Henrywood and Agarwal, Equation (13)

Average Accuracy: 7.1% → 70.1%

Time: 47.2s

Precision: binary64

Cost: 20873

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;D \cdot D \leq 2 \cdot 10^{-256} \lor \neg \left(D \cdot D \leq 2 \cdot 10^{-65}\right):\\ \;\;\;\;0.25 \cdot \frac{\frac{D}{d} \cdot \left(h \cdot M\right)}{\frac{\frac{d}{D}}{M}}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\sqrt[3]{D} \cdot \left(\left(\left(D \cdot \frac{M}{d}\right) \cdot \frac{M}{\frac{d}{h}}\right) \cdot {\left(\sqrt[3]{D}\right)}^{2}\right)\right)\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (*
  (/ c0 (* 2.0 w))
  (+
   (/ (* c0 (* d d)) (* (* w h) (* D D)))
   (sqrt
    (-
     (*
      (/ (* c0 (* d d)) (* (* w h) (* D D)))
      (/ (* c0 (* d d)) (* (* w h) (* D D))))
     (* M M))))))

↓

(FPCore (c0 w h D d M)
 :precision binary64
 (if (or (<= (* D D) 2e-256) (not (<= (* D D) 2e-65)))
   (* 0.25 (/ (* (/ D d) (* h M)) (/ (/ d D) M)))
   (*
    0.25
    (* (cbrt D) (* (* (* D (/ M d)) (/ M (/ d h))) (pow (cbrt D) 2.0))))))

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 tmp;
	if (((D * D) <= 2e-256) || !((D * D) <= 2e-65)) {
		tmp = 0.25 * (((D / d) * (h * M)) / ((d / D) / M));
	} else {
		tmp = 0.25 * (cbrt(D) * (((D * (M / d)) * (M / (d / h))) * pow(cbrt(D), 2.0)));
	}
	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 tmp;
	if (((D * D) <= 2e-256) || !((D * D) <= 2e-65)) {
		tmp = 0.25 * (((D / d) * (h * M)) / ((d / D) / M));
	} else {
		tmp = 0.25 * (Math.cbrt(D) * (((D * (M / d)) * (M / (d / h))) * Math.pow(Math.cbrt(D), 2.0)));
	}
	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)
	tmp = 0.0
	if ((Float64(D * D) <= 2e-256) || !(Float64(D * D) <= 2e-65))
		tmp = Float64(0.25 * Float64(Float64(Float64(D / d) * Float64(h * M)) / Float64(Float64(d / D) / M)));
	else
		tmp = Float64(0.25 * Float64(cbrt(D) * Float64(Float64(Float64(D * Float64(M / d)) * Float64(M / Float64(d / h))) * (cbrt(D) ^ 2.0))));
	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_] := If[Or[LessEqual[N[(D * D), $MachinePrecision], 2e-256], N[Not[LessEqual[N[(D * D), $MachinePrecision], 2e-65]], $MachinePrecision]], N[(0.25 * N[(N[(N[(D / d), $MachinePrecision] * N[(h * M), $MachinePrecision]), $MachinePrecision] / N[(N[(d / D), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[Power[D, 1/3], $MachinePrecision] * N[(N[(N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision] * N[(M / N[(d / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[D, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 D D) < 1.99999999999999995e-256 or 1.99999999999999985e-65 < (*.f64 D D)

   1. Initial program 5.7%

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

   2. Simplified 6.1%

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

      [Start] 5.7	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      times-frac [=>] 4.5	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      fma-neg [=>] 4.5	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\color{blue}{\mathsf{fma}\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, -M \cdot M\right)}}\right) \]
      times-frac [=>] 4.8	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\mathsf{fma}\left(\color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}}, \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, -M \cdot M\right)}\right) \]
      times-frac [=>] 6.1	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}, \color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}}, -M \cdot M\right)}\right) \]

   3. Taylor expanded in c0 around -inf 4.6%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w} + 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]

   4. Simplified 39.8%

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

      [Start] 4.6	\[ -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w} + 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \]

   5. Taylor expanded in w around 0 55.2%

      \[\leadsto \color{blue}{0} + 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

