Henrywood and Agarwal, Equation (13)

Average Error: 59.5 → 18.3

Time: 29.1s

Precision: binary64

Cost: 30540

Math

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

↓

\[\begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_2 := t_0 \cdot \left(t_1 + \sqrt{t_1 \cdot t_1 - M \cdot M}\right)\\ t_3 := 0.25 \cdot \left(D \cdot \frac{M \cdot \frac{D}{d}}{\frac{\frac{d}{M}}{h}}\right)\\ \mathbf{if}\;t_2 \leq -2 \cdot 10^{-22}:\\ \;\;\;\;\frac{c0 \cdot d}{w \cdot D} \cdot \frac{d \cdot \frac{c0}{h}}{w \cdot D}\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \frac{d \cdot \frac{c0 \cdot \frac{d}{D}}{w}}{h \cdot D}\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

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 (* 2.0 w)))
        (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_2 (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M))))))
        (t_3 (* 0.25 (* D (/ (* M (/ D d)) (/ (/ d M) h))))))
   (if (<= t_2 -2e-22)
     (* (/ (* c0 d) (* w D)) (/ (* d (/ c0 h)) (* w D)))
     (if (<= t_2 0.0)
       t_3
       (if (<= t_2 INFINITY)
         (* t_0 (* 2.0 (/ (* d (/ (* c0 (/ d D)) w)) (* h D))))
         t_3)))))

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 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_2 = t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	double t_3 = 0.25 * (D * ((M * (D / d)) / ((d / M) / h)));
	double tmp;
	if (t_2 <= -2e-22) {
		tmp = ((c0 * d) / (w * D)) * ((d * (c0 / h)) / (w * D));
	} else if (t_2 <= 0.0) {
		tmp = t_3;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_0 * (2.0 * ((d * ((c0 * (d / D)) / w)) / (h * D)));
	} else {
		tmp = t_3;
	}
	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 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_2 = t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))));
	double t_3 = 0.25 * (D * ((M * (D / d)) / ((d / M) / h)));
	double tmp;
	if (t_2 <= -2e-22) {
		tmp = ((c0 * d) / (w * D)) * ((d * (c0 / h)) / (w * D));
	} else if (t_2 <= 0.0) {
		tmp = t_3;
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * (2.0 * ((d * ((c0 * (d / D)) / w)) / (h * D)));
	} else {
		tmp = t_3;
	}
	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 / (2.0 * w)
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	t_2 = t_0 * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))
	t_3 = 0.25 * (D * ((M * (D / d)) / ((d / M) / h)))
	tmp = 0
	if t_2 <= -2e-22:
		tmp = ((c0 * d) / (w * D)) * ((d * (c0 / h)) / (w * D))
	elif t_2 <= 0.0:
		tmp = t_3
	elif t_2 <= math.inf:
		tmp = t_0 * (2.0 * ((d * ((c0 * (d / D)) / w)) / (h * D)))
	else:
		tmp = t_3
	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(c0 / Float64(2.0 * w))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	t_2 = Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M)))))
	t_3 = Float64(0.25 * Float64(D * Float64(Float64(M * Float64(D / d)) / Float64(Float64(d / M) / h))))
	tmp = 0.0
	if (t_2 <= -2e-22)
		tmp = Float64(Float64(Float64(c0 * d) / Float64(w * D)) * Float64(Float64(d * Float64(c0 / h)) / Float64(w * D)));
	elseif (t_2 <= 0.0)
		tmp = t_3;
	elseif (t_2 <= Inf)
		tmp = Float64(t_0 * Float64(2.0 * Float64(Float64(d * Float64(Float64(c0 * Float64(d / D)) / w)) / Float64(h * D))));
	else
		tmp = t_3;
	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 / (2.0 * w);
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	t_2 = t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	t_3 = 0.25 * (D * ((M * (D / d)) / ((d / M) / h)));
	tmp = 0.0;
	if (t_2 <= -2e-22)
		tmp = ((c0 * d) / (w * D)) * ((d * (c0 / h)) / (w * D));
	elseif (t_2 <= 0.0)
		tmp = t_3;
	elseif (t_2 <= Inf)
		tmp = t_0 * (2.0 * ((d * ((c0 * (d / D)) / w)) / (h * D)));
	else
		tmp = t_3;
	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[(c0 / N[(2.0 * w), $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[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(0.25 * N[(D * N[(N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision] / N[(N[(d / M), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-22], N[(N[(N[(c0 * d), $MachinePrecision] / N[(w * D), $MachinePrecision]), $MachinePrecision] * N[(N[(d * N[(c0 / h), $MachinePrecision]), $MachinePrecision] / N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], t$95$3, If[LessEqual[t$95$2, Infinity], N[(t$95$0 * N[(2.0 * N[(N[(d * N[(N[(c0 * N[(d / D), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision] / N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]

