Henrywood and Agarwal, Equation (13)

Average Accuracy: 6.8% → 65.0%

Time: 49.4s

Precision: binary64

Cost: 76301

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{\frac{D}{d} \cdot \sqrt{\frac{w \cdot h}{c0}}}\\ t_1 := \sqrt{h \cdot \frac{w}{c0}}\\ t_2 := \sqrt[3]{w \cdot \frac{2}{c0}}\\ t_3 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_4 := \frac{c0}{2 \cdot w} \cdot \left(t_3 + \sqrt{t_3 \cdot t_3 - M \cdot M}\right)\\ \mathbf{if}\;t_4 \leq -4 \cdot 10^{-117}:\\ \;\;\;\;\left(d \cdot \frac{\sqrt{2}}{D \cdot t_1}\right) \cdot \left(\frac{\sqrt{2}}{\frac{D}{d}} \cdot \frac{c0 \cdot \frac{0.5}{w}}{t_1}\right)\\ \mathbf{elif}\;t_4 \leq 0 \lor \neg \left(t_4 \leq \infty\right):\\ \;\;\;\;\frac{c0}{c0 \cdot \left(\frac{d}{w} \cdot \left(\frac{\frac{d}{h \cdot \left(D \cdot M\right)}}{D \cdot M} \cdot \left(w \cdot 4\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{{t_2}^{2}} \cdot \frac{t_0}{t_2}\\ \end{array} \]

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 (/ (sqrt 2.0) (* (/ D d) (sqrt (/ (* w h) c0)))))
        (t_1 (sqrt (* h (/ w c0))))
        (t_2 (cbrt (* w (/ 2.0 c0))))
        (t_3 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_4 (* (/ c0 (* 2.0 w)) (+ t_3 (sqrt (- (* t_3 t_3) (* M M)))))))
   (if (<= t_4 -4e-117)
     (*
      (* d (/ (sqrt 2.0) (* D t_1)))
      (* (/ (sqrt 2.0) (/ D d)) (/ (* c0 (/ 0.5 w)) t_1)))
     (if (or (<= t_4 0.0) (not (<= t_4 INFINITY)))
       (/ c0 (* c0 (* (/ d w) (* (/ (/ d (* h (* D M))) (* D M)) (* w 4.0)))))
       (* (/ t_0 (pow t_2 2.0)) (/ t_0 t_2))))))

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 = sqrt(2.0) / ((D / d) * sqrt(((w * h) / c0)));
	double t_1 = sqrt((h * (w / c0)));
	double t_2 = cbrt((w * (2.0 / c0)));
	double t_3 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_4 = (c0 / (2.0 * w)) * (t_3 + sqrt(((t_3 * t_3) - (M * M))));
	double tmp;
	if (t_4 <= -4e-117) {
		tmp = (d * (sqrt(2.0) / (D * t_1))) * ((sqrt(2.0) / (D / d)) * ((c0 * (0.5 / w)) / t_1));
	} else if ((t_4 <= 0.0) || !(t_4 <= ((double) INFINITY))) {
		tmp = c0 / (c0 * ((d / w) * (((d / (h * (D * M))) / (D * M)) * (w * 4.0))));
	} else {
		tmp = (t_0 / pow(t_2, 2.0)) * (t_0 / t_2);
	}
	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 = Math.sqrt(2.0) / ((D / d) * Math.sqrt(((w * h) / c0)));
	double t_1 = Math.sqrt((h * (w / c0)));
	double t_2 = Math.cbrt((w * (2.0 / c0)));
	double t_3 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_4 = (c0 / (2.0 * w)) * (t_3 + Math.sqrt(((t_3 * t_3) - (M * M))));
	double tmp;
	if (t_4 <= -4e-117) {
		tmp = (d * (Math.sqrt(2.0) / (D * t_1))) * ((Math.sqrt(2.0) / (D / d)) * ((c0 * (0.5 / w)) / t_1));
	} else if ((t_4 <= 0.0) || !(t_4 <= Double.POSITIVE_INFINITY)) {
		tmp = c0 / (c0 * ((d / w) * (((d / (h * (D * M))) / (D * M)) * (w * 4.0))));
	} else {
		tmp = (t_0 / Math.pow(t_2, 2.0)) * (t_0 / t_2);
	}
	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(sqrt(2.0) / Float64(Float64(D / d) * sqrt(Float64(Float64(w * h) / c0))))
	t_1 = sqrt(Float64(h * Float64(w / c0)))
	t_2 = cbrt(Float64(w * Float64(2.0 / c0)))
	t_3 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	t_4 = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_3 + sqrt(Float64(Float64(t_3 * t_3) - Float64(M * M)))))
	tmp = 0.0
	if (t_4 <= -4e-117)
		tmp = Float64(Float64(d * Float64(sqrt(2.0) / Float64(D * t_1))) * Float64(Float64(sqrt(2.0) / Float64(D / d)) * Float64(Float64(c0 * Float64(0.5 / w)) / t_1)));
	elseif ((t_4 <= 0.0) || !(t_4 <= Inf))
		tmp = Float64(c0 / Float64(c0 * Float64(Float64(d / w) * Float64(Float64(Float64(d / Float64(h * Float64(D * M))) / Float64(D * M)) * Float64(w * 4.0)))));
	else
		tmp = Float64(Float64(t_0 / (t_2 ^ 2.0)) * Float64(t_0 / t_2));
	end
	return 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[Sqrt[2.0], $MachinePrecision] / N[(N[(D / d), $MachinePrecision] * N[Sqrt[N[(N[(w * h), $MachinePrecision] / c0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(h * N[(w / c0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(w * N[(2.0 / c0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$3 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * 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, -4e-117], N[(N[(d * N[(N[Sqrt[2.0], $MachinePrecision] / N[(D * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(c0 * N[(0.5 / w), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$4, 0.0], N[Not[LessEqual[t$95$4, Infinity]], $MachinePrecision]], N[(c0 / N[(c0 * N[(N[(d / w), $MachinePrecision] * N[(N[(N[(d / N[(h * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(D * M), $MachinePrecision]), $MachinePrecision] * N[(w * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / t$95$2), $MachinePrecision]), $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))))) < -4.00000000000000012e-117

