Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.7% → 41.1%

Time: 31.6s

Precision: binary64

Cost: 9812

• could not determine a ground truth for program body

  1. c0 = 2.588831892730052e+250
  2. w = -3.622814202560256e-296
  3. h = 2.3361209464310794e-252
  4. D = 7.002386713264074e-247
  5. d = 7.819664664553314e+236
  6. M = 1.2709663287493348e+174

• could not determine a ground truth for program body

  1. c0 = -6.40676779922915e+281
  2. w = 9.913856762329484e-279
  3. h = 7.722957424393044e-238
  4. D = -9.532905939784468e-286
  5. d = 1.2332118866302464e+246
  6. M = -2.732707537871162e+307

• could not determine a ground truth for program body

  1. c0 = -1.2598285734444896e+308
  2. w = 5.283047732455752e-303
  3. h = 1.3847192065786743e-295
  4. D = -1.2662497240238284e-213
  5. d = 2.2404969996517862e+204
  6. M = 3.885044648975163e+284

• could not determine a ground truth for program body

  1. c0 = -3.088392685653731e+265
  2. w = -1.899040859328017e-298
  3. h = -1.1833140113374948e+45
  4. D = -3.533894867130813e-305
  5. d = 1.0009362451745088e+254
  6. M = 4.2035537749222144e-153

• could not determine a ground truth for program body

  1. c0 = -6.426148777806251e+159
  2. w = -4.438223872941767e-254
  3. h = -1.903978784912602e-285
  4. D = 7.540524664338302e-285
  5. d = 5.52668266754298e+247
  6. M = -4.18356417136122e+252

• could not determine a ground truth for program body

  1. c0 = -1.4594073643275457e+279
  2. w = -5.574272504325534e-177
  3. h = -2.5691627109132856e-237
  4. D = 1.7822168975502565e-300
  5. d = 8.68871761548951e+272
  6. M = -1.5724717072871344e-195

Math

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

↓

\[\begin{array}{l} t_0 := {\left(\frac{d}{D}\right)}^{2}\\ t_1 := \frac{c0 \cdot \left(2 \cdot \left(t_0 \cdot \frac{c0}{w \cdot h}\right)\right)}{2 \cdot w}\\ t_2 := \frac{c0}{2 \cdot w}\\ \mathbf{if}\;D \cdot D \leq 5 \cdot 10^{-302}:\\ \;\;\;\;t_2 \cdot \left(2 \cdot \frac{\frac{c0 \cdot t_0}{w}}{h}\right)\\ \mathbf{elif}\;D \cdot D \leq 10^{-230}:\\ \;\;\;\;0\\ \mathbf{elif}\;D \cdot D \leq 5 \cdot 10^{-153}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;D \cdot D \leq 1000000000000:\\ \;\;\;\;0\\ \mathbf{elif}\;D \cdot D \leq 10^{+294}:\\ \;\;\;\;t_2 \cdot \mathsf{fma}\left(2, d \cdot \left(d \cdot \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right), 0.5 \cdot \frac{D \cdot D}{\frac{c0}{\left(w \cdot h\right) \cdot 0}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (pow (/ d D) 2.0))
        (t_1 (/ (* c0 (* 2.0 (* t_0 (/ c0 (* w h))))) (* 2.0 w)))
        (t_2 (/ c0 (* 2.0 w))))
   (if (<= (* D D) 5e-302)
     (* t_2 (* 2.0 (/ (/ (* c0 t_0) w) h)))
     (if (<= (* D D) 1e-230)
       0.0
       (if (<= (* D D) 5e-153)
         t_1
         (if (<= (* D D) 1000000000000.0)
           0.0
           (if (<= (* D D) 1e+294)
             (*
              t_2
              (fma
               2.0
               (* d (* d (/ c0 (* (* w h) (* D D)))))
               (* 0.5 (/ (* D D) (/ c0 (* (* w h) 0.0))))))
             t_1)))))))

