Henrywood and Agarwal, Equation (13)

Average Error: 59.8 → 30.1

Time: 45.4s

Precision: binary64

Cost: 23492

Math

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

↓

\[\begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t_0 + \sqrt{t_0 \cdot t_0 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{\frac{\frac{1}{h}}{{\left(D \cdot w\right)}^{2}}}{\frac{1}{{\left(d \cdot c0\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

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)
     (/ (/ (/ 1.0 h) (pow (* D w) 2.0)) (/ 1.0 (pow (* d c0) 2.0)))
     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 = ((1.0 / h) / pow((D * w), 2.0)) / (1.0 / pow((d * c0), 2.0));
	} 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 = ((1.0 / h) / Math.pow((D * w), 2.0)) / (1.0 / Math.pow((d * c0), 2.0));
	} 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 = ((1.0 / h) / math.pow((D * w), 2.0)) / (1.0 / math.pow((d * c0), 2.0))
	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(1.0 / h) / (Float64(D * w) ^ 2.0)) / Float64(1.0 / (Float64(d * c0) ^ 2.0)));
	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 = ((1.0 / h) / ((D * w) ^ 2.0)) / (1.0 / ((d * c0) ^ 2.0));
	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[(1.0 / h), $MachinePrecision] / N[Power[N[(D * w), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[Power[N[(d * c0), $MachinePrecision], 2.0], $MachinePrecision]), $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 41.6

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

   2. Simplified 43.5

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

      [Start] 41.6	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      rational.json-simplify-49 [=>] 44.6	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(d \cdot d\right) \cdot \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      rational.json-simplify-2 [=>] 44.6	\[ \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \frac{c0}{\color{blue}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      rational.json-simplify-2 [=>] 44.6	\[ \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \frac{c0}{\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot w\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      rational.json-simplify-43 [=>] 47.4	\[ \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \frac{c0}{\color{blue}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Taylor expanded in c0 around inf 50.6

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

   4. Simplified 39.5

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

      [Start] 50.6	\[ \frac{{d}^{2} \cdot {c0}^{2}}{{D}^{2} \cdot \left({w}^{2} \cdot h\right)} \]
      exponential.json-simplify-27 [=>] 45.0	\[ \frac{\color{blue}{{\left(d \cdot c0\right)}^{2}}}{{D}^{2} \cdot \left({w}^{2} \cdot h\right)} \]
      rational.json-simplify-43 [=>] 46.1	\[ \frac{{\left(d \cdot c0\right)}^{2}}{\color{blue}{{w}^{2} \cdot \left(h \cdot {D}^{2}\right)}} \]
      rational.json-simplify-43 [=>] 44.3	\[ \frac{{\left(d \cdot c0\right)}^{2}}{\color{blue}{h \cdot \left({D}^{2} \cdot {w}^{2}\right)}} \]
      exponential.json-simplify-27 [=>] 39.5	\[ \frac{{\left(d \cdot c0\right)}^{2}}{h \cdot \color{blue}{{\left(D \cdot w\right)}^{2}}} \]

   5. Applied egg-rr 38.5

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

   6. Applied egg-rr 39.6

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

   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 64.0

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

   2. Simplified 63.0

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

      [Start] 64.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      rational.json-simplify-46 [=>] 64.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{c0 \cdot \left(d \cdot d\right)}{w \cdot h}}{D \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      rational.json-simplify-2 [=>] 64.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{\color{blue}{\left(d \cdot d\right) \cdot c0}}{w \cdot h}}{D \cdot D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      rational.json-simplify-49 [=>] 64.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{c0 \cdot \frac{d \cdot d}{w \cdot h}}}{D \cdot D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      rational.json-simplify-49 [=>] 64.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{d \cdot d}{w \cdot h} \cdot \frac{c0}{D \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      rational.json-simplify-46 [=>] 64.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{d \cdot d}{w}}{h}} \cdot \frac{c0}{D \cdot D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Taylor expanded in c0 around -inf 63.9