   6. Applied egg-rr 70.1%

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

      [Start] 55.2	\[ 0 + 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]
      associate-*l* [=>] 57.3	\[ 0 + 0.25 \cdot \color{blue}{\left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)} \]
      clear-num [=>] 57.3	\[ 0 + 0.25 \cdot \left(\color{blue}{\frac{1}{\frac{d}{D}}} \cdot \left(\frac{D}{d} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right) \]
      associate-*l/ [=>] 57.3	\[ 0 + 0.25 \cdot \color{blue}{\frac{1 \cdot \left(\frac{D}{d} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\frac{d}{D}}} \]
      *-un-lft-identity [<=] 57.3	\[ 0 + 0.25 \cdot \frac{\color{blue}{\frac{D}{d} \cdot \left(h \cdot \left(M \cdot M\right)\right)}}{\frac{d}{D}} \]
      associate-*r* [=>] 62.6	\[ 0 + 0.25 \cdot \frac{\frac{D}{d} \cdot \color{blue}{\left(\left(h \cdot M\right) \cdot M\right)}}{\frac{d}{D}} \]
      associate-*r* [=>] 68.7	\[ 0 + 0.25 \cdot \frac{\color{blue}{\left(\frac{D}{d} \cdot \left(h \cdot M\right)\right) \cdot M}}{\frac{d}{D}} \]
      associate-/l* [=>] 70.1	\[ 0 + 0.25 \cdot \color{blue}{\frac{\frac{D}{d} \cdot \left(h \cdot M\right)}{\frac{\frac{d}{D}}{M}}} \]

   if 1.99999999999999995e-256 < (*.f64 D D) < 1.99999999999999985e-65

   1. Initial program 13.0%

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

   2. Simplified 3.8%

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

      [Start] 13.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      associate-*l/ [<=] 10.7	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \left(d \cdot d\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      *-commutative [=>] 10.7	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(d \cdot d\right) \cdot \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      fma-def [=>] 9.3	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(d \cdot d, \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)} \]
      associate-*l* [=>] 7.8	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{\color{blue}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      associate-/r* [=>] 7.8	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \color{blue}{\frac{\frac{c0}{w}}{h \cdot \left(D \cdot D\right)}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      associate-*r* [=>] 7.8	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{\color{blue}{\left(h \cdot D\right) \cdot D}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      *-commutative [=>] 7.8	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{\color{blue}{D \cdot \left(h \cdot D\right)}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Taylor expanded in c0 around -inf 13.7%

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

   4. Simplified 45.7%

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

      [Start] 13.7	\[ \frac{c0}{2 \cdot w} \cdot \left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0} + -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right) \]
      fma-def [=>] 13.7	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0}, -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right)} \]