   1. Initial program 22.7%

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

   2. Taylor expanded in c0 around inf 34.8%

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

   3. Simplified 39.1%

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

      [Start] 34.8	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]
      associate-/r* [=>] 33.7	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{\frac{{d}^{2} \cdot c0}{{D}^{2}}}{w \cdot h}}\right) \]
      *-commutative [=>] 33.7	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{\color{blue}{c0 \cdot {d}^{2}}}{{D}^{2}}}{w \cdot h}\right) \]
      unpow2 [=>] 33.7	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{{D}^{2}}}{w \cdot h}\right) \]
      unpow2 [=>] 33.7	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{D \cdot D}}}{w \cdot h}\right) \]
      times-frac [=>] 37.5	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\color{blue}{\frac{c0}{D} \cdot \frac{d \cdot d}{D}}}{w \cdot h}\right) \]
      associate-/l* [=>] 39.1	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{D} \cdot \color{blue}{\frac{d}{\frac{D}{d}}}}{w \cdot h}\right) \]

   4. Applied egg-rr 39.2%

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

      [Start] 39.1	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{D} \cdot \frac{d}{\frac{D}{d}}}{w \cdot h}\right) \]
      clear-num [=>] 38.9	\[ \color{blue}{\frac{1}{\frac{2 \cdot w}{c0}}} \cdot \left(2 \cdot \frac{\frac{c0}{D} \cdot \frac{d}{\frac{D}{d}}}{w \cdot h}\right) \]
      associate-*l/ [=>] 38.9	\[ \color{blue}{\frac{1 \cdot \left(2 \cdot \frac{\frac{c0}{D} \cdot \frac{d}{\frac{D}{d}}}{w \cdot h}\right)}{\frac{2 \cdot w}{c0}}} \]
      *-un-lft-identity [<=] 38.9	\[ \frac{\color{blue}{2 \cdot \frac{\frac{c0}{D} \cdot \frac{d}{\frac{D}{d}}}{w \cdot h}}}{\frac{2 \cdot w}{c0}} \]
      associate-*r/ [=>] 38.9	\[ \frac{\color{blue}{\frac{2 \cdot \left(\frac{c0}{D} \cdot \frac{d}{\frac{D}{d}}\right)}{w \cdot h}}}{\frac{2 \cdot w}{c0}} \]
      associate-/l* [=>] 38.9	\[ \frac{\color{blue}{\frac{2}{\frac{w \cdot h}{\frac{c0}{D} \cdot \frac{d}{\frac{D}{d}}}}}}{\frac{2 \cdot w}{c0}} \]
      frac-times [=>] 41.9	\[ \frac{\frac{2}{\frac{w \cdot h}{\color{blue}{\frac{c0 \cdot d}{D \cdot \frac{D}{d}}}}}}{\frac{2 \cdot w}{c0}} \]
      associate-/l* [=>] 38.5	\[ \frac{\frac{2}{\frac{w \cdot h}{\color{blue}{\frac{c0}{\frac{D \cdot \frac{D}{d}}{d}}}}}}{\frac{2 \cdot w}{c0}} \]
      associate-*l/ [<=] 39.2	\[ \frac{\frac{2}{\frac{w \cdot h}{\frac{c0}{\color{blue}{\frac{D}{d} \cdot \frac{D}{d}}}}}}{\frac{2 \cdot w}{c0}} \]
      pow2 [=>] 39.2	\[ \frac{\frac{2}{\frac{w \cdot h}{\frac{c0}{\color{blue}{{\left(\frac{D}{d}\right)}^{2}}}}}}{\frac{2 \cdot w}{c0}} \]
      associate-/l* [=>] 39.2	\[ \frac{\frac{2}{\frac{w \cdot h}{\frac{c0}{{\left(\frac{D}{d}\right)}^{2}}}}}{\color{blue}{\frac{2}{\frac{c0}{w}}}} \]

