\right)\\ t_1 := 0.25 \cdot \frac{t_0}{\frac{d}{D}}\\ \mathbf{if}\;c0 \leq -4.4 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c0 \leq -1.7 \cdot 10^{-43}:\\ \;\;\;\;\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot c0}{h \cdot \left(w \cdot w\right)}\\ \mathbf{elif}\;c0 \leq 1.16 \cdot 10^{-97}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c0 \leq 27000000000 \lor \neg \left(c0 \leq 6.8 \cdot 10^{+52}\right):\\ \;\;\;\;\left(\frac{c0}{w} \cdot \frac{c0}{w}\right) \cdot \frac{\frac{\frac{d}{D}}{\frac{D}{d}}}{h}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot t_0}{d}\\ \end{array} \]

Alternative 15
Accuracy	49.4%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;h \leq -2 \cdot 10^{-216}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \left(M \cdot \left(\frac{D}{d} \cdot \left(h \cdot M\right)\right)\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(h \cdot \left(\frac{D}{d} \cdot \left(M \cdot \left(M \cdot \frac{D}{d}\right)\right)\right)\right)\\ \end{array} \]

Alternative 16
Accuracy	50.2%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;M \leq -1.4 \cdot 10^{-28}:\\ \;\;\;\;0.25 \cdot \left(h \cdot \frac{\left(M \cdot \frac{D}{d}\right) \cdot \left(D \cdot M\right)}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{M \cdot \left(\frac{D}{d} \cdot \left(h \cdot M\right)\right)}{\frac{d}{D}}\\ \end{array} \]

Alternative 17
Accuracy	49.0%
Cost	960

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

Alternative 18
Accuracy	49.1%
Cost	960

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

Alternative 19
Accuracy	33.1%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023163 
(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))))))