Henrywood and Agarwal, Equation (13)

Average Accuracy: 7.0% → 79.8%

Time: 36.3s

Precision: binary64

Cost: 42636

Math

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

↓

\[\begin{array}{l} t_0 := \frac{D}{\frac{d}{M}}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_2 := \frac{c0}{2 \cdot w} \cdot \left(t_1 + \sqrt{t_1 \cdot t_1 - M \cdot M}\right)\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{-208}:\\ \;\;\;\;\frac{c0 \cdot d}{w \cdot \left(\left(w \cdot h\right) \cdot D\right)} \cdot \frac{c0 \cdot d}{D}\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(0.25, t_0 \cdot \frac{h \cdot D}{\frac{d}{M}}, 0\right)\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;{\left(\frac{\frac{c0}{w}}{\sqrt{h}} \cdot \frac{d}{D}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.25, t_0 \cdot \left(h \cdot \left(D \cdot \frac{M}{d}\right)\right), 0\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ D (/ d M)))
        (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_2 (* (/ c0 (* 2.0 w)) (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))))
   (if (<= t_2 -5e-208)
     (* (/ (* c0 d) (* w (* (* w h) D))) (/ (* c0 d) D))
     (if (<= t_2 0.0)
       (fma 0.25 (* t_0 (/ (* h D) (/ d M))) 0.0)
       (if (<= t_2 INFINITY)
         (pow (* (/ (/ c0 w) (sqrt h)) (/ d D)) 2.0)
         (fma 0.25 (* t_0 (* h (* D (/ M d)))) 0.0))))))

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 t_0 = D / (d / M);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_2 = (c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= -5e-208) {
		tmp = ((c0 * d) / (w * ((w * h) * D))) * ((c0 * d) / D);
	} else if (t_2 <= 0.0) {
		tmp = fma(0.25, (t_0 * ((h * D) / (d / M))), 0.0);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = pow((((c0 / w) / sqrt(h)) * (d / D)), 2.0);
	} else {
		tmp = fma(0.25, (t_0 * (h * (D * (M / d)))), 0.0);
	}
	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)
	t_0 = Float64(D / Float64(d / M))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	t_2 = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M)))))
	tmp = 0.0
	if (t_2 <= -5e-208)
		tmp = Float64(Float64(Float64(c0 * d) / Float64(w * Float64(Float64(w * h) * D))) * Float64(Float64(c0 * d) / D));
	elseif (t_2 <= 0.0)
		tmp = fma(0.25, Float64(t_0 * Float64(Float64(h * D) / Float64(d / M))), 0.0);
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(Float64(c0 / w) / sqrt(h)) * Float64(d / D)) ^ 2.0;
	else
		tmp = fma(0.25, Float64(t_0 * Float64(h * Float64(D * Float64(M / d)))), 0.0);
	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_] := Block[{t$95$0 = N[(D / N[(d / M), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-208], N[(N[(N[(c0 * d), $MachinePrecision] / N[(w * N[(N[(w * h), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(c0 * d), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(0.25 * N[(t$95$0 * N[(N[(h * D), $MachinePrecision] / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Power[N[(N[(N[(c0 / w), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(0.25 * N[(t$95$0 * N[(h * N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < -4.99999999999999963e-208

   1. Initial program 24.2%

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

   2. Simplified 24.3%

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

      [Start] 24.2	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      times-frac [=>] 20.6	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      fma-def [=>] 20.6	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d \cdot d}{D \cdot D}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)} \]
      associate-/r* [=>] 20.6	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \color{blue}{\frac{\frac{d \cdot d}{D}}{D}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      difference-of-squares [=>] 20.6	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{\frac{d \cdot d}{D}}{D}, \sqrt{\color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + M\right) \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M\right)}}\right) \]

   3. Taylor expanded in c0 around inf 12.8%

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

   4. Simplified 11.6%

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

      [Start] 12.8	\[ \frac{{d}^{2} \cdot {c0}^{2}}{{D}^{2} \cdot \left({w}^{2} \cdot h\right)} \]
      times-frac [=>] 11.6	\[ \color{blue}{\frac{{d}^{2}}{{D}^{2}} \cdot \frac{{c0}^{2}}{{w}^{2} \cdot h}} \]
      unpow2 [=>] 11.6	\[ \frac{\color{blue}{d \cdot d}}{{D}^{2}} \cdot \frac{{c0}^{2}}{{w}^{2} \cdot h} \]
      unpow2 [=>] 11.6	\[ \frac{d \cdot d}{\color{blue}{D \cdot D}} \cdot \frac{{c0}^{2}}{{w}^{2} \cdot h} \]
      unpow2 [=>] 11.6	\[ \frac{d \cdot d}{D \cdot D} \cdot \frac{\color{blue}{c0 \cdot c0}}{{w}^{2} \cdot h} \]
      *-commutative [=>] 11.6	\[ \frac{d \cdot d}{D \cdot D} \cdot \frac{c0 \cdot c0}{\color{blue}{h \cdot {w}^{2}}} \]
      unpow2 [=>] 11.6	\[ \frac{d \cdot d}{D \cdot D} \cdot \frac{c0 \cdot c0}{h \cdot \color{blue}{\left(w \cdot w\right)}} \]