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 = pow((d / D), 2.0);
	double t_1 = (c0 * (2.0 * (t_0 * (c0 / (w * h))))) / (2.0 * w);
	double t_2 = c0 / (2.0 * w);
	double tmp;
	if ((D * D) <= 5e-302) {
		tmp = t_2 * (2.0 * (((c0 * t_0) / w) / h));
	} else if ((D * D) <= 1e-230) {
		tmp = 0.0;
	} else if ((D * D) <= 5e-153) {
		tmp = t_1;
	} else if ((D * D) <= 1000000000000.0) {
		tmp = 0.0;
	} else if ((D * D) <= 1e+294) {
		tmp = t_2 * fma(2.0, (d * (d * (c0 / ((w * h) * (D * D))))), (0.5 * ((D * D) / (c0 / ((w * h) * 0.0)))));
	} else {
		tmp = t_1;
	}
	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(d / D) ^ 2.0
	t_1 = Float64(Float64(c0 * Float64(2.0 * Float64(t_0 * Float64(c0 / Float64(w * h))))) / Float64(2.0 * w))
	t_2 = Float64(c0 / Float64(2.0 * w))
	tmp = 0.0
	if (Float64(D * D) <= 5e-302)
		tmp = Float64(t_2 * Float64(2.0 * Float64(Float64(Float64(c0 * t_0) / w) / h)));
	elseif (Float64(D * D) <= 1e-230)
		tmp = 0.0;
	elseif (Float64(D * D) <= 5e-153)
		tmp = t_1;
	elseif (Float64(D * D) <= 1000000000000.0)
		tmp = 0.0;
	elseif (Float64(D * D) <= 1e+294)
		tmp = Float64(t_2 * fma(2.0, Float64(d * Float64(d * Float64(c0 / Float64(Float64(w * h) * Float64(D * D))))), Float64(0.5 * Float64(Float64(D * D) / Float64(c0 / Float64(Float64(w * h) * 0.0))))));
	else
		tmp = t_1;
	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[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(2.0 * N[(t$95$0 * N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(D * D), $MachinePrecision], 5e-302], N[(t$95$2 * N[(2.0 * N[(N[(N[(c0 * t$95$0), $MachinePrecision] / w), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(D * D), $MachinePrecision], 1e-230], 0.0, If[LessEqual[N[(D * D), $MachinePrecision], 5e-153], t$95$1, If[LessEqual[N[(D * D), $MachinePrecision], 1000000000000.0], 0.0, If[LessEqual[N[(D * D), $MachinePrecision], 1e+294], N[(t$95$2 * N[(2.0 * N[(d * N[(d * N[(c0 / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(D * D), $MachinePrecision] / N[(c0 / N[(N[(w * h), $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 D D) < 5.00000000000000033e-302

   1. Initial program 26.7%

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

   2. Simplified 36.3%

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

      [Start] 26.7	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      times-frac [=>] 26.7	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      fma-def [=>] 26.7	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d \cdot d}{D \cdot D}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)} \]
      associate-/r* [=>] 26.7	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \color{blue}{\frac{\frac{d \cdot d}{D}}{D}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      difference-of-squares [=>] 32.0	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{\frac{d \cdot d}{D}}{D}, \sqrt{\color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + M\right) \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M\right)}}\right) \]

   3. Applied egg-rr 34.9%

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

      [Start] 36.3	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{\frac{d \cdot d}{D}}{D}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{\frac{d \cdot d}{D}}{D}, M\right) \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{\frac{d \cdot d}{D}}{D} - M\right)}\right) \]
      frac-times [=>] 34.9	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{\frac{d \cdot d}{D}}{D}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{\frac{d \cdot d}{D}}{D}, M\right) \cdot \left(\color{blue}{\frac{c0 \cdot \frac{d \cdot d}{D}}{\left(w \cdot h\right) \cdot D}} - M\right)}\right) \]
      associate-/l* [=>] 35.9	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{\frac{d \cdot d}{D}}{D}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{\frac{d \cdot d}{D}}{D}, M\right) \cdot \left(\frac{c0 \cdot \color{blue}{\frac{d}{\frac{D}{d}}}}{\left(w \cdot h\right) \cdot D} - M\right)}\right) \]
      div-inv [=>] 35.9	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{\frac{d \cdot d}{D}}{D}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{\frac{d \cdot d}{D}}{D}, M\right) \cdot \left(\frac{c0 \cdot \color{blue}{\left(d \cdot \frac{1}{\frac{D}{d}}\right)}}{\left(w \cdot h\right) \cdot D} - M\right)}\right) \]
      clear-num [<=] 35.9	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{\frac{d \cdot d}{D}}{D}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{\frac{d \cdot d}{D}}{D}, M\right) \cdot \left(\frac{c0 \cdot \left(d \cdot \color{blue}{\frac{d}{D}}\right)}{\left(w \cdot h\right) \cdot D} - M\right)}\right) \]
      associate-*l* [=>] 34.9	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{\frac{d \cdot d}{D}}{D}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{\frac{d \cdot d}{D}}{D}, M\right) \cdot \left(\frac{c0 \cdot \left(d \cdot \frac{d}{D}\right)}{\color{blue}{w \cdot \left(h \cdot D\right)}} - M\right)}\right) \]