   5. Applied egg-rr 46.7%

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

      [Start] 45.7	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(0.5, \frac{\frac{w \cdot \left(M \cdot \left(M \cdot h\right)\right)}{{\left(\frac{d}{D}\right)}^{2}}}{c0}, c0 \cdot 0\right) \]
      associate-/r/ [<=] 51.4	\[ \color{blue}{\frac{c0}{\frac{2 \cdot w}{\mathsf{fma}\left(0.5, \frac{\frac{w \cdot \left(M \cdot \left(M \cdot h\right)\right)}{{\left(\frac{d}{D}\right)}^{2}}}{c0}, c0 \cdot 0\right)}}} \]
      mul0-rgt [=>] 51.4	\[ \frac{c0}{\frac{2 \cdot w}{\mathsf{fma}\left(0.5, \frac{\frac{w \cdot \left(M \cdot \left(M \cdot h\right)\right)}{{\left(\frac{d}{D}\right)}^{2}}}{c0}, \color{blue}{0}\right)}} \]
      fma-udef [=>] 51.4	\[ \frac{c0}{\frac{2 \cdot w}{\color{blue}{0.5 \cdot \frac{\frac{w \cdot \left(M \cdot \left(M \cdot h\right)\right)}{{\left(\frac{d}{D}\right)}^{2}}}{c0} + 0}}} \]
      +-rgt-identity [=>] 51.4	\[ \frac{c0}{\frac{2 \cdot w}{\color{blue}{0.5 \cdot \frac{\frac{w \cdot \left(M \cdot \left(M \cdot h\right)\right)}{{\left(\frac{d}{D}\right)}^{2}}}{c0}}}} \]
      clear-num [=>] 51.4	\[ \frac{c0}{\frac{2 \cdot w}{0.5 \cdot \color{blue}{\frac{1}{\frac{c0}{\frac{w \cdot \left(M \cdot \left(M \cdot h\right)\right)}{{\left(\frac{d}{D}\right)}^{2}}}}}}} \]
      un-div-inv [=>] 51.4	\[ \frac{c0}{\frac{2 \cdot w}{\color{blue}{\frac{0.5}{\frac{c0}{\frac{w \cdot \left(M \cdot \left(M \cdot h\right)\right)}{{\left(\frac{d}{D}\right)}^{2}}}}}}} \]
      associate-/r/ [=>] 51.4	\[ \frac{c0}{\color{blue}{\frac{2 \cdot w}{0.5} \cdot \frac{c0}{\frac{w \cdot \left(M \cdot \left(M \cdot h\right)\right)}{{\left(\frac{d}{D}\right)}^{2}}}}} \]
      *-commutative [=>] 51.4	\[ \frac{c0}{\frac{\color{blue}{w \cdot 2}}{0.5} \cdot \frac{c0}{\frac{w \cdot \left(M \cdot \left(M \cdot h\right)\right)}{{\left(\frac{d}{D}\right)}^{2}}}} \]
      div-inv [=>] 51.4	\[ \frac{c0}{\frac{w \cdot 2}{0.5} \cdot \color{blue}{\left(c0 \cdot \frac{1}{\frac{w \cdot \left(M \cdot \left(M \cdot h\right)\right)}{{\left(\frac{d}{D}\right)}^{2}}}\right)}} \]
      clear-num [<=] 51.4	\[ \frac{c0}{\frac{w \cdot 2}{0.5} \cdot \left(c0 \cdot \color{blue}{\frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot \left(M \cdot \left(M \cdot h\right)\right)}}\right)} \]
      *-commutative [=>] 51.4	\[ \frac{c0}{\frac{w \cdot 2}{0.5} \cdot \left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{\color{blue}{\left(M \cdot \left(M \cdot h\right)\right) \cdot w}}\right)} \]
      associate-*r* [=>] 49.5	\[ \frac{c0}{\frac{w \cdot 2}{0.5} \cdot \left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{\color{blue}{\left(\left(M \cdot M\right) \cdot h\right)} \cdot w}\right)} \]

   6. Simplified 51.4%

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

      [Start] 46.7	\[ \frac{c0}{\frac{w \cdot 2}{0.5} \cdot \left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{\left(M \cdot M\right) \cdot \left(h \cdot w\right)}\right)} \]
      associate-/l* [=>] 46.7	\[ \frac{c0}{\color{blue}{\frac{w}{\frac{0.5}{2}}} \cdot \left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{\left(M \cdot M\right) \cdot \left(h \cdot w\right)}\right)} \]
      metadata-eval [=>] 46.7	\[ \frac{c0}{\frac{w}{\color{blue}{0.25}} \cdot \left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{\left(M \cdot M\right) \cdot \left(h \cdot w\right)}\right)} \]
      associate-*r* [=>] 49.5	\[ \frac{c0}{\frac{w}{0.25} \cdot \left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{\color{blue}{\left(\left(M \cdot M\right) \cdot h\right) \cdot w}}\right)} \]
      associate-*r* [<=] 51.4	\[ \frac{c0}{\frac{w}{0.25} \cdot \left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{\color{blue}{\left(M \cdot \left(M \cdot h\right)\right)} \cdot w}\right)} \]
      *-commutative [<=] 51.4	\[ \frac{c0}{\frac{w}{0.25} \cdot \left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{\color{blue}{w \cdot \left(M \cdot \left(M \cdot h\right)\right)}}\right)} \]