   5. Applied egg-rr 15.3%

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

      [Start] 11.6	\[ \frac{d \cdot d}{D \cdot D} \cdot \frac{c0 \cdot c0}{h \cdot \left(w \cdot w\right)} \]
      frac-2neg [=>] 11.6	\[ \color{blue}{\frac{-d \cdot d}{-D \cdot D}} \cdot \frac{c0 \cdot c0}{h \cdot \left(w \cdot w\right)} \]
      frac-2neg [=>] 11.6	\[ \frac{-d \cdot d}{-D \cdot D} \cdot \color{blue}{\frac{-c0 \cdot c0}{-h \cdot \left(w \cdot w\right)}} \]
      frac-times [=>] 12.8	\[ \color{blue}{\frac{\left(-d \cdot d\right) \cdot \left(-c0 \cdot c0\right)}{\left(-D \cdot D\right) \cdot \left(-h \cdot \left(w \cdot w\right)\right)}} \]
      distribute-rgt-neg-in [=>] 12.8	\[ \frac{\left(-d \cdot d\right) \cdot \left(-c0 \cdot c0\right)}{\color{blue}{\left(D \cdot \left(-D\right)\right)} \cdot \left(-h \cdot \left(w \cdot w\right)\right)} \]
      associate-*r* [=>] 15.3	\[ \frac{\left(-d \cdot d\right) \cdot \left(-c0 \cdot c0\right)}{\left(D \cdot \left(-D\right)\right) \cdot \left(-\color{blue}{\left(h \cdot w\right) \cdot w}\right)} \]
      distribute-lft-neg-in [=>] 15.3	\[ \frac{\left(-d \cdot d\right) \cdot \left(-c0 \cdot c0\right)}{\left(D \cdot \left(-D\right)\right) \cdot \color{blue}{\left(\left(-h \cdot w\right) \cdot w\right)}} \]

   6. Simplified 39.8%

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

      [Start] 15.3	\[ \frac{\left(-d \cdot d\right) \cdot \left(-c0 \cdot c0\right)}{\left(D \cdot \left(-D\right)\right) \cdot \left(\left(-h \cdot w\right) \cdot w\right)} \]
      distribute-lft-neg-out [=>] 15.3	\[ \frac{\color{blue}{-\left(d \cdot d\right) \cdot \left(-c0 \cdot c0\right)}}{\left(D \cdot \left(-D\right)\right) \cdot \left(\left(-h \cdot w\right) \cdot w\right)} \]
      distribute-rgt-neg-out [=>] 15.3	\[ \frac{-\color{blue}{\left(-\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)\right)}}{\left(D \cdot \left(-D\right)\right) \cdot \left(\left(-h \cdot w\right) \cdot w\right)} \]
      remove-double-neg [=>] 15.3	\[ \frac{\color{blue}{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}}{\left(D \cdot \left(-D\right)\right) \cdot \left(\left(-h \cdot w\right) \cdot w\right)} \]
      associate-*l* [=>] 18.5	\[ \frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{\color{blue}{D \cdot \left(\left(-D\right) \cdot \left(\left(-h \cdot w\right) \cdot w\right)\right)}} \]
      associate-/r* [=>] 19.9	\[ \color{blue}{\frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{D}}{\left(-D\right) \cdot \left(\left(-h \cdot w\right) \cdot w\right)}} \]
      swap-sqr [<=] 35.6	\[ \frac{\frac{\color{blue}{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}}{D}}{\left(-D\right) \cdot \left(\left(-h \cdot w\right) \cdot w\right)} \]
      unpow2 [<=] 35.6	\[ \frac{\frac{\color{blue}{{\left(d \cdot c0\right)}^{2}}}{D}}{\left(-D\right) \cdot \left(\left(-h \cdot w\right) \cdot w\right)} \]
      *-commutative [=>] 35.6	\[ \frac{\frac{{\left(d \cdot c0\right)}^{2}}{D}}{\left(-D\right) \cdot \color{blue}{\left(w \cdot \left(-h \cdot w\right)\right)}} \]
      associate-*r* [=>] 39.8	\[ \frac{\frac{{\left(d \cdot c0\right)}^{2}}{D}}{\color{blue}{\left(\left(-D\right) \cdot w\right) \cdot \left(-h \cdot w\right)}} \]
      *-commutative [=>] 39.8	\[ \frac{\frac{{\left(d \cdot c0\right)}^{2}}{D}}{\left(\left(-D\right) \cdot w\right) \cdot \left(-\color{blue}{w \cdot h}\right)} \]
      distribute-rgt-neg-in [=>] 39.8	\[ \frac{\frac{{\left(d \cdot c0\right)}^{2}}{D}}{\left(\left(-D\right) \cdot w\right) \cdot \color{blue}{\left(w \cdot \left(-h\right)\right)}} \]