   4. Taylor expanded in c0 around inf 33.4%

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

   5. Simplified 47.8%

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

      [Start] 33.4	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]
      *-commutative [=>] 33.4	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\color{blue}{c0 \cdot {d}^{2}}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]
      associate-*r* [=>] 34.4	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\color{blue}{\left({D}^{2} \cdot w\right) \cdot h}}\right) \]
      *-commutative [=>] 34.4	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\color{blue}{\left(w \cdot {D}^{2}\right)} \cdot h}\right) \]
      associate-*r* [<=] 34.4	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\color{blue}{w \cdot \left({D}^{2} \cdot h\right)}}\right) \]
      unpow2 [=>] 34.4	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{w \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot h\right)}\right) \]
      times-frac [=>] 35.4	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{c0}{w} \cdot \frac{{d}^{2}}{\left(D \cdot D\right) \cdot h}\right)}\right) \]
      unpow2 [<=] 35.4	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{{d}^{2}}{\color{blue}{{D}^{2}} \cdot h}\right)\right) \]
      associate-*r/ [=>] 35.3	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{\frac{c0}{w} \cdot {d}^{2}}{{D}^{2} \cdot h}}\right) \]
      unpow2 [=>] 35.3	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{w} \cdot \color{blue}{\left(d \cdot d\right)}}{{D}^{2} \cdot h}\right) \]
      associate-/r* [=>] 35.4	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{\frac{\frac{c0}{w} \cdot \left(d \cdot d\right)}{{D}^{2}}}{h}}\right) \]

   6. Applied egg-rr 48.2%

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

      [Start] 47.8	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w}}{h}\right) \]
      associate-*r/ [=>] 48.2	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\color{blue}{\frac{{\left(\frac{d}{D}\right)}^{2} \cdot c0}{w}}}{h}\right) \]

   if 5.00000000000000033e-302 < (*.f64 D D) < 1.00000000000000005e-230 or 5.00000000000000033e-153 < (*.f64 D D) < 1e12

   1. Initial program 11.3%

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

   2. Simplified 5.2%

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

      [Start] 11.3	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      times-frac [=>] 4.9	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      fma-def [=>] 2.7	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d \cdot d}{D \cdot D}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)} \]
      associate-/r* [=>] 2.8	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \color{blue}{\frac{\frac{d \cdot d}{D}}{D}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      difference-of-squares [=>] 5.1	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{\frac{d \cdot d}{D}}{D}, \sqrt{\color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + M\right) \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M\right)}}\right) \]

   3. Taylor expanded in c0 around -inf 10.8%

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

   4. Simplified 41.8%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]
      Step-by-step derivation

      [Start] 10.8	\[ \frac{c0}{2 \cdot w} \cdot \left(-1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right) \]
      associate-*r* [=>] 10.8	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\left(-1 \cdot \left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)\right) \cdot c0\right)} \]
      distribute-rgt1-in [=>] 10.8	\[ \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}\right) \cdot c0\right) \]
      metadata-eval [=>] 10.8	\[ \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)\right) \cdot c0\right) \]
      mul0-lft [=>] 41.8	\[ \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot \color{blue}{0}\right) \cdot c0\right) \]
      metadata-eval [=>] 41.8	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{0} \cdot c0\right) \]
      mul0-lft [<=] 10.8	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(0 \cdot \frac{{d}^{2} \cdot M}{{D}^{2} \cdot \left(w \cdot h\right)}\right)} \cdot c0\right) \]
      metadata-eval [<=] 10.8	\[ \frac{c0}{2 \cdot w} \cdot \left(\left(\color{blue}{\left(-1 + 1\right)} \cdot \frac{{d}^{2} \cdot M}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right) \]
      distribute-lft1-in [<=] 10.8	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(-1 \cdot \frac{{d}^{2} \cdot M}{{D}^{2} \cdot \left(w \cdot h\right)} + \frac{{d}^{2} \cdot M}{{D}^{2} \cdot \left(w \cdot h\right)}\right)} \cdot c0\right) \]
      *-commutative [=>] 10.8	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot \left(-1 \cdot \frac{{d}^{2} \cdot M}{{D}^{2} \cdot \left(w \cdot h\right)} + \frac{{d}^{2} \cdot M}{{D}^{2} \cdot \left(w \cdot h\right)}\right)\right)} \]
      distribute-lft1-in [=>] 10.8	\[ \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2} \cdot M}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}\right) \]
      metadata-eval [=>] 10.8	\[ \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2} \cdot M}{{D}^{2} \cdot \left(w \cdot h\right)}\right)\right) \]
      mul0-lft [=>] 41.8	\[ \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{0}\right) \]