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

   4. Simplified 63.4

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

      [Start] 63.9	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot d}{w}}{h} \cdot \frac{c0}{D \cdot D} + -1 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]
      rational.json-simplify-2 [=>] 63.9	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot d}{w}}{h} \cdot \frac{c0}{D \cdot D} + \color{blue}{\frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)} \cdot -1}\right) \]
      rational.json-simplify-9 [=>] 63.9	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot d}{w}}{h} \cdot \frac{c0}{D \cdot D} + \color{blue}{\left(-\frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}\right) \]
      rational.json-simplify-46 [=>] 63.7	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot d}{w}}{h} \cdot \frac{c0}{D \cdot D} + \left(-\color{blue}{\frac{\frac{{d}^{2} \cdot c0}{{D}^{2}}}{w \cdot h}}\right)\right) \]
      rational.json-simplify-44 [=>] 63.7	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot d}{w}}{h} \cdot \frac{c0}{D \cdot D} + \left(-\color{blue}{\frac{\frac{{d}^{2} \cdot c0}{w \cdot h}}{{D}^{2}}}\right)\right) \]
      rational.json-simplify-2 [=>] 63.7	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot d}{w}}{h} \cdot \frac{c0}{D \cdot D} + \left(-\frac{\frac{\color{blue}{c0 \cdot {d}^{2}}}{w \cdot h}}{{D}^{2}}\right)\right) \]
      rational.json-simplify-49 [=>] 63.6	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot d}{w}}{h} \cdot \frac{c0}{D \cdot D} + \left(-\frac{\color{blue}{{d}^{2} \cdot \frac{c0}{w \cdot h}}}{{D}^{2}}\right)\right) \]
      rational.json-simplify-46 [=>] 63.4	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d \cdot d}{w}}{h} \cdot \frac{c0}{D \cdot D} + \left(-\frac{{d}^{2} \cdot \color{blue}{\frac{\frac{c0}{w}}{h}}}{{D}^{2}}\right)\right) \]

   5. Taylor expanded in d around 0 33.0

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

   6. Taylor expanded in c0 around 0 27.8

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

4. Final simplification 30.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{\frac{\frac{1}{h}}{{\left(D \cdot w\right)}^{2}}}{\frac{1}{{\left(d \cdot c0\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternatives

Alternative 1
Error	29.8
Cost	23236

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

Alternative 2
Error	30.0
Cost	23236

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

Alternative 3
Error	30.5
Cost	19396

\[\begin{array}{l} t_0 := d \cdot \frac{c0}{\left(D \cdot D\right) \cdot \left(w \cdot \frac{h}{d}\right)}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t_1 + \sqrt{t_1 \cdot t_1 - M \cdot M}\right) \leq 2 \cdot 10^{+75}:\\ \;\;\;\;c0 \cdot \frac{t_0 + \sqrt{\left(M + t_0\right) \cdot \left(t_0 - M\right)}}{w \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 4
Error	30.5
Cost	19396

\[\begin{array}{l} t_0 := d \cdot \left(\frac{\frac{d}{w}}{h} \cdot \frac{\frac{c0}{D}}{D}\right)\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t_1 + \sqrt{t_1 \cdot t_1 - M \cdot M}\right) \leq 10^{+148}:\\ \;\;\;\;\frac{t_0 + \sqrt{t_0 \cdot t_0 - M \cdot M}}{w \cdot \frac{2}{c0}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5
Error	30.2
Cost	19396

\[\begin{array}{l} t_0 := d \cdot \frac{\frac{\frac{c0}{D}}{h \cdot \frac{w}{d}}}{D}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t_1 + \sqrt{t_1 \cdot t_1 - M \cdot M}\right) \leq 10^{+148}:\\ \;\;\;\;\frac{t_0 + \sqrt{t_0 \cdot t_0 - M \cdot M}}{w \cdot \frac{2}{c0}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6
Error	31.5
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023068 
(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))))))