   7. Applied egg-rr 58.3%

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

      [Start] 39.8	\[ \frac{\frac{{\left(d \cdot c0\right)}^{2}}{D}}{\left(\left(-D\right) \cdot w\right) \cdot \left(w \cdot \left(-h\right)\right)} \]
      associate-/l/ [=>] 41.6	\[ \color{blue}{\frac{{\left(d \cdot c0\right)}^{2}}{\left(\left(\left(-D\right) \cdot w\right) \cdot \left(w \cdot \left(-h\right)\right)\right) \cdot D}} \]
      unpow2 [=>] 41.6	\[ \frac{\color{blue}{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}}{\left(\left(\left(-D\right) \cdot w\right) \cdot \left(w \cdot \left(-h\right)\right)\right) \cdot D} \]
      times-frac [=>] 57.3	\[ \color{blue}{\frac{d \cdot c0}{\left(\left(-D\right) \cdot w\right) \cdot \left(w \cdot \left(-h\right)\right)} \cdot \frac{d \cdot c0}{D}} \]
      *-commutative [=>] 57.3	\[ \frac{d \cdot c0}{\color{blue}{\left(w \cdot \left(-D\right)\right)} \cdot \left(w \cdot \left(-h\right)\right)} \cdot \frac{d \cdot c0}{D} \]
      associate-*l* [=>] 58.3	\[ \frac{d \cdot c0}{\color{blue}{w \cdot \left(\left(-D\right) \cdot \left(w \cdot \left(-h\right)\right)\right)}} \cdot \frac{d \cdot c0}{D} \]
      add-sqr-sqrt [=>] 29.2	\[ \frac{d \cdot c0}{w \cdot \left(\color{blue}{\left(\sqrt{-D} \cdot \sqrt{-D}\right)} \cdot \left(w \cdot \left(-h\right)\right)\right)} \cdot \frac{d \cdot c0}{D} \]
      sqrt-unprod [=>] 24.8	\[ \frac{d \cdot c0}{w \cdot \left(\color{blue}{\sqrt{\left(-D\right) \cdot \left(-D\right)}} \cdot \left(w \cdot \left(-h\right)\right)\right)} \cdot \frac{d \cdot c0}{D} \]
      sqr-neg [=>] 24.8	\[ \frac{d \cdot c0}{w \cdot \left(\sqrt{\color{blue}{D \cdot D}} \cdot \left(w \cdot \left(-h\right)\right)\right)} \cdot \frac{d \cdot c0}{D} \]
      sqrt-unprod [<=] 0.9	\[ \frac{d \cdot c0}{w \cdot \left(\color{blue}{\left(\sqrt{D} \cdot \sqrt{D}\right)} \cdot \left(w \cdot \left(-h\right)\right)\right)} \cdot \frac{d \cdot c0}{D} \]
      add-sqr-sqrt [<=] 1.4	\[ \frac{d \cdot c0}{w \cdot \left(\color{blue}{D} \cdot \left(w \cdot \left(-h\right)\right)\right)} \cdot \frac{d \cdot c0}{D} \]
      add-sqr-sqrt [=>] 0.9	\[ \frac{d \cdot c0}{w \cdot \left(D \cdot \left(w \cdot \color{blue}{\left(\sqrt{-h} \cdot \sqrt{-h}\right)}\right)\right)} \cdot \frac{d \cdot c0}{D} \]
      sqrt-unprod [=>] 2.1	\[ \frac{d \cdot c0}{w \cdot \left(D \cdot \left(w \cdot \color{blue}{\sqrt{\left(-h\right) \cdot \left(-h\right)}}\right)\right)} \cdot \frac{d \cdot c0}{D} \]
      sqr-neg [=>] 2.1	\[ \frac{d \cdot c0}{w \cdot \left(D \cdot \left(w \cdot \sqrt{\color{blue}{h \cdot h}}\right)\right)} \cdot \frac{d \cdot c0}{D} \]
      sqrt-unprod [<=] 0.6	\[ \frac{d \cdot c0}{w \cdot \left(D \cdot \left(w \cdot \color{blue}{\left(\sqrt{h} \cdot \sqrt{h}\right)}\right)\right)} \cdot \frac{d \cdot c0}{D} \]
      add-sqr-sqrt [<=] 58.3	\[ \frac{d \cdot c0}{w \cdot \left(D \cdot \left(w \cdot \color{blue}{h}\right)\right)} \cdot \frac{d \cdot c0}{D} \]

   if -4.99999999999999963e-208 < (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < -0.0