   5. Taylor expanded in c0 around 0 54.7%

      \[\leadsto \color{blue}{0} \]

   if 1.00000000000000005e-230 < (*.f64 D D) < 5.00000000000000033e-153 or 1.00000000000000007e294 < (*.f64 D D)

   1. Initial program 16.9%

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

   2. Simplified 41.1%

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

      [Start] 16.9	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      times-frac [=>] 16.9	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      fma-def [=>] 17.0	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d \cdot d}{D \cdot D}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)} \]
      associate-/r* [=>] 17.0	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \color{blue}{\frac{\frac{d \cdot d}{D}}{D}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      difference-of-squares [=>] 17.0	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{\frac{d \cdot d}{D}}{D}, \sqrt{\color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + M\right) \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M\right)}}\right) \]

   3. Applied egg-rr 37.4%

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

      [Start] 41.1	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{\frac{d \cdot d}{D}}{D}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{\frac{d \cdot d}{D}}{D}, M\right) \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{\frac{d \cdot d}{D}}{D} - M\right)}\right) \]
      frac-times [=>] 39.2	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{\frac{d \cdot d}{D}}{D}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{\frac{d \cdot d}{D}}{D}, M\right) \cdot \left(\color{blue}{\frac{c0 \cdot \frac{d \cdot d}{D}}{\left(w \cdot h\right) \cdot D}} - M\right)}\right) \]
      associate-/l* [=>] 39.2	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{\frac{d \cdot d}{D}}{D}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{\frac{d \cdot d}{D}}{D}, M\right) \cdot \left(\frac{c0 \cdot \color{blue}{\frac{d}{\frac{D}{d}}}}{\left(w \cdot h\right) \cdot D} - M\right)}\right) \]
      div-inv [=>] 39.2	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{\frac{d \cdot d}{D}}{D}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{\frac{d \cdot d}{D}}{D}, M\right) \cdot \left(\frac{c0 \cdot \color{blue}{\left(d \cdot \frac{1}{\frac{D}{d}}\right)}}{\left(w \cdot h\right) \cdot D} - M\right)}\right) \]
      clear-num [<=] 39.2	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{\frac{d \cdot d}{D}}{D}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{\frac{d \cdot d}{D}}{D}, M\right) \cdot \left(\frac{c0 \cdot \left(d \cdot \color{blue}{\frac{d}{D}}\right)}{\left(w \cdot h\right) \cdot D} - M\right)}\right) \]
      associate-*l* [=>] 37.4	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{\frac{d \cdot d}{D}}{D}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{\frac{d \cdot d}{D}}{D}, M\right) \cdot \left(\frac{c0 \cdot \left(d \cdot \frac{d}{D}\right)}{\color{blue}{w \cdot \left(h \cdot D\right)}} - M\right)}\right) \]

   4. Taylor expanded in c0 around inf 18.9%

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

   5. Simplified 56.7%

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

      [Start] 18.9	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]
      *-commutative [=>] 18.9	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\color{blue}{c0 \cdot {d}^{2}}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]
      associate-*r* [=>] 18.9	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\color{blue}{\left({D}^{2} \cdot w\right) \cdot h}}\right) \]
      *-commutative [=>] 18.9	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\color{blue}{\left(w \cdot {D}^{2}\right)} \cdot h}\right) \]
      associate-*r* [<=] 17.1	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\color{blue}{w \cdot \left({D}^{2} \cdot h\right)}}\right) \]
      unpow2 [=>] 17.1	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{w \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot h\right)}\right) \]
      times-frac [=>] 20.7	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{c0}{w} \cdot \frac{{d}^{2}}{\left(D \cdot D\right) \cdot h}\right)}\right) \]
      unpow2 [<=] 20.7	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{{d}^{2}}{\color{blue}{{D}^{2}} \cdot h}\right)\right) \]
      associate-*r/ [=>] 18.8	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{\frac{c0}{w} \cdot {d}^{2}}{{D}^{2} \cdot h}}\right) \]
      unpow2 [=>] 18.8	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{w} \cdot \color{blue}{\left(d \cdot d\right)}}{{D}^{2} \cdot h}\right) \]
      associate-/r* [=>] 22.5	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{\frac{\frac{c0}{w} \cdot \left(d \cdot d\right)}{{D}^{2}}}{h}}\right) \]

