Henrywood and Agarwal, Equation (13)

Average Accuracy: 4.4% → 35.7%

Time: 32.9s

Precision: binary64

Cost: 10820

Math

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

↓

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

FPCore

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

↓

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
        INFINITY)
     (/ (* (/ (* c0 d) (* w h)) (/ (* c0 d) w)) (* D 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 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= ((double) INFINITY)) {
		tmp = (((c0 * d) / (w * h)) * ((c0 * d) / w)) / (D * D);
	} else {
		tmp = 0.0;
	}
	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 t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = (((c0 * d) / (w * h)) * ((c0 * d) / w)) / (D * D);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	return (c0 / (2.0 * w)) * (((c0 * (d * d)) / ((w * h) * (D * D))) + math.sqrt(((((c0 * (d * d)) / ((w * h) * (D * D))) * ((c0 * (d * d)) / ((w * h) * (D * D)))) - (M * M))))

↓

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))) <= math.inf:
		tmp = (((c0 * d) / (w * h)) * ((c0 * d) / w)) / (D * D)
	else:
		tmp = 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(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M))))) <= Inf)
		tmp = Float64(Float64(Float64(Float64(c0 * d) / Float64(w * h)) * Float64(Float64(c0 * d) / w)) / Float64(D * D));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = (c0 / (2.0 * w)) * (((c0 * (d * d)) / ((w * h) * (D * D))) + sqrt(((((c0 * (d * d)) / ((w * h) * (D * D))) * ((c0 * (d * d)) / ((w * h) * (D * D)))) - (M * M))));
end

↓

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= Inf)
		tmp = (((c0 * d) / (w * h)) * ((c0 * d) / w)) / (D * D);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[(N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(N[(c0 * d), $MachinePrecision] / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[(N[(c0 * d), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 13.5%

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

   2. Simplified 12.7%

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

      [Start] 13.5	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      associate-*l* [=>] 11.1	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      difference-of-squares [=>] 11.1	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)} + \sqrt{\color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + M\right) \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M\right)}}\right) \]
      associate-*l* [=>] 11.3	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)} + \sqrt{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)}} + M\right) \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M\right)}\right) \]
      associate-*l* [=>] 12.7	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)} + \sqrt{\left(\frac{c0 \cdot \left(d \cdot d\right)}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)} + M\right) \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)}} - M\right)}\right) \]

   3. Taylor expanded in c0 around inf 12.7%

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

   4. Simplified 11.4%

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

      [Start] 12.7	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]
      unpow2 [=>] 12.7	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\color{blue}{\left(d \cdot d\right)} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]
      associate-*l/ [<=] 12.0	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{d \cdot d}{{D}^{2} \cdot \left(w \cdot h\right)} \cdot c0\right)}\right) \]
      *-commutative [=>] 12.0	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{d \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot {D}^{2}}} \cdot c0\right)\right) \]
      associate-*r* [<=] 11.0	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{d \cdot d}{\color{blue}{w \cdot \left(h \cdot {D}^{2}\right)}} \cdot c0\right)\right) \]
      unpow2 [=>] 11.0	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{d \cdot d}{w \cdot \left(h \cdot \color{blue}{\left(D \cdot D\right)}\right)} \cdot c0\right)\right) \]
      times-frac [=>] 11.9	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\color{blue}{\left(\frac{d}{w} \cdot \frac{d}{h \cdot \left(D \cdot D\right)}\right)} \cdot c0\right)\right) \]
      unpow2 [<=] 11.9	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\left(\frac{d}{w} \cdot \frac{d}{h \cdot \color{blue}{{D}^{2}}}\right) \cdot c0\right)\right) \]
      associate-/r* [=>] 11.4	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\left(\frac{d}{w} \cdot \color{blue}{\frac{\frac{d}{h}}{{D}^{2}}}\right) \cdot c0\right)\right) \]
      unpow2 [=>] 11.4	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\left(\frac{d}{w} \cdot \frac{\frac{d}{h}}{\color{blue}{D \cdot D}}\right) \cdot c0\right)\right) \]

   5. Taylor expanded in c0 around 0 8.1%

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

   6. Simplified 7.7%

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

      [Start] 8.1	\[ \frac{{d}^{2} \cdot {c0}^{2}}{{D}^{2} \cdot \left({w}^{2} \cdot h\right)} \]
      unpow2 [=>] 8.1	\[ \frac{{d}^{2} \cdot {c0}^{2}}{\color{blue}{\left(D \cdot D\right)} \cdot \left({w}^{2} \cdot h\right)} \]
      times-frac [=>] 7.7	\[ \color{blue}{\frac{{d}^{2}}{D \cdot D} \cdot \frac{{c0}^{2}}{{w}^{2} \cdot h}} \]
      unpow2 [=>] 7.7	\[ \frac{\color{blue}{d \cdot d}}{D \cdot D} \cdot \frac{{c0}^{2}}{{w}^{2} \cdot h} \]
      unpow2 [=>] 7.7	\[ \frac{d \cdot d}{D \cdot D} \cdot \frac{\color{blue}{c0 \cdot c0}}{{w}^{2} \cdot h} \]
      *-commutative [=>] 7.7	\[ \frac{d \cdot d}{D \cdot D} \cdot \frac{c0 \cdot c0}{\color{blue}{h \cdot {w}^{2}}} \]
      unpow2 [=>] 7.7	\[ \frac{d \cdot d}{D \cdot D} \cdot \frac{c0 \cdot c0}{h \cdot \color{blue}{\left(w \cdot w\right)}} \]