   1. Initial program 54.9%

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

   2. Taylor expanded in c0 around -inf 49.2%

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

   3. Simplified 51.7%

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

      [Start] 49.2	\[ -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w} + 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \]
      +-commutative [=>] 49.2	\[ \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} + -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}} \]
      fma-def [=>] 49.2	\[ \color{blue}{\mathsf{fma}\left(0.25, \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}, -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}\right)} \]

   4. Applied egg-rr 65.1%

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

      [Start] 51.7	\[ \mathsf{fma}\left(0.25, \frac{\left(D \cdot D\right) \cdot h}{\frac{d \cdot d}{M \cdot M}}, \frac{c0 \cdot c0}{\frac{w}{0}}\right) \]
      associate-*l* [=>] 53.9	\[ \mathsf{fma}\left(0.25, \frac{\color{blue}{D \cdot \left(D \cdot h\right)}}{\frac{d \cdot d}{M \cdot M}}, \frac{c0 \cdot c0}{\frac{w}{0}}\right) \]
      add-sqr-sqrt [=>] 53.9	\[ \mathsf{fma}\left(0.25, \frac{D \cdot \left(D \cdot h\right)}{\color{blue}{\sqrt{\frac{d \cdot d}{M \cdot M}} \cdot \sqrt{\frac{d \cdot d}{M \cdot M}}}}, \frac{c0 \cdot c0}{\frac{w}{0}}\right) \]
      times-frac [=>] 55.8	\[ \mathsf{fma}\left(0.25, \color{blue}{\frac{D}{\sqrt{\frac{d \cdot d}{M \cdot M}}} \cdot \frac{D \cdot h}{\sqrt{\frac{d \cdot d}{M \cdot M}}}}, \frac{c0 \cdot c0}{\frac{w}{0}}\right) \]
      times-frac [=>] 55.8	\[ \mathsf{fma}\left(0.25, \frac{D}{\sqrt{\color{blue}{\frac{d}{M} \cdot \frac{d}{M}}}} \cdot \frac{D \cdot h}{\sqrt{\frac{d \cdot d}{M \cdot M}}}, \frac{c0 \cdot c0}{\frac{w}{0}}\right) \]
      sqrt-prod [=>] 26.5	\[ \mathsf{fma}\left(0.25, \frac{D}{\color{blue}{\sqrt{\frac{d}{M}} \cdot \sqrt{\frac{d}{M}}}} \cdot \frac{D \cdot h}{\sqrt{\frac{d \cdot d}{M \cdot M}}}, \frac{c0 \cdot c0}{\frac{w}{0}}\right) \]
      add-sqr-sqrt [<=] 50.2	\[ \mathsf{fma}\left(0.25, \frac{D}{\color{blue}{\frac{d}{M}}} \cdot \frac{D \cdot h}{\sqrt{\frac{d \cdot d}{M \cdot M}}}, \frac{c0 \cdot c0}{\frac{w}{0}}\right) \]
      times-frac [=>] 53.2	\[ \mathsf{fma}\left(0.25, \frac{D}{\frac{d}{M}} \cdot \frac{D \cdot h}{\sqrt{\color{blue}{\frac{d}{M} \cdot \frac{d}{M}}}}, \frac{c0 \cdot c0}{\frac{w}{0}}\right) \]
      sqrt-prod [=>] 31.9	\[ \mathsf{fma}\left(0.25, \frac{D}{\frac{d}{M}} \cdot \frac{D \cdot h}{\color{blue}{\sqrt{\frac{d}{M}} \cdot \sqrt{\frac{d}{M}}}}, \frac{c0 \cdot c0}{\frac{w}{0}}\right) \]
      add-sqr-sqrt [<=] 65.1	\[ \mathsf{fma}\left(0.25, \frac{D}{\frac{d}{M}} \cdot \frac{D \cdot h}{\color{blue}{\frac{d}{M}}}, \frac{c0 \cdot c0}{\frac{w}{0}}\right) \]