   5. Applied egg-rr 64.8%

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

      [Start] 39.2	\[ \frac{\frac{2}{\frac{w \cdot h}{\frac{c0}{{\left(\frac{D}{d}\right)}^{2}}}}}{\frac{2}{\frac{c0}{w}}} \]
      div-inv [=>] 39.2	\[ \color{blue}{\frac{2}{\frac{w \cdot h}{\frac{c0}{{\left(\frac{D}{d}\right)}^{2}}}} \cdot \frac{1}{\frac{2}{\frac{c0}{w}}}} \]
      add-sqr-sqrt [=>] 38.8	\[ \color{blue}{\left(\sqrt{\frac{2}{\frac{w \cdot h}{\frac{c0}{{\left(\frac{D}{d}\right)}^{2}}}}} \cdot \sqrt{\frac{2}{\frac{w \cdot h}{\frac{c0}{{\left(\frac{D}{d}\right)}^{2}}}}}\right)} \cdot \frac{1}{\frac{2}{\frac{c0}{w}}} \]
      associate-*l* [=>] 38.9	\[ \color{blue}{\sqrt{\frac{2}{\frac{w \cdot h}{\frac{c0}{{\left(\frac{D}{d}\right)}^{2}}}}} \cdot \left(\sqrt{\frac{2}{\frac{w \cdot h}{\frac{c0}{{\left(\frac{D}{d}\right)}^{2}}}}} \cdot \frac{1}{\frac{2}{\frac{c0}{w}}}\right)} \]
      sqrt-div [=>] 38.8	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{w \cdot h}{\frac{c0}{{\left(\frac{D}{d}\right)}^{2}}}}}} \cdot \left(\sqrt{\frac{2}{\frac{w \cdot h}{\frac{c0}{{\left(\frac{D}{d}\right)}^{2}}}}} \cdot \frac{1}{\frac{2}{\frac{c0}{w}}}\right) \]
      associate-/r/ [=>] 38.2	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{w \cdot h}{c0} \cdot {\left(\frac{D}{d}\right)}^{2}}}} \cdot \left(\sqrt{\frac{2}{\frac{w \cdot h}{\frac{c0}{{\left(\frac{D}{d}\right)}^{2}}}}} \cdot \frac{1}{\frac{2}{\frac{c0}{w}}}\right) \]
      sqrt-prod [=>] 38.2	\[ \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{w \cdot h}{c0}} \cdot \sqrt{{\left(\frac{D}{d}\right)}^{2}}}} \cdot \left(\sqrt{\frac{2}{\frac{w \cdot h}{\frac{c0}{{\left(\frac{D}{d}\right)}^{2}}}}} \cdot \frac{1}{\frac{2}{\frac{c0}{w}}}\right) \]
      sqrt-pow1 [=>] 19.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{w \cdot h}{c0}} \cdot \color{blue}{{\left(\frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}}} \cdot \left(\sqrt{\frac{2}{\frac{w \cdot h}{\frac{c0}{{\left(\frac{D}{d}\right)}^{2}}}}} \cdot \frac{1}{\frac{2}{\frac{c0}{w}}}\right) \]
      metadata-eval [=>] 19.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{w \cdot h}{c0}} \cdot {\left(\frac{D}{d}\right)}^{\color{blue}{1}}} \cdot \left(\sqrt{\frac{2}{\frac{w \cdot h}{\frac{c0}{{\left(\frac{D}{d}\right)}^{2}}}}} \cdot \frac{1}{\frac{2}{\frac{c0}{w}}}\right) \]
      pow1 [<=] 19.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{w \cdot h}{c0}} \cdot \color{blue}{\frac{D}{d}}} \cdot \left(\sqrt{\frac{2}{\frac{w \cdot h}{\frac{c0}{{\left(\frac{D}{d}\right)}^{2}}}}} \cdot \frac{1}{\frac{2}{\frac{c0}{w}}}\right) \]

   6. Simplified 61.6%

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

      [Start] 64.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{w \cdot h}{c0}} \cdot \frac{D}{d}} \cdot \left(\frac{\sqrt{2}}{\sqrt{\frac{w \cdot h}{c0}} \cdot \frac{D}{d}} \cdot \frac{0.5}{\frac{w}{c0}}\right) \]
      associate-*r/ [=>] 64.6	\[ \frac{\sqrt{2}}{\color{blue}{\frac{\sqrt{\frac{w \cdot h}{c0}} \cdot D}{d}}} \cdot \left(\frac{\sqrt{2}}{\sqrt{\frac{w \cdot h}{c0}} \cdot \frac{D}{d}} \cdot \frac{0.5}{\frac{w}{c0}}\right) \]
      associate-/r/ [=>] 64.5	\[ \color{blue}{\left(\frac{\sqrt{2}}{\sqrt{\frac{w \cdot h}{c0}} \cdot D} \cdot d\right)} \cdot \left(\frac{\sqrt{2}}{\sqrt{\frac{w \cdot h}{c0}} \cdot \frac{D}{d}} \cdot \frac{0.5}{\frac{w}{c0}}\right) \]
      *-commutative [=>] 64.5	\[ \left(\frac{\sqrt{2}}{\color{blue}{D \cdot \sqrt{\frac{w \cdot h}{c0}}}} \cdot d\right) \cdot \left(\frac{\sqrt{2}}{\sqrt{\frac{w \cdot h}{c0}} \cdot \frac{D}{d}} \cdot \frac{0.5}{\frac{w}{c0}}\right) \]
      associate-*l/ [<=] 64.6	\[ \left(\frac{\sqrt{2}}{D \cdot \sqrt{\color{blue}{\frac{w}{c0} \cdot h}}} \cdot d\right) \cdot \left(\frac{\sqrt{2}}{\sqrt{\frac{w \cdot h}{c0}} \cdot \frac{D}{d}} \cdot \frac{0.5}{\frac{w}{c0}}\right) \]
      *-commutative [=>] 64.6	\[ \left(\frac{\sqrt{2}}{D \cdot \sqrt{\color{blue}{h \cdot \frac{w}{c0}}}} \cdot d\right) \cdot \left(\frac{\sqrt{2}}{\sqrt{\frac{w \cdot h}{c0}} \cdot \frac{D}{d}} \cdot \frac{0.5}{\frac{w}{c0}}\right) \]
      associate-*l/ [=>] 64.6	\[ \left(\frac{\sqrt{2}}{D \cdot \sqrt{h \cdot \frac{w}{c0}}} \cdot d\right) \cdot \color{blue}{\frac{\sqrt{2} \cdot \frac{0.5}{\frac{w}{c0}}}{\sqrt{\frac{w \cdot h}{c0}} \cdot \frac{D}{d}}} \]
      *-commutative [=>] 64.6	\[ \left(\frac{\sqrt{2}}{D \cdot \sqrt{h \cdot \frac{w}{c0}}} \cdot d\right) \cdot \frac{\sqrt{2} \cdot \frac{0.5}{\frac{w}{c0}}}{\color{blue}{\frac{D}{d} \cdot \sqrt{\frac{w \cdot h}{c0}}}} \]
      times-frac [=>] 58.9	\[ \left(\frac{\sqrt{2}}{D \cdot \sqrt{h \cdot \frac{w}{c0}}} \cdot d\right) \cdot \color{blue}{\left(\frac{\sqrt{2}}{\frac{D}{d}} \cdot \frac{\frac{0.5}{\frac{w}{c0}}}{\sqrt{\frac{w \cdot h}{c0}}}\right)} \]
      associate-/r/ [=>] 58.9	\[ \left(\frac{\sqrt{2}}{D \cdot \sqrt{h \cdot \frac{w}{c0}}} \cdot d\right) \cdot \left(\frac{\sqrt{2}}{\frac{D}{d}} \cdot \frac{\color{blue}{\frac{0.5}{w} \cdot c0}}{\sqrt{\frac{w \cdot h}{c0}}}\right) \]
      *-commutative [=>] 58.9	\[ \left(\frac{\sqrt{2}}{D \cdot \sqrt{h \cdot \frac{w}{c0}}} \cdot d\right) \cdot \left(\frac{\sqrt{2}}{\frac{D}{d}} \cdot \frac{\color{blue}{c0 \cdot \frac{0.5}{w}}}{\sqrt{\frac{w \cdot h}{c0}}}\right) \]
      associate-*l/ [<=] 61.6	\[ \left(\frac{\sqrt{2}}{D \cdot \sqrt{h \cdot \frac{w}{c0}}} \cdot d\right) \cdot \left(\frac{\sqrt{2}}{\frac{D}{d}} \cdot \frac{c0 \cdot \frac{0.5}{w}}{\sqrt{\color{blue}{\frac{w}{c0} \cdot h}}}\right) \]
      *-commutative [=>] 61.6	\[ \left(\frac{\sqrt{2}}{D \cdot \sqrt{h \cdot \frac{w}{c0}}} \cdot d\right) \cdot \left(\frac{\sqrt{2}}{\frac{D}{d}} \cdot \frac{c0 \cdot \frac{0.5}{w}}{\sqrt{\color{blue}{h \cdot \frac{w}{c0}}}}\right) \]