   7. Taylor expanded in c0 around 0 53.3%

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

   8. Simplified 58.1%

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

      [Start] 53.3	\[ 0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}} \]
      *-commutative [=>] 53.3	\[ 0.25 \cdot \frac{\color{blue}{\left(h \cdot {M}^{2}\right) \cdot {D}^{2}}}{{d}^{2}} \]
      unpow2 [=>] 53.3	\[ 0.25 \cdot \frac{\left(h \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot {D}^{2}}{{d}^{2}} \]
      associate-/l* [=>] 52.8	\[ 0.25 \cdot \color{blue}{\frac{h \cdot \left(M \cdot M\right)}{\frac{{d}^{2}}{{D}^{2}}}} \]
      unpow2 [=>] 52.8	\[ 0.25 \cdot \frac{h \cdot \left(M \cdot M\right)}{\frac{\color{blue}{d \cdot d}}{{D}^{2}}} \]
      unpow2 [=>] 52.8	\[ 0.25 \cdot \frac{h \cdot \left(M \cdot M\right)}{\frac{d \cdot d}{\color{blue}{D \cdot D}}} \]
      times-frac [=>] 54.6	\[ 0.25 \cdot \frac{h \cdot \left(M \cdot M\right)}{\color{blue}{\frac{d}{D} \cdot \frac{d}{D}}} \]
      unpow2 [<=] 54.6	\[ 0.25 \cdot \frac{h \cdot \left(M \cdot M\right)}{\color{blue}{{\left(\frac{d}{D}\right)}^{2}}} \]
      *-commutative [=>] 54.6	\[ 0.25 \cdot \frac{\color{blue}{\left(M \cdot M\right) \cdot h}}{{\left(\frac{d}{D}\right)}^{2}} \]
      associate-*l* [=>] 58.1	\[ 0.25 \cdot \frac{\color{blue}{M \cdot \left(M \cdot h\right)}}{{\left(\frac{d}{D}\right)}^{2}} \]

   9. Applied egg-rr 62.3%

      \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{\frac{M \cdot \left(M \cdot h\right)}{d} \cdot D}{\frac{d}{{\left(\sqrt[3]{D}\right)}^{2}}} \cdot \sqrt[3]{D}\right)} \]
      Proof

      [Start] 58.1	\[ 0.25 \cdot \frac{M \cdot \left(M \cdot h\right)}{{\left(\frac{d}{D}\right)}^{2}} \]
      unpow2 [=>] 58.1	\[ 0.25 \cdot \frac{M \cdot \left(M \cdot h\right)}{\color{blue}{\frac{d}{D} \cdot \frac{d}{D}}} \]
      associate-/r* [=>] 62.8	\[ 0.25 \cdot \color{blue}{\frac{\frac{M \cdot \left(M \cdot h\right)}{\frac{d}{D}}}{\frac{d}{D}}} \]
      add-cube-cbrt [=>] 62.6	\[ 0.25 \cdot \frac{\frac{M \cdot \left(M \cdot h\right)}{\frac{d}{D}}}{\frac{d}{\color{blue}{\left(\sqrt[3]{D} \cdot \sqrt[3]{D}\right) \cdot \sqrt[3]{D}}}} \]
      associate-/r* [=>] 62.6	\[ 0.25 \cdot \frac{\frac{M \cdot \left(M \cdot h\right)}{\frac{d}{D}}}{\color{blue}{\frac{\frac{d}{\sqrt[3]{D} \cdot \sqrt[3]{D}}}{\sqrt[3]{D}}}} \]
      associate-/r/ [=>] 62.5	\[ 0.25 \cdot \color{blue}{\left(\frac{\frac{M \cdot \left(M \cdot h\right)}{\frac{d}{D}}}{\frac{d}{\sqrt[3]{D} \cdot \sqrt[3]{D}}} \cdot \sqrt[3]{D}\right)} \]
      associate-/r/ [=>] 62.3	\[ 0.25 \cdot \left(\frac{\color{blue}{\frac{M \cdot \left(M \cdot h\right)}{d} \cdot D}}{\frac{d}{\sqrt[3]{D} \cdot \sqrt[3]{D}}} \cdot \sqrt[3]{D}\right) \]
      pow2 [=>] 62.3	\[ 0.25 \cdot \left(\frac{\frac{M \cdot \left(M \cdot h\right)}{d} \cdot D}{\frac{d}{\color{blue}{{\left(\sqrt[3]{D}\right)}^{2}}}} \cdot \sqrt[3]{D}\right) \]

   10. Simplified 70.0%

       \[\leadsto 0.25 \cdot \color{blue}{\left(\sqrt[3]{D} \cdot \left(\left(\left(\frac{M}{d} \cdot D\right) \cdot \frac{M}{\frac{d}{h}}\right) \cdot {\left(\sqrt[3]{D}\right)}^{2}\right)\right)} \]
       Proof