   6. Applied egg-rr 60.3%

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

      [Start] 56.7	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w}}{h}\right) \]
      associate-*l/ [=>] 56.8	\[ \color{blue}{\frac{c0 \cdot \left(2 \cdot \frac{{\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w}}{h}\right)}{2 \cdot w}} \]
      associate-*r/ [=>] 58.8	\[ \frac{c0 \cdot \left(2 \cdot \frac{\color{blue}{\frac{{\left(\frac{d}{D}\right)}^{2} \cdot c0}{w}}}{h}\right)}{2 \cdot w} \]
      *-commutative [<=] 58.8	\[ \frac{c0 \cdot \left(2 \cdot \frac{\frac{\color{blue}{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}}{w}}{h}\right)}{2 \cdot w} \]
      associate-/r* [<=] 58.4	\[ \frac{c0 \cdot \left(2 \cdot \color{blue}{\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w \cdot h}}\right)}{2 \cdot w} \]
      associate-*l/ [<=] 60.3	\[ \frac{c0 \cdot \left(2 \cdot \color{blue}{\left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\right)}\right)}{2 \cdot w} \]
      *-commutative [=>] 60.3	\[ \frac{c0 \cdot \left(2 \cdot \color{blue}{\left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h}\right)}\right)}{2 \cdot w} \]

   if 1e12 < (*.f64 D D) < 1.00000000000000007e294

   1. Initial program 25.9%

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

   2. Simplified 20.9%

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

      [Start] 25.9	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      associate-*l* [=>] 22.7	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      difference-of-squares [=>] 22.9	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)} + \sqrt{\color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + M\right) \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M\right)}}\right) \]
      associate-*l* [=>] 20.5	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)} + \sqrt{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)}} + M\right) \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M\right)}\right) \]
      associate-*l* [=>] 20.9	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)} + \sqrt{\left(\frac{c0 \cdot \left(d \cdot d\right)}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)} + M\right) \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)}} - M\right)}\right) \]

   3. Applied egg-rr 20.8%

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

      [Start] 20.9	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)} + \sqrt{\left(\frac{c0 \cdot \left(d \cdot d\right)}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)} + M\right) \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)} - M\right)}\right) \]
      times-frac [=>] 20.8	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{w} \cdot \frac{d \cdot d}{h \cdot \left(D \cdot D\right)}} + \sqrt{\left(\frac{c0 \cdot \left(d \cdot d\right)}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)} + M\right) \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)} - M\right)}\right) \]

   4. Taylor expanded in d around inf 25.4%

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

   5. Simplified 57.5%

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

      [Start] 25.4	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)} + 0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left(\left(\frac{c0 \cdot M}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{c0 \cdot M}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot h\right)\right)}{c0}\right) \]
      fma-def [=>] 25.4	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(2, \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}, 0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left(\left(\frac{c0 \cdot M}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{c0 \cdot M}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot h\right)\right)}{c0}\right)} \]

3. Recombined 4 regimes into one program.

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \cdot D \leq 5 \cdot 10^{-302}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w}}{h}\right)\\ \mathbf{elif}\;D \cdot D \leq 10^{-230}:\\ \;\;\;\;0\\ \mathbf{elif}\;D \cdot D \leq 5 \cdot 10^{-153}:\\ \;\;\;\;\frac{c0 \cdot \left(2 \cdot \left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h}\right)\right)}{2 \cdot w}\\ \mathbf{elif}\;D \cdot D \leq 1000000000000:\\ \;\;\;\;0\\ \mathbf{elif}\;D \cdot D \leq 10^{+294}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(2, d \cdot \left(d \cdot \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right), 0.5 \cdot \frac{D \cdot D}{\frac{c0}{\left(w \cdot h\right) \cdot 0}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \left(2 \cdot \left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h}\right)\right)}{2 \cdot w}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	54.2%
Cost	18372

\[\begin{array}{l} t_0 := \left(w \cdot h\right) \cdot \left(D \cdot D\right)\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{t_0}\\ \mathbf{if}\;t_1 \cdot \left(t_2 + \sqrt{t_2 \cdot t_2 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t_1 \cdot \mathsf{fma}\left(2, d \cdot \left(d \cdot \frac{c0}{t_0}\right), 0.5 \cdot \frac{D \cdot D}{\frac{d \cdot d}{w \cdot \left(h \cdot 0\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 2
Accuracy	40.7%
Cost	8852