   if -4.00000000000000012e-117 < (*.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 4.6%

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

   2. Simplified 1.6%

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

      [Start] 4.6	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      associate-*l/ [<=] 3.8	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \left(d \cdot d\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      *-commutative [=>] 3.8	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(d \cdot d\right) \cdot \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      fma-def [=>] 2.8	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(d \cdot d, \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)} \]
      associate-*l* [=>] 2.3	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{\color{blue}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      associate-/r* [=>] 2.2	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \color{blue}{\frac{\frac{c0}{w}}{h \cdot \left(D \cdot D\right)}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      associate-*r* [=>] 2.1	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{\color{blue}{\left(h \cdot D\right) \cdot D}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      *-commutative [=>] 2.1	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{\color{blue}{D \cdot \left(h \cdot D\right)}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Taylor expanded in c0 around -inf 5.4%

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

   4. Simplified 49.6%

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

      [Start] 5.4	\[ \frac{c0}{2 \cdot w} \cdot \left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0} + -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right) \]
      fma-def [=>] 5.4	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0}, -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right)} \]

   5. Applied egg-rr 51.0%

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

      [Start] 49.6	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(0.5, \frac{\frac{w \cdot \left(M \cdot \left(M \cdot h\right)\right)}{{\left(\frac{d}{D}\right)}^{2}}}{c0}, c0 \cdot 0\right) \]
      associate-/r/ [<=] 56.7	\[ \color{blue}{\frac{c0}{\frac{2 \cdot w}{\mathsf{fma}\left(0.5, \frac{\frac{w \cdot \left(M \cdot \left(M \cdot h\right)\right)}{{\left(\frac{d}{D}\right)}^{2}}}{c0}, c0 \cdot 0\right)}}} \]
      mul0-rgt [=>] 56.7	\[ \frac{c0}{\frac{2 \cdot w}{\mathsf{fma}\left(0.5, \frac{\frac{w \cdot \left(M \cdot \left(M \cdot h\right)\right)}{{\left(\frac{d}{D}\right)}^{2}}}{c0}, \color{blue}{0}\right)}} \]
      fma-udef [=>] 56.7	\[ \frac{c0}{\frac{2 \cdot w}{\color{blue}{0.5 \cdot \frac{\frac{w \cdot \left(M \cdot \left(M \cdot h\right)\right)}{{\left(\frac{d}{D}\right)}^{2}}}{c0} + 0}}} \]
      +-rgt-identity [=>] 56.7	\[ \frac{c0}{\frac{2 \cdot w}{\color{blue}{0.5 \cdot \frac{\frac{w \cdot \left(M \cdot \left(M \cdot h\right)\right)}{{\left(\frac{d}{D}\right)}^{2}}}{c0}}}} \]
      clear-num [=>] 56.6	\[ \frac{c0}{\frac{2 \cdot w}{0.5 \cdot \color{blue}{\frac{1}{\frac{c0}{\frac{w \cdot \left(M \cdot \left(M \cdot h\right)\right)}{{\left(\frac{d}{D}\right)}^{2}}}}}}} \]
      un-div-inv [=>] 56.6	\[ \frac{c0}{\frac{2 \cdot w}{\color{blue}{\frac{0.5}{\frac{c0}{\frac{w \cdot \left(M \cdot \left(M \cdot h\right)\right)}{{\left(\frac{d}{D}\right)}^{2}}}}}}} \]
      associate-/r/ [=>] 56.6	\[ \frac{c0}{\color{blue}{\frac{2 \cdot w}{0.5} \cdot \frac{c0}{\frac{w \cdot \left(M \cdot \left(M \cdot h\right)\right)}{{\left(\frac{d}{D}\right)}^{2}}}}} \]
      *-commutative [=>] 56.6	\[ \frac{c0}{\frac{\color{blue}{w \cdot 2}}{0.5} \cdot \frac{c0}{\frac{w \cdot \left(M \cdot \left(M \cdot h\right)\right)}{{\left(\frac{d}{D}\right)}^{2}}}} \]
      div-inv [=>] 56.6	\[ \frac{c0}{\frac{w \cdot 2}{0.5} \cdot \color{blue}{\left(c0 \cdot \frac{1}{\frac{w \cdot \left(M \cdot \left(M \cdot h\right)\right)}{{\left(\frac{d}{D}\right)}^{2}}}\right)}} \]
      clear-num [<=] 56.6	\[ \frac{c0}{\frac{w \cdot 2}{0.5} \cdot \left(c0 \cdot \color{blue}{\frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot \left(M \cdot \left(M \cdot h\right)\right)}}\right)} \]
      *-commutative [=>] 56.6	\[ \frac{c0}{\frac{w \cdot 2}{0.5} \cdot \left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{\color{blue}{\left(M \cdot \left(M \cdot h\right)\right) \cdot w}}\right)} \]
      associate-*r* [=>] 53.9	\[ \frac{c0}{\frac{w \cdot 2}{0.5} \cdot \left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{\color{blue}{\left(\left(M \cdot M\right) \cdot h\right)} \cdot w}\right)} \]