       [Start] 62.3	\[ 0.25 \cdot \left(\frac{\frac{M \cdot \left(M \cdot h\right)}{d} \cdot D}{\frac{d}{{\left(\sqrt[3]{D}\right)}^{2}}} \cdot \sqrt[3]{D}\right) \]
       *-commutative [=>] 62.3	\[ 0.25 \cdot \color{blue}{\left(\sqrt[3]{D} \cdot \frac{\frac{M \cdot \left(M \cdot h\right)}{d} \cdot D}{\frac{d}{{\left(\sqrt[3]{D}\right)}^{2}}}\right)} \]
       associate-/r/ [=>] 61.8	\[ 0.25 \cdot \left(\sqrt[3]{D} \cdot \color{blue}{\left(\frac{\frac{M \cdot \left(M \cdot h\right)}{d} \cdot D}{d} \cdot {\left(\sqrt[3]{D}\right)}^{2}\right)}\right) \]
       associate-/l* [=>] 61.8	\[ 0.25 \cdot \left(\sqrt[3]{D} \cdot \left(\color{blue}{\frac{\frac{M \cdot \left(M \cdot h\right)}{d}}{\frac{d}{D}}} \cdot {\left(\sqrt[3]{D}\right)}^{2}\right)\right) \]
       associate-/l/ [=>] 58.2	\[ 0.25 \cdot \left(\sqrt[3]{D} \cdot \left(\color{blue}{\frac{M \cdot \left(M \cdot h\right)}{\frac{d}{D} \cdot d}} \cdot {\left(\sqrt[3]{D}\right)}^{2}\right)\right) \]
       times-frac [=>] 68.1	\[ 0.25 \cdot \left(\sqrt[3]{D} \cdot \left(\color{blue}{\left(\frac{M}{\frac{d}{D}} \cdot \frac{M \cdot h}{d}\right)} \cdot {\left(\sqrt[3]{D}\right)}^{2}\right)\right) \]
       associate-/l* [=>] 68.4	\[ 0.25 \cdot \left(\sqrt[3]{D} \cdot \left(\left(\frac{M}{\frac{d}{D}} \cdot \color{blue}{\frac{M}{\frac{d}{h}}}\right) \cdot {\left(\sqrt[3]{D}\right)}^{2}\right)\right) \]
       associate-/r/ [=>] 70.0	\[ 0.25 \cdot \left(\sqrt[3]{D} \cdot \left(\left(\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \frac{M}{\frac{d}{h}}\right) \cdot {\left(\sqrt[3]{D}\right)}^{2}\right)\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \cdot D \leq 2 \cdot 10^{-256} \lor \neg \left(D \cdot D \leq 2 \cdot 10^{-65}\right):\\ \;\;\;\;0.25 \cdot \frac{\frac{D}{d} \cdot \left(h \cdot M\right)}{\frac{\frac{d}{D}}{M}}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\sqrt[3]{D} \cdot \left(\left(\left(D \cdot \frac{M}{d}\right) \cdot \frac{M}{\frac{d}{h}}\right) \cdot {\left(\sqrt[3]{D}\right)}^{2}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	67.5%
Cost	1481

\[\begin{array}{l} \mathbf{if}\;D \cdot D \leq 5 \cdot 10^{-277} \lor \neg \left(D \cdot D \leq 5 \cdot 10^{-143}\right):\\ \;\;\;\;0.25 \cdot \left(\frac{M}{\frac{d}{D}} \cdot \left(D \cdot \frac{h \cdot M}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(\frac{M}{d} \cdot \frac{M}{\frac{d}{h}}\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	69.8%
Cost	1481

\[\begin{array}{l} \mathbf{if}\;D \cdot D \leq 2 \cdot 10^{-256} \lor \neg \left(D \cdot D \leq 10^{-154}\right):\\ \;\;\;\;0.25 \cdot \frac{\frac{D}{d} \cdot \left(h \cdot M\right)}{\frac{\frac{d}{D}}{M}}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(\frac{M}{d} \cdot \frac{M}{\frac{d}{h}}\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	60.3%
Cost	960

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

Alternative 4
Accuracy	67.6%
Cost	960

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

Alternative 5
Accuracy	49.6%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023142 
(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))))))