\[\begin{array}{l} t_0 := {\left(\frac{d}{D}\right)}^{2}\\ t_1 := \frac{c0 \cdot \left(2 \cdot \left(t_0 \cdot \frac{c0}{w \cdot h}\right)\right)}{2 \cdot w}\\ \mathbf{if}\;D \cdot D \leq 5 \cdot 10^{-302}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0 \cdot t_0}{w}}{h}\right)\\ \mathbf{elif}\;D \cdot D \leq 10^{-230}:\\ \;\;\;\;0\\ \mathbf{elif}\;D \cdot D \leq 5 \cdot 10^{-153}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;D \cdot D \leq 1000000000000:\\ \;\;\;\;0\\ \mathbf{elif}\;D \cdot D \leq 10^{+91}:\\ \;\;\;\;d \cdot \left(\frac{d}{D \cdot D} \cdot \left(\frac{c0}{h} \cdot \frac{c0}{w \cdot w}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Accuracy	40.3%
Cost	8852

\[\begin{array}{l} t_0 := {\left(\frac{d}{D}\right)}^{2}\\ t_1 := \frac{c0 \cdot \left(2 \cdot \left(t_0 \cdot \frac{c0}{w \cdot h}\right)\right)}{2 \cdot w}\\ \mathbf{if}\;D \cdot D \leq 5 \cdot 10^{-302}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0 \cdot t_0}{w}}{h}\right)\\ \mathbf{elif}\;D \cdot D \leq 10^{-230}:\\ \;\;\;\;0\\ \mathbf{elif}\;D \cdot D \leq 5 \cdot 10^{-153}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;D \cdot D \leq 1000000000000:\\ \;\;\;\;0\\ \mathbf{elif}\;D \cdot D \leq 10^{+294}:\\ \;\;\;\;d \cdot \left(\frac{d}{D \cdot D} \cdot \frac{c0 \cdot c0}{w \cdot \left(w \cdot h\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Accuracy	42.3%
Cost	8072

\[\begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ \mathbf{if}\;d \cdot d \leq 10^{+134}:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h}\right)\\ \mathbf{elif}\;d \cdot d \leq 5 \cdot 10^{+302}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h}\right)\right)\\ \end{array} \]

Alternative 5
Accuracy	43.0%
Cost	8072

\[\begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ \mathbf{if}\;d \cdot d \leq 10^{+134}:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h}\right)\\ \mathbf{elif}\;d \cdot d \leq 5 \cdot 10^{+302}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \frac{\frac{c0 \cdot {\left(\frac{d}{D}\right)}^{2}}{w}}{h}\right)\\ \end{array} \]

Alternative 6
Accuracy	43.2%
Cost	1865

\[\begin{array}{l} \mathbf{if}\;d \cdot d \leq 10^{+134} \lor \neg \left(d \cdot d \leq 5 \cdot 10^{+302}\right):\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7
Accuracy	42.6%
Cost	1609

\[\begin{array}{l} \mathbf{if}\;c0 \leq -24500000 \lor \neg \left(c0 \leq 3.3 \cdot 10^{+80}\right):\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{w} \cdot \left(d \cdot \frac{d}{D \cdot D}\right)}{h}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8
Accuracy	41.2%
Cost	1353

\[\begin{array}{l} \mathbf{if}\;c0 \leq -26500000 \lor \neg \left(c0 \leq 2.5 \cdot 10^{+81}\right):\\ \;\;\;\;\left(d \cdot \frac{d}{D \cdot D}\right) \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{c0}{w}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9
Accuracy	40.7%
Cost	1352

\[\begin{array}{l} \mathbf{if}\;c0 \leq -9600000:\\ \;\;\;\;\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot c0}{h \cdot \left(w \cdot w\right)}\\ \mathbf{elif}\;c0 \leq 5 \cdot 10^{+82}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(d \cdot \frac{d}{D \cdot D}\right) \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{c0}{w}\right)\\ \end{array} \]

Alternative 10
Accuracy	36.1%
Cost	1220

\[\begin{array}{l} \mathbf{if}\;M \leq 7.2 \cdot 10^{-21}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c0}{h} \cdot \frac{c0}{w \cdot w}\right) \cdot \left(d \cdot \frac{d}{D \cdot D}\right)\\ \end{array} \]

Alternative 11
Accuracy	32.8%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023162 
(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))))))