   7. Applied egg-rr 11.7%

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

      [Start] 7.7	\[ \frac{d \cdot d}{D \cdot D} \cdot \frac{c0 \cdot c0}{h \cdot \left(w \cdot w\right)} \]
      associate-*l/ [=>] 8.7	\[ \color{blue}{\frac{\left(d \cdot d\right) \cdot \frac{c0 \cdot c0}{h \cdot \left(w \cdot w\right)}}{D \cdot D}} \]
      associate-*r/ [=>] 8.3	\[ \frac{\color{blue}{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{h \cdot \left(w \cdot w\right)}}}{D \cdot D} \]
      pow2 [=>] 8.3	\[ \frac{\frac{\color{blue}{{d}^{2}} \cdot \left(c0 \cdot c0\right)}{h \cdot \left(w \cdot w\right)}}{D \cdot D} \]
      pow2 [=>] 8.3	\[ \frac{\frac{{d}^{2} \cdot \color{blue}{{c0}^{2}}}{h \cdot \left(w \cdot w\right)}}{D \cdot D} \]
      pow-prod-down [=>] 11.7	\[ \frac{\frac{\color{blue}{{\left(d \cdot c0\right)}^{2}}}{h \cdot \left(w \cdot w\right)}}{D \cdot D} \]

   8. Applied egg-rr 17.2%

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

      [Start] 11.7	\[ \frac{\frac{{\left(d \cdot c0\right)}^{2}}{h \cdot \left(w \cdot w\right)}}{D \cdot D} \]
      unpow2 [=>] 11.7	\[ \frac{\frac{\color{blue}{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}}{h \cdot \left(w \cdot w\right)}}{D \cdot D} \]
      associate-*r* [=>] 12.8	\[ \frac{\frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{\color{blue}{\left(h \cdot w\right) \cdot w}}}{D \cdot D} \]
      times-frac [=>] 17.2	\[ \frac{\color{blue}{\frac{d \cdot c0}{h \cdot w} \cdot \frac{d \cdot c0}{w}}}{D \cdot D} \]

   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. Simplified 0.3%

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

      [Start] 0.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      associate-*l* [=>] 0.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      difference-of-squares [=>] 0.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)} + \sqrt{\color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + M\right) \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M\right)}}\right) \]
      associate-*l* [=>] 0.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)} + \sqrt{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)}} + M\right) \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M\right)}\right) \]
      associate-*l* [=>] 0.3	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)} + \sqrt{\left(\frac{c0 \cdot \left(d \cdot d\right)}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)} + M\right) \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)}} - M\right)}\right) \]

   3. Taylor expanded in c0 around -inf 1.4%

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

   4. Simplified 38.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{0} \]
      Proof

      [Start] 1.4	\[ \frac{c0}{2 \cdot w} \cdot \left(-1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right) \]
      *-commutative [=>] 1.4	\[ \frac{c0}{2 \cdot w} \cdot \left(-1 \cdot \color{blue}{\left(c0 \cdot \left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)\right)}\right) \]
      distribute-rgt-in [=>] 0.9	\[ \frac{c0}{2 \cdot w} \cdot \left(-1 \cdot \color{blue}{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} \cdot c0 + \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)}\right) \]
      mul-1-neg [=>] 0.9	\[ \frac{c0}{2 \cdot w} \cdot \left(-1 \cdot \left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} \cdot c0 + \color{blue}{\left(-\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)} \cdot c0\right)\right) \]
      distribute-lft-neg-in [<=] 0.9	\[ \frac{c0}{2 \cdot w} \cdot \left(-1 \cdot \left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} \cdot c0 + \color{blue}{\left(-\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} \cdot c0\right)}\right)\right) \]
      unpow2 [=>] 0.9	\[ \frac{c0}{2 \cdot w} \cdot \left(-1 \cdot \left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} \cdot c0 + \left(-\frac{\color{blue}{d \cdot d}}{{D}^{2} \cdot \left(w \cdot h\right)} \cdot c0\right)\right)\right) \]
      associate-*l/ [=>] 0.1	\[ \frac{c0}{2 \cdot w} \cdot \left(-1 \cdot \left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} \cdot c0 + \left(-\color{blue}{\frac{\left(d \cdot d\right) \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}}\right)\right)\right) \]
      unpow2 [<=] 0.1	\[ \frac{c0}{2 \cdot w} \cdot \left(-1 \cdot \left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} \cdot c0 + \left(-\frac{\color{blue}{{d}^{2}} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right)\right)\right) \]
      mul-1-neg [<=] 0.1	\[ \frac{c0}{2 \cdot w} \cdot \left(-1 \cdot \left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} \cdot c0 + \color{blue}{-1 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}}\right)\right) \]

   5. Taylor expanded in c0 around 0 44.8%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.7%

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

Alternatives

Alternative 1
Accuracy	31.8%
Cost	2129

\[\begin{array}{l} \mathbf{if}\;d \cdot d \leq 10^{-302}:\\ \;\;\;\;0\\ \mathbf{elif}\;d \cdot d \leq 5 \cdot 10^{-116} \lor \neg \left(d \cdot d \leq 10^{-34}\right) \land d \cdot d \leq 4 \cdot 10^{+29}:\\ \;\;\;\;\frac{c0}{\frac{h}{\frac{\left(d \cdot \frac{c0}{D}\right) \cdot \frac{d}{w \cdot D}}{w}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 2
Accuracy	32.6%
Cost	1352

\[\begin{array}{l} \mathbf{if}\;d \leq -22000000:\\ \;\;\;\;0\\ \mathbf{elif}\;d \leq -2.3 \cdot 10^{-172}:\\ \;\;\;\;\frac{c0}{w} \cdot \frac{\frac{c0}{\frac{\frac{D}{d}}{d}}}{\left(w \cdot h\right) \cdot D}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 3
Accuracy	33.6%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023153 
(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))))))