   5. Applied egg-rr 82.4%

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

      [Start] 65.1	\[ \mathsf{fma}\left(0.25, \frac{D}{\frac{d}{M}} \cdot \frac{D \cdot h}{\frac{d}{M}}, \frac{c0 \cdot c0}{\frac{w}{0}}\right) \]
      associate-/r/ [=>] 63.6	\[ \mathsf{fma}\left(0.25, \frac{D}{\frac{d}{M}} \cdot \frac{D \cdot h}{\frac{d}{M}}, \color{blue}{\frac{c0 \cdot c0}{w} \cdot 0}\right) \]
      mul0-rgt [=>] 82.4	\[ \mathsf{fma}\left(0.25, \frac{D}{\frac{d}{M}} \cdot \frac{D \cdot h}{\frac{d}{M}}, \color{blue}{0}\right) \]

   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 24.2%

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

   2. Simplified 24.3%

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

      [Start] 24.2	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      times-frac [=>] 20.1	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      fma-def [=>] 20.2	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d \cdot d}{D \cdot D}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)} \]
      associate-/r* [=>] 20.1	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \color{blue}{\frac{\frac{d \cdot d}{D}}{D}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      difference-of-squares [=>] 20.1	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{\frac{d \cdot d}{D}}{D}, \sqrt{\color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + M\right) \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M\right)}}\right) \]

   3. Taylor expanded in c0 around inf 13.9%

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

   4. Simplified 13.7%

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

      [Start] 13.9	\[ \frac{{d}^{2} \cdot {c0}^{2}}{{D}^{2} \cdot \left({w}^{2} \cdot h\right)} \]
      times-frac [=>] 13.7	\[ \color{blue}{\frac{{d}^{2}}{{D}^{2}} \cdot \frac{{c0}^{2}}{{w}^{2} \cdot h}} \]
      unpow2 [=>] 13.7	\[ \frac{\color{blue}{d \cdot d}}{{D}^{2}} \cdot \frac{{c0}^{2}}{{w}^{2} \cdot h} \]
      unpow2 [=>] 13.7	\[ \frac{d \cdot d}{\color{blue}{D \cdot D}} \cdot \frac{{c0}^{2}}{{w}^{2} \cdot h} \]
      unpow2 [=>] 13.7	\[ \frac{d \cdot d}{D \cdot D} \cdot \frac{\color{blue}{c0 \cdot c0}}{{w}^{2} \cdot h} \]
      *-commutative [=>] 13.7	\[ \frac{d \cdot d}{D \cdot D} \cdot \frac{c0 \cdot c0}{\color{blue}{h \cdot {w}^{2}}} \]
      unpow2 [=>] 13.7	\[ \frac{d \cdot d}{D \cdot D} \cdot \frac{c0 \cdot c0}{h \cdot \color{blue}{\left(w \cdot w\right)}} \]

   5. Applied egg-rr 70.7%

      \[\leadsto \color{blue}{0 + {\left(\frac{c0}{w \cdot \sqrt{h}} \cdot \frac{d}{D}\right)}^{2}} \]
      Proof

      [Start] 13.7	\[ \frac{d \cdot d}{D \cdot D} \cdot \frac{c0 \cdot c0}{h \cdot \left(w \cdot w\right)} \]
      add-log-exp [=>] 2.5	\[ \color{blue}{\log \left(e^{\frac{d \cdot d}{D \cdot D} \cdot \frac{c0 \cdot c0}{h \cdot \left(w \cdot w\right)}}\right)} \]
      *-un-lft-identity [=>] 2.5	\[ \log \color{blue}{\left(1 \cdot e^{\frac{d \cdot d}{D \cdot D} \cdot \frac{c0 \cdot c0}{h \cdot \left(w \cdot w\right)}}\right)} \]
      log-prod [=>] 2.5	\[ \color{blue}{\log 1 + \log \left(e^{\frac{d \cdot d}{D \cdot D} \cdot \frac{c0 \cdot c0}{h \cdot \left(w \cdot w\right)}}\right)} \]
      metadata-eval [=>] 2.5	\[ \color{blue}{0} + \log \left(e^{\frac{d \cdot d}{D \cdot D} \cdot \frac{c0 \cdot c0}{h \cdot \left(w \cdot w\right)}}\right) \]
      add-log-exp [<=] 13.7	\[ 0 + \color{blue}{\frac{d \cdot d}{D \cdot D} \cdot \frac{c0 \cdot c0}{h \cdot \left(w \cdot w\right)}} \]
      add-sqr-sqrt [=>] 13.5	\[ 0 + \color{blue}{\sqrt{\frac{d \cdot d}{D \cdot D} \cdot \frac{c0 \cdot c0}{h \cdot \left(w \cdot w\right)}} \cdot \sqrt{\frac{d \cdot d}{D \cdot D} \cdot \frac{c0 \cdot c0}{h \cdot \left(w \cdot w\right)}}} \]
      pow2 [=>] 13.5	\[ 0 + \color{blue}{{\left(\sqrt{\frac{d \cdot d}{D \cdot D} \cdot \frac{c0 \cdot c0}{h \cdot \left(w \cdot w\right)}}\right)}^{2}} \]