   6. Simplified 56.6%

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

      [Start] 51.0	\[ \frac{c0}{\frac{w \cdot 2}{0.5} \cdot \left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{\left(M \cdot M\right) \cdot \left(h \cdot w\right)}\right)} \]
      associate-/l* [=>] 51.0	\[ \frac{c0}{\color{blue}{\frac{w}{\frac{0.5}{2}}} \cdot \left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{\left(M \cdot M\right) \cdot \left(h \cdot w\right)}\right)} \]
      metadata-eval [=>] 51.0	\[ \frac{c0}{\frac{w}{\color{blue}{0.25}} \cdot \left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{\left(M \cdot M\right) \cdot \left(h \cdot w\right)}\right)} \]
      associate-*r* [=>] 53.9	\[ \frac{c0}{\frac{w}{0.25} \cdot \left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{\color{blue}{\left(\left(M \cdot M\right) \cdot h\right) \cdot w}}\right)} \]
      associate-*r* [<=] 56.6	\[ \frac{c0}{\frac{w}{0.25} \cdot \left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{\color{blue}{\left(M \cdot \left(M \cdot h\right)\right)} \cdot w}\right)} \]
      *-commutative [<=] 56.6	\[ \frac{c0}{\frac{w}{0.25} \cdot \left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{\color{blue}{w \cdot \left(M \cdot \left(M \cdot h\right)\right)}}\right)} \]

   7. Applied egg-rr 60.1%

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

      [Start] 56.6	\[ \frac{c0}{\frac{w}{0.25} \cdot \left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot \left(M \cdot \left(M \cdot h\right)\right)}\right)} \]
      unpow2 [=>] 56.6	\[ \frac{c0}{\frac{w}{0.25} \cdot \left(c0 \cdot \frac{\color{blue}{\frac{d}{D} \cdot \frac{d}{D}}}{w \cdot \left(M \cdot \left(M \cdot h\right)\right)}\right)} \]
      associate-*r* [=>] 57.5	\[ \frac{c0}{\frac{w}{0.25} \cdot \left(c0 \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{\color{blue}{\left(w \cdot M\right) \cdot \left(M \cdot h\right)}}\right)} \]
      times-frac [=>] 62.6	\[ \frac{c0}{\frac{w}{0.25} \cdot \left(c0 \cdot \color{blue}{\left(\frac{\frac{d}{D}}{w \cdot M} \cdot \frac{\frac{d}{D}}{M \cdot h}\right)}\right)} \]
      associate-/l/ [=>] 60.9	\[ \frac{c0}{\frac{w}{0.25} \cdot \left(c0 \cdot \left(\color{blue}{\frac{d}{\left(w \cdot M\right) \cdot D}} \cdot \frac{\frac{d}{D}}{M \cdot h}\right)\right)} \]
      associate-/l/ [=>] 60.1	\[ \frac{c0}{\frac{w}{0.25} \cdot \left(c0 \cdot \left(\frac{d}{\left(w \cdot M\right) \cdot D} \cdot \color{blue}{\frac{d}{\left(M \cdot h\right) \cdot D}}\right)\right)} \]

   8. Applied egg-rr 29.9%

      \[\leadsto \frac{c0}{\color{blue}{e^{\mathsf{log1p}\left(c0 \cdot \left(\left(\frac{d}{w} \cdot \frac{\frac{d}{M \cdot \left(D \cdot h\right)}}{M \cdot D}\right) \cdot \left(w \cdot 4\right)\right)\right)} - 1}} \]
      Proof

      [Start] 60.1	\[ \frac{c0}{\frac{w}{0.25} \cdot \left(c0 \cdot \left(\frac{d}{\left(w \cdot M\right) \cdot D} \cdot \frac{d}{\left(M \cdot h\right) \cdot D}\right)\right)} \]
      expm1-log1p-u [=>] 31.7	\[ \frac{c0}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{w}{0.25} \cdot \left(c0 \cdot \left(\frac{d}{\left(w \cdot M\right) \cdot D} \cdot \frac{d}{\left(M \cdot h\right) \cdot D}\right)\right)\right)\right)}} \]
      expm1-udef [=>] 28.5	\[ \frac{c0}{\color{blue}{e^{\mathsf{log1p}\left(\frac{w}{0.25} \cdot \left(c0 \cdot \left(\frac{d}{\left(w \cdot M\right) \cdot D} \cdot \frac{d}{\left(M \cdot h\right) \cdot D}\right)\right)\right)} - 1}} \]

   9. Simplified 65.4%

      \[\leadsto \frac{c0}{\color{blue}{c0 \cdot \left(\frac{d}{w} \cdot \left(\frac{\frac{d}{\left(D \cdot M\right) \cdot h}}{D \cdot M} \cdot \left(w \cdot 4\right)\right)\right)}} \]
      Proof