   6. Simplified 71.2%

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

      [Start] 70.7	\[ 0 + {\left(\frac{c0}{w \cdot \sqrt{h}} \cdot \frac{d}{D}\right)}^{2} \]
      +-lft-identity [=>] 70.7	\[ \color{blue}{{\left(\frac{c0}{w \cdot \sqrt{h}} \cdot \frac{d}{D}\right)}^{2}} \]
      associate-/r* [=>] 71.2	\[ {\left(\color{blue}{\frac{\frac{c0}{w}}{\sqrt{h}}} \cdot \frac{d}{D}\right)}^{2} \]

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

   1. Initial program 0.0%

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

   2. Taylor expanded in c0 around -inf 0.9%

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

   3. Simplified 34.5%

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

      [Start] 0.9	\[ -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w} + 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \]
      +-commutative [=>] 0.9	\[ \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} + -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}} \]
      fma-def [=>] 0.9	\[ \color{blue}{\mathsf{fma}\left(0.25, \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}, -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}\right)} \]

   4. Applied egg-rr 54.0%

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

      [Start] 34.5	\[ \mathsf{fma}\left(0.25, \frac{\left(D \cdot D\right) \cdot h}{\frac{d \cdot d}{M \cdot M}}, \frac{c0 \cdot c0}{\frac{w}{0}}\right) \]
      associate-*l* [=>] 36.7	\[ \mathsf{fma}\left(0.25, \frac{\color{blue}{D \cdot \left(D \cdot h\right)}}{\frac{d \cdot d}{M \cdot M}}, \frac{c0 \cdot c0}{\frac{w}{0}}\right) \]
      add-sqr-sqrt [=>] 36.7	\[ \mathsf{fma}\left(0.25, \frac{D \cdot \left(D \cdot h\right)}{\color{blue}{\sqrt{\frac{d \cdot d}{M \cdot M}} \cdot \sqrt{\frac{d \cdot d}{M \cdot M}}}}, \frac{c0 \cdot c0}{\frac{w}{0}}\right) \]
      times-frac [=>] 38.5	\[ \mathsf{fma}\left(0.25, \color{blue}{\frac{D}{\sqrt{\frac{d \cdot d}{M \cdot M}}} \cdot \frac{D \cdot h}{\sqrt{\frac{d \cdot d}{M \cdot M}}}}, \frac{c0 \cdot c0}{\frac{w}{0}}\right) \]
      times-frac [=>] 38.6	\[ \mathsf{fma}\left(0.25, \frac{D}{\sqrt{\color{blue}{\frac{d}{M} \cdot \frac{d}{M}}}} \cdot \frac{D \cdot h}{\sqrt{\frac{d \cdot d}{M \cdot M}}}, \frac{c0 \cdot c0}{\frac{w}{0}}\right) \]
      sqrt-prod [=>] 19.2	\[ \mathsf{fma}\left(0.25, \frac{D}{\color{blue}{\sqrt{\frac{d}{M}} \cdot \sqrt{\frac{d}{M}}}} \cdot \frac{D \cdot h}{\sqrt{\frac{d \cdot d}{M \cdot M}}}, \frac{c0 \cdot c0}{\frac{w}{0}}\right) \]
      add-sqr-sqrt [<=] 36.7	\[ \mathsf{fma}\left(0.25, \frac{D}{\color{blue}{\frac{d}{M}}} \cdot \frac{D \cdot h}{\sqrt{\frac{d \cdot d}{M \cdot M}}}, \frac{c0 \cdot c0}{\frac{w}{0}}\right) \]
      times-frac [=>] 44.8	\[ \mathsf{fma}\left(0.25, \frac{D}{\frac{d}{M}} \cdot \frac{D \cdot h}{\sqrt{\color{blue}{\frac{d}{M} \cdot \frac{d}{M}}}}, \frac{c0 \cdot c0}{\frac{w}{0}}\right) \]
      sqrt-prod [=>] 26.8	\[ \mathsf{fma}\left(0.25, \frac{D}{\frac{d}{M}} \cdot \frac{D \cdot h}{\color{blue}{\sqrt{\frac{d}{M}} \cdot \sqrt{\frac{d}{M}}}}, \frac{c0 \cdot c0}{\frac{w}{0}}\right) \]
      add-sqr-sqrt [<=] 54.0	\[ \mathsf{fma}\left(0.25, \frac{D}{\frac{d}{M}} \cdot \frac{D \cdot h}{\color{blue}{\frac{d}{M}}}, \frac{c0 \cdot c0}{\frac{w}{0}}\right) \]