      [Start] 29.9	\[ \frac{c0}{e^{\mathsf{log1p}\left(c0 \cdot \left(\left(\frac{d}{w} \cdot \frac{\frac{d}{M \cdot \left(D \cdot h\right)}}{M \cdot D}\right) \cdot \left(w \cdot 4\right)\right)\right)} - 1} \]
      expm1-def [=>] 32.8	\[ \frac{c0}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(c0 \cdot \left(\left(\frac{d}{w} \cdot \frac{\frac{d}{M \cdot \left(D \cdot h\right)}}{M \cdot D}\right) \cdot \left(w \cdot 4\right)\right)\right)\right)}} \]
      expm1-log1p [=>] 62.6	\[ \frac{c0}{\color{blue}{c0 \cdot \left(\left(\frac{d}{w} \cdot \frac{\frac{d}{M \cdot \left(D \cdot h\right)}}{M \cdot D}\right) \cdot \left(w \cdot 4\right)\right)}} \]
      associate-*l* [=>] 62.8	\[ \frac{c0}{c0 \cdot \color{blue}{\left(\frac{d}{w} \cdot \left(\frac{\frac{d}{M \cdot \left(D \cdot h\right)}}{M \cdot D} \cdot \left(w \cdot 4\right)\right)\right)}} \]
      associate-*r* [=>] 65.4	\[ \frac{c0}{c0 \cdot \left(\frac{d}{w} \cdot \left(\frac{\frac{d}{\color{blue}{\left(M \cdot D\right) \cdot h}}}{M \cdot D} \cdot \left(w \cdot 4\right)\right)\right)} \]
      *-commutative [=>] 65.4	\[ \frac{c0}{c0 \cdot \left(\frac{d}{w} \cdot \left(\frac{\frac{d}{\color{blue}{\left(D \cdot M\right)} \cdot h}}{M \cdot D} \cdot \left(w \cdot 4\right)\right)\right)} \]
      *-commutative [=>] 65.4	\[ \frac{c0}{c0 \cdot \left(\frac{d}{w} \cdot \left(\frac{\frac{d}{\left(D \cdot M\right) \cdot h}}{\color{blue}{D \cdot M}} \cdot \left(w \cdot 4\right)\right)\right)} \]

   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 23.3%

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

   2. Taylor expanded in c0 around inf 31.7%

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

   3. Simplified 36.2%

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

      [Start] 31.7	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]
      associate-/r* [=>] 30.7	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{\frac{{d}^{2} \cdot c0}{{D}^{2}}}{w \cdot h}}\right) \]
      *-commutative [=>] 30.7	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{\color{blue}{c0 \cdot {d}^{2}}}{{D}^{2}}}{w \cdot h}\right) \]
      unpow2 [=>] 30.7	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{{D}^{2}}}{w \cdot h}\right) \]
      unpow2 [=>] 30.7	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{D \cdot D}}}{w \cdot h}\right) \]
      times-frac [=>] 34.4	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\color{blue}{\frac{c0}{D} \cdot \frac{d \cdot d}{D}}}{w \cdot h}\right) \]
      associate-/l* [=>] 36.2	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{D} \cdot \color{blue}{\frac{d}{\frac{D}{d}}}}{w \cdot h}\right) \]

   4. Applied egg-rr 35.9%

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

      [Start] 36.2	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{D} \cdot \frac{d}{\frac{D}{d}}}{w \cdot h}\right) \]
      clear-num [=>] 36.3	\[ \color{blue}{\frac{1}{\frac{2 \cdot w}{c0}}} \cdot \left(2 \cdot \frac{\frac{c0}{D} \cdot \frac{d}{\frac{D}{d}}}{w \cdot h}\right) \]
      associate-*l/ [=>] 36.3	\[ \color{blue}{\frac{1 \cdot \left(2 \cdot \frac{\frac{c0}{D} \cdot \frac{d}{\frac{D}{d}}}{w \cdot h}\right)}{\frac{2 \cdot w}{c0}}} \]
      *-un-lft-identity [<=] 36.3	\[ \frac{\color{blue}{2 \cdot \frac{\frac{c0}{D} \cdot \frac{d}{\frac{D}{d}}}{w \cdot h}}}{\frac{2 \cdot w}{c0}} \]
      associate-*r/ [=>] 36.3	\[ \frac{\color{blue}{\frac{2 \cdot \left(\frac{c0}{D} \cdot \frac{d}{\frac{D}{d}}\right)}{w \cdot h}}}{\frac{2 \cdot w}{c0}} \]
      associate-/l* [=>] 36.3	\[ \frac{\color{blue}{\frac{2}{\frac{w \cdot h}{\frac{c0}{D} \cdot \frac{d}{\frac{D}{d}}}}}}{\frac{2 \cdot w}{c0}} \]
      frac-times [=>] 39.4	\[ \frac{\frac{2}{\frac{w \cdot h}{\color{blue}{\frac{c0 \cdot d}{D \cdot \frac{D}{d}}}}}}{\frac{2 \cdot w}{c0}} \]
      associate-/l* [=>] 34.8	\[ \frac{\frac{2}{\frac{w \cdot h}{\color{blue}{\frac{c0}{\frac{D \cdot \frac{D}{d}}{d}}}}}}{\frac{2 \cdot w}{c0}} \]
      associate-*l/ [<=] 35.9	\[ \frac{\frac{2}{\frac{w \cdot h}{\frac{c0}{\color{blue}{\frac{D}{d} \cdot \frac{D}{d}}}}}}{\frac{2 \cdot w}{c0}} \]
      pow2 [=>] 35.9	\[ \frac{\frac{2}{\frac{w \cdot h}{\frac{c0}{\color{blue}{{\left(\frac{D}{d}\right)}^{2}}}}}}{\frac{2 \cdot w}{c0}} \]
      associate-/l* [=>] 35.9	\[ \frac{\frac{2}{\frac{w \cdot h}{\frac{c0}{{\left(\frac{D}{d}\right)}^{2}}}}}{\color{blue}{\frac{2}{\frac{c0}{w}}}} \]