   5. Applied egg-rr 77.6%

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

      [Start] 54.0	\[ \mathsf{fma}\left(0.25, \frac{D}{\frac{d}{M}} \cdot \frac{D \cdot h}{\frac{d}{M}}, \frac{c0 \cdot c0}{\frac{w}{0}}\right) \]
      associate-/r/ [=>] 47.5	\[ \mathsf{fma}\left(0.25, \frac{D}{\frac{d}{M}} \cdot \frac{D \cdot h}{\frac{d}{M}}, \color{blue}{\frac{c0 \cdot c0}{w} \cdot 0}\right) \]
      mul0-rgt [=>] 77.6	\[ \mathsf{fma}\left(0.25, \frac{D}{\frac{d}{M}} \cdot \frac{D \cdot h}{\frac{d}{M}}, \color{blue}{0}\right) \]

   6. Applied egg-rr 82.0%

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

      [Start] 77.6	\[ \mathsf{fma}\left(0.25, \frac{D}{\frac{d}{M}} \cdot \frac{D \cdot h}{\frac{d}{M}}, 0\right) \]
      associate-/l* [=>] 78.5	\[ \mathsf{fma}\left(0.25, \frac{D}{\frac{d}{M}} \cdot \color{blue}{\frac{D}{\frac{\frac{d}{M}}{h}}}, 0\right) \]
      associate-/r/ [=>] 82.0	\[ \mathsf{fma}\left(0.25, \frac{D}{\frac{d}{M}} \cdot \color{blue}{\left(\frac{D}{\frac{d}{M}} \cdot h\right)}, 0\right) \]
      div-inv [=>] 81.9	\[ \mathsf{fma}\left(0.25, \frac{D}{\frac{d}{M}} \cdot \left(\color{blue}{\left(D \cdot \frac{1}{\frac{d}{M}}\right)} \cdot h\right), 0\right) \]
      clear-num [<=] 82.0	\[ \mathsf{fma}\left(0.25, \frac{D}{\frac{d}{M}} \cdot \left(\left(D \cdot \color{blue}{\frac{M}{d}}\right) \cdot h\right), 0\right) \]

3. Recombined 4 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq -5 \cdot 10^{-208}:\\ \;\;\;\;\frac{c0 \cdot d}{w \cdot \left(\left(w \cdot h\right) \cdot D\right)} \cdot \frac{c0 \cdot d}{D}\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 0:\\ \;\;\;\;\mathsf{fma}\left(0.25, \frac{D}{\frac{d}{M}} \cdot \frac{h \cdot D}{\frac{d}{M}}, 0\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;{\left(\frac{\frac{c0}{w}}{\sqrt{h}} \cdot \frac{d}{D}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.25, \frac{D}{\frac{d}{M}} \cdot \left(h \cdot \left(D \cdot \frac{M}{d}\right)\right), 0\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	76.5%
Cost	36556

\[\begin{array}{l} t_0 := c0 \cdot \left(d \cdot d\right)\\ t_1 := \frac{c0 \cdot d}{w \cdot \left(\left(w \cdot h\right) \cdot D\right)} \cdot \frac{c0 \cdot d}{D}\\ t_2 := \frac{t_0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_3 := \frac{c0}{2 \cdot w}\\ t_4 := t_3 \cdot \left(t_2 + \sqrt{t_2 \cdot t_2 - M \cdot M}\right)\\ t_5 := \frac{t_0}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)}\\ \mathbf{if}\;t_4 \leq -4 \cdot 10^{+248}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_4 \leq 2 \cdot 10^{-280}:\\ \;\;\;\;t_3 \cdot \left(t_5 + \sqrt{\left(M + t_5\right) \cdot \left(t_5 - M\right)}\right)\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.25, \frac{D}{\frac{d}{M}} \cdot \left(h \cdot \left(D \cdot \frac{M}{d}\right)\right), 0\right)\\ \end{array} \]

Alternative 2
Accuracy	73.6%
Cost	7625

\[\begin{array}{l} \mathbf{if}\;c0 \leq 3.6 \cdot 10^{-215} \lor \neg \left(c0 \leq 4.6 \cdot 10^{-167}\right):\\ \;\;\;\;\mathsf{fma}\left(0.25, \frac{D}{\frac{d}{M}} \cdot \left(h \cdot \left(D \cdot \frac{M}{d}\right)\right), 0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot d}{w \cdot \left(\left(w \cdot h\right) \cdot D\right)} \cdot \frac{c0 \cdot d}{D}\\ \end{array} \]

Alternative 3
Accuracy	64.2%
Cost	7564

\[\begin{array}{l} t_0 := M \cdot \left(h \cdot \left(0.25 \cdot \frac{M}{{\left(\frac{d}{D}\right)}^{2}}\right)\right)\\ \mathbf{if}\;D \leq -1.12 \cdot 10^{+91}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;D \leq -9 \cdot 10^{+37}:\\ \;\;\;\;\left(d \cdot \frac{\frac{d}{D}}{D}\right) \cdot \left(c0 \cdot \frac{c0}{w \cdot \left(w \cdot h\right)}\right)\\ \mathbf{elif}\;D \leq -9.8 \cdot 10^{-210}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;D \leq 6.7 \cdot 10^{-184}:\\ \;\;\;\;0.25 \cdot \left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d} \cdot \frac{h}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 \cdot \left(D \cdot \left(M \cdot \frac{D}{\frac{d}{h}}\right)\right)}{\frac{d}{M}}\\ \end{array} \]

Alternative 4
Accuracy	64.2%
Cost	7564

\[\begin{array}{l} t_0 := {\left(\frac{d}{D}\right)}^{2}\\ \mathbf{if}\;D \leq -1.12 \cdot 10^{+91}:\\ \;\;\;\;\frac{M \cdot 0.25}{\frac{t_0}{h \cdot M}}\\ \mathbf{elif}\;D \leq -9 \cdot 10^{+37}:\\ \;\;\;\;\left(d \cdot \frac{\frac{d}{D}}{D}\right) \cdot \left(c0 \cdot \frac{c0}{w \cdot \left(w \cdot h\right)}\right)\\ \mathbf{elif}\;D \leq -3.7 \cdot 10^{-209}:\\ \;\;\;\;M \cdot \left(h \cdot \left(0.25 \cdot \frac{M}{t_0}\right)\right)\\ \mathbf{elif}\;D \leq 2 \cdot 10^{-184}:\\ \;\;\;\;0.25 \cdot \left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d} \cdot \frac{h}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 \cdot \left(D \cdot \left(M \cdot \frac{D}{\frac{d}{h}}\right)\right)}{\frac{d}{M}}\\ \end{array} \]

Alternative 5
Accuracy	68.4%
Cost	1353

\[\begin{array}{l} \mathbf{if}\;c0 \leq 3.6 \cdot 10^{-215} \lor \neg \left(c0 \leq 4.6 \cdot 10^{-167}\right):\\ \;\;\;\;\left(M \cdot \frac{D}{\frac{d}{h}}\right) \cdot \left(0.25 \cdot \frac{D \cdot M}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot d}{w \cdot \left(\left(w \cdot h\right) \cdot D\right)} \cdot \frac{c0 \cdot d}{D}\\ \end{array} \]

Alternative 6
Accuracy	68.2%
Cost	1352

\[\begin{array}{l} \mathbf{if}\;h \leq -4 \cdot 10^{+161}:\\ \;\;\;\;0.25 \cdot \left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d} \cdot \frac{h}{d}\right)\\ \mathbf{elif}\;h \leq -1.12 \cdot 10^{+142}:\\ \;\;\;\;\left(\frac{d}{w} \cdot \left(\frac{d}{h} \cdot \frac{c0}{D}\right)\right) \cdot \frac{\frac{c0}{w}}{D}\\ \mathbf{else}:\\ \;\;\;\;\left(M \cdot \frac{D}{\frac{d}{h}}\right) \cdot \left(0.25 \cdot \frac{D \cdot M}{d}\right)\\ \end{array} \]

Alternative 7
Accuracy	68.1%
Cost	1225

\[\begin{array}{l} \mathbf{if}\;D \leq -2 \cdot 10^{-167} \lor \neg \left(D \leq 5 \cdot 10^{-98}\right):\\ \;\;\;\;\left(M \cdot \frac{D}{\frac{d}{h}}\right) \cdot \left(0.25 \cdot \frac{D \cdot M}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d} \cdot \frac{h}{d}\right)\\ \end{array} \]

Alternative 8
Accuracy	67.8%
Cost	1224

\[\begin{array}{l} t_0 := M \cdot \frac{D}{\frac{d}{h}}\\ \mathbf{if}\;D \leq -1.4 \cdot 10^{-167}:\\ \;\;\;\;t_0 \cdot \left(0.25 \cdot \frac{D \cdot M}{d}\right)\\ \mathbf{elif}\;D \leq 3 \cdot 10^{-183}:\\ \;\;\;\;0.25 \cdot \left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d} \cdot \frac{h}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 \cdot \left(D \cdot t_0\right)}{\frac{d}{M}}\\ \end{array} \]

Alternative 9
Accuracy	63.6%
Cost	960

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

Alternative 10
Accuracy	50.8%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023146 
(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))))))