   5. Applied egg-rr 61.5%

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

      [Start] 35.9	\[ \frac{\frac{2}{\frac{w \cdot h}{\frac{c0}{{\left(\frac{D}{d}\right)}^{2}}}}}{\frac{2}{\frac{c0}{w}}} \]
      add-sqr-sqrt [=>] 35.5	\[ \frac{\color{blue}{\sqrt{\frac{2}{\frac{w \cdot h}{\frac{c0}{{\left(\frac{D}{d}\right)}^{2}}}}} \cdot \sqrt{\frac{2}{\frac{w \cdot h}{\frac{c0}{{\left(\frac{D}{d}\right)}^{2}}}}}}}{\frac{2}{\frac{c0}{w}}} \]
      add-cube-cbrt [=>] 35.1	\[ \frac{\sqrt{\frac{2}{\frac{w \cdot h}{\frac{c0}{{\left(\frac{D}{d}\right)}^{2}}}}} \cdot \sqrt{\frac{2}{\frac{w \cdot h}{\frac{c0}{{\left(\frac{D}{d}\right)}^{2}}}}}}{\color{blue}{\left(\sqrt[3]{\frac{2}{\frac{c0}{w}}} \cdot \sqrt[3]{\frac{2}{\frac{c0}{w}}}\right) \cdot \sqrt[3]{\frac{2}{\frac{c0}{w}}}}} \]
      times-frac [=>] 35.1	\[ \color{blue}{\frac{\sqrt{\frac{2}{\frac{w \cdot h}{\frac{c0}{{\left(\frac{D}{d}\right)}^{2}}}}}}{\sqrt[3]{\frac{2}{\frac{c0}{w}}} \cdot \sqrt[3]{\frac{2}{\frac{c0}{w}}}} \cdot \frac{\sqrt{\frac{2}{\frac{w \cdot h}{\frac{c0}{{\left(\frac{D}{d}\right)}^{2}}}}}}{\sqrt[3]{\frac{2}{\frac{c0}{w}}}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq -4 \cdot 10^{-117}:\\ \;\;\;\;\left(d \cdot \frac{\sqrt{2}}{D \cdot \sqrt{h \cdot \frac{w}{c0}}}\right) \cdot \left(\frac{\sqrt{2}}{\frac{D}{d}} \cdot \frac{c0 \cdot \frac{0.5}{w}}{\sqrt{h \cdot \frac{w}{c0}}}\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 0 \lor \neg \left(\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty\right):\\ \;\;\;\;\frac{c0}{c0 \cdot \left(\frac{d}{w} \cdot \left(\frac{\frac{d}{h \cdot \left(D \cdot M\right)}}{D \cdot M} \cdot \left(w \cdot 4\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{\frac{D}{d} \cdot \sqrt{\frac{w \cdot h}{c0}}}}{{\left(\sqrt[3]{w \cdot \frac{2}{c0}}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{D}{d} \cdot \sqrt{\frac{w \cdot h}{c0}}}}{\sqrt[3]{w \cdot \frac{2}{c0}}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	63.5%
Cost	43277

\[\begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_1 := \frac{c0}{2 \cdot w} \cdot \left(t_0 + \sqrt{t_0 \cdot t_0 - M \cdot M}\right)\\ \mathbf{if}\;t_1 \leq -4 \cdot 10^{-117}:\\ \;\;\;\;\frac{\frac{2}{\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \frac{w \cdot h}{c0}\right)}}{\frac{2}{\frac{c0}{w}}}\\ \mathbf{elif}\;t_1 \leq 0 \lor \neg \left(t_1 \leq \infty\right):\\ \;\;\;\;\frac{c0}{c0 \cdot \left(\frac{d}{w} \cdot \left(\frac{\frac{d}{h \cdot \left(D \cdot M\right)}}{D \cdot M} \cdot \left(w \cdot 4\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{M + {\left(\frac{d}{D} \cdot \sqrt{\frac{c0 \cdot 2}{w \cdot h}}\right)}^{2}}{2 \cdot w}\\ \end{array} \]

Alternative 2
Accuracy	63.9%
Cost	43277

\[\begin{array}{l} t_0 := \sqrt{h \cdot \frac{w}{c0}}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_2 := \frac{c0}{2 \cdot w} \cdot \left(t_1 + \sqrt{t_1 \cdot t_1 - M \cdot M}\right)\\ \mathbf{if}\;t_2 \leq -4 \cdot 10^{-117}:\\ \;\;\;\;\left(d \cdot \frac{\sqrt{2}}{D \cdot t_0}\right) \cdot \left(\frac{\sqrt{2}}{\frac{D}{d}} \cdot \frac{c0 \cdot \frac{0.5}{w}}{t_0}\right)\\ \mathbf{elif}\;t_2 \leq 0 \lor \neg \left(t_2 \leq \infty\right):\\ \;\;\;\;\frac{c0}{c0 \cdot \left(\frac{d}{w} \cdot \left(\frac{\frac{d}{h \cdot \left(D \cdot M\right)}}{D \cdot M} \cdot \left(w \cdot 4\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{M + {\left(\frac{d}{D} \cdot \sqrt{\frac{c0 \cdot 2}{w \cdot h}}\right)}^{2}}{2 \cdot w}\\ \end{array} \]

Alternative 3
Accuracy	63.9%
Cost	30669

\[\begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_2 := t_0 \cdot \left(t_1 + \sqrt{t_1 \cdot t_1 - M \cdot M}\right)\\ \mathbf{if}\;t_2 \leq -4 \cdot 10^{-117}:\\ \;\;\;\;\frac{\frac{2}{\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \frac{w \cdot h}{c0}\right)}}{\frac{2}{\frac{c0}{w}}}\\ \mathbf{elif}\;t_2 \leq 0 \lor \neg \left(t_2 \leq \infty\right):\\ \;\;\;\;\frac{c0}{c0 \cdot \left(\frac{d}{w} \cdot \left(\frac{\frac{d}{h \cdot \left(D \cdot M\right)}}{D \cdot M} \cdot \left(w \cdot 4\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \frac{\frac{d}{D}}{D \cdot \frac{w \cdot h}{c0 \cdot d}}\right)\\ \end{array} \]

Alternative 4
Accuracy	55.0%
Cost	2517

\[\begin{array}{l} \mathbf{if}\;M \cdot M \leq 2 \cdot 10^{-294}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \cdot M \leq 5 \cdot 10^{-251}:\\ \;\;\;\;\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \left(\frac{c0}{h} \cdot \frac{c0}{w \cdot w}\right)\\ \mathbf{elif}\;M \cdot M \leq 5 \cdot 10^{-166}:\\ \;\;\;\;\frac{0.25}{d} \cdot \frac{\left(M \cdot M\right) \cdot \left(h \cdot \left(D \cdot D\right)\right)}{d}\\ \mathbf{elif}\;M \cdot M \leq 5 \cdot 10^{-147} \lor \neg \left(M \cdot M \leq 5 \cdot 10^{+171}\right):\\ \;\;\;\;\frac{c0}{4 \cdot \left(c0 \cdot \frac{d \cdot d}{h \cdot \left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\frac{d \cdot d}{\left(M \cdot M\right) \cdot \left(D \cdot \left(h \cdot D\right)\right)}}\\ \end{array} \]

Alternative 5
Accuracy	55.2%
Cost	2517

\[\begin{array}{l} \mathbf{if}\;M \cdot M \leq 4 \cdot 10^{-293}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \cdot M \leq 5 \cdot 10^{-251}:\\ \;\;\;\;\frac{\frac{c0}{w} \cdot \left(d \cdot \frac{d}{w \cdot \frac{h}{c0}}\right)}{D \cdot D}\\ \mathbf{elif}\;M \cdot M \leq 5 \cdot 10^{-166}:\\ \;\;\;\;\frac{0.25}{d} \cdot \frac{\left(M \cdot M\right) \cdot \left(h \cdot \left(D \cdot D\right)\right)}{d}\\ \mathbf{elif}\;M \cdot M \leq 5 \cdot 10^{-147} \lor \neg \left(M \cdot M \leq 5 \cdot 10^{+171}\right):\\ \;\;\;\;\frac{c0}{4 \cdot \left(c0 \cdot \frac{d \cdot d}{h \cdot \left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\frac{d \cdot d}{\left(M \cdot M\right) \cdot \left(D \cdot \left(h \cdot D\right)\right)}}\\ \end{array} \]

Alternative 6
Accuracy	55.0%
Cost	2257

\[\begin{array}{l} \mathbf{if}\;M \cdot M \leq 2 \cdot 10^{-294}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \cdot M \leq 5 \cdot 10^{-251}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{d \cdot \frac{d}{D}}{D \cdot \left(w \cdot \frac{h}{c0}\right)}\right)\\ \mathbf{elif}\;M \cdot M \leq 5 \cdot 10^{-147} \lor \neg \left(M \cdot M \leq 5 \cdot 10^{+171}\right):\\ \;\;\;\;\frac{c0}{4 \cdot \left(c0 \cdot \frac{d \cdot d}{h \cdot \left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\frac{d \cdot d}{\left(M \cdot M\right) \cdot \left(D \cdot \left(h \cdot D\right)\right)}}\\ \end{array} \]

Alternative 7
Accuracy	55.2%
Cost	2257

\[\begin{array}{l} \mathbf{if}\;M \cdot M \leq 2 \cdot 10^{-294}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \cdot M \leq 5 \cdot 10^{-251}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{d}{D} \cdot \left(d \cdot \frac{\frac{c0}{h}}{w}\right)}{D}\right)\\ \mathbf{elif}\;M \cdot M \leq 5 \cdot 10^{-147} \lor \neg \left(M \cdot M \leq 5 \cdot 10^{+171}\right):\\ \;\;\;\;\frac{c0}{4 \cdot \left(c0 \cdot \frac{d \cdot d}{h \cdot \left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\frac{d \cdot d}{\left(M \cdot M\right) \cdot \left(D \cdot \left(h \cdot D\right)\right)}}\\ \end{array} \]

Alternative 8
Accuracy	55.3%
Cost	2257

\[\begin{array}{l} \mathbf{if}\;M \cdot M \leq 2 \cdot 10^{-294}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \cdot M \leq 5 \cdot 10^{-251}:\\ \;\;\;\;\frac{\frac{d}{D} \cdot \left(2 \cdot \frac{d}{w \cdot \frac{h}{c0}}\right)}{D \cdot \left(2 \cdot \frac{w}{c0}\right)}\\ \mathbf{elif}\;M \cdot M \leq 5 \cdot 10^{-147} \lor \neg \left(M \cdot M \leq 5 \cdot 10^{+171}\right):\\ \;\;\;\;\frac{c0}{4 \cdot \left(c0 \cdot \frac{d \cdot d}{h \cdot \left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\frac{d \cdot d}{\left(M \cdot M\right) \cdot \left(D \cdot \left(h \cdot D\right)\right)}}\\ \end{array} \]

Alternative 9
Accuracy	54.8%
Cost	1480

\[\begin{array}{l} \mathbf{if}\;D \cdot D \leq 0:\\ \;\;\;\;0\\ \mathbf{elif}\;D \cdot D \leq 1.4 \cdot 10^{+200}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{M \cdot \left(h \cdot M\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 10
Accuracy	55.2%
Cost	1480

\[\begin{array}{l} \mathbf{if}\;d \cdot d \leq 4 \cdot 10^{-281}:\\ \;\;\;\;\frac{0.25}{d} \cdot \frac{\left(M \cdot M\right) \cdot \left(h \cdot \left(D \cdot D\right)\right)}{d}\\ \mathbf{elif}\;d \cdot d \leq 2 \cdot 10^{+251}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{M \cdot \left(h \cdot M\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 11
Accuracy	50.5%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023129 
(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))))))