Derivation

1. Split input into 3 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))))) < -2.0000000000000001e-22

   1. Initial program 55.2

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

   2. Taylor expanded in c0 around inf 47.0

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

   3. Simplified 44.6

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \left(c0 \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w \cdot h}\right)\right)} \]
      Proof
      (*.f64 2 (*.f64 c0 (/.f64 (*.f64 (/.f64 d D) (/.f64 d D)) (*.f64 w h)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 c0 (/.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 d d) (*.f64 D D))) (*.f64 w h)))): 42 points increase in error, 15 points decrease in error
      (*.f64 2 (*.f64 c0 (/.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 d 2)) (*.f64 D D)) (*.f64 w h)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 c0 (/.f64 (/.f64 (pow.f64 d 2) (Rewrite<= unpow2_binary64 (pow.f64 D 2))) (*.f64 w h)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 c0 (Rewrite<= associate-/r*_binary64 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h)))))): 6 points increase in error, 8 points decrease in error
      (*.f64 2 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 c0 (pow.f64 d 2)) (*.f64 (pow.f64 D 2) (*.f64 w h))))): 14 points increase in error, 7 points decrease in error
      (*.f64 2 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 d 2) c0)) (*.f64 (pow.f64 D 2) (*.f64 w h)))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in c0 around 0 56.6

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

   5. Simplified 52.9

      \[\leadsto \color{blue}{d \cdot \left(\frac{d}{w \cdot w} \cdot \frac{c0 \cdot \frac{c0}{h}}{D \cdot D}\right)} \]
      Proof
      (*.f64 d (*.f64 (/.f64 d (*.f64 w w)) (/.f64 (*.f64 c0 (/.f64 c0 h)) (*.f64 D D)))): 0 points increase in error, 0 points decrease in error
      (*.f64 d (*.f64 (/.f64 d (Rewrite<= unpow2_binary64 (pow.f64 w 2))) (/.f64 (*.f64 c0 (/.f64 c0 h)) (*.f64 D D)))): 0 points increase in error, 0 points decrease in error
      (*.f64 d (*.f64 (/.f64 d (pow.f64 w 2)) (/.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 c0 c0) h)) (*.f64 D D)))): 1 points increase in error, 4 points decrease in error
      (*.f64 d (*.f64 (/.f64 d (pow.f64 w 2)) (/.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 c0 2)) h) (*.f64 D D)))): 0 points increase in error, 0 points decrease in error
      (*.f64 d (*.f64 (/.f64 d (pow.f64 w 2)) (/.f64 (/.f64 (pow.f64 c0 2) h) (Rewrite<= unpow2_binary64 (pow.f64 D 2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 d (*.f64 (/.f64 d (pow.f64 w 2)) (Rewrite=> associate-/l/_binary64 (/.f64 (pow.f64 c0 2) (*.f64 (pow.f64 D 2) h))))): 10 points increase in error, 5 points decrease in error
      (*.f64 d (*.f64 (/.f64 d (pow.f64 w 2)) (/.f64 (pow.f64 c0 2) (Rewrite=> *-commutative_binary64 (*.f64 h (pow.f64 D 2)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 d (Rewrite<= associate-/r/_binary64 (/.f64 d (/.f64 (pow.f64 w 2) (/.f64 (pow.f64 c0 2) (*.f64 h (pow.f64 D 2))))))): 4 points increase in error, 8 points decrease in error
      (*.f64 d (/.f64 d (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 w 2) (*.f64 h (pow.f64 D 2))) (pow.f64 c0 2))))): 4 points increase in error, 7 points decrease in error
      (*.f64 d (/.f64 d (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (pow.f64 w 2) h) (pow.f64 D 2))) (pow.f64 c0 2)))): 4 points increase in error, 10 points decrease in error
      (*.f64 d (/.f64 d (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 w 2) h))) (pow.f64 c0 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 d (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 w 2) h)) (pow.f64 c0 2))) d)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 d (/.f64 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 w 2) h)) (pow.f64 c0 2)) d))): 1 points increase in error, 1 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 d d) (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 w 2) h)) (pow.f64 c0 2)))): 41 points increase in error, 1 points decrease in error
      (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 d 2)) (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 w 2) h)) (pow.f64 c0 2))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 d 2) (pow.f64 c0 2)) (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 w 2) h)))): 6 points increase in error, 2 points decrease in error

   6. Applied egg-rr 42.0

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

   7. Applied egg-rr 27.3

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

   if -2.0000000000000001e-22 < (*.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 or +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 60.7

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

   2. Simplified 62.9

      \[\leadsto \color{blue}{\frac{\frac{c0}{2}}{w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{4}, M \cdot \left(-M\right)\right)}\right)} \]
      Proof
      (*.f64 (/.f64 (/.f64 c0 2) w) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 w (*.f64 h (*.f64 D D)))) (sqrt.f64 (fma.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 c0 (*.f64 w h)) (pow.f64 (/.f64 d D) 4)) (*.f64 M (neg.f64 M)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 c0 (*.f64 2 w))) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 w (*.f64 h (*.f64 D D)))) (sqrt.f64 (fma.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 c0 (*.f64 w h)) (pow.f64 (/.f64 d D) 4)) (*.f64 M (neg.f64 M)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 w h) (*.f64 D D)))) (sqrt.f64 (fma.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 c0 (*.f64 w h)) (pow.f64 (/.f64 d D) 4)) (*.f64 M (neg.f64 M)))))): 1 points increase in error, 3 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (fma.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 c0 (*.f64 w h)) (pow.f64 (/.f64 d D) (Rewrite<= metadata-eval (+.f64 3 1)))) (*.f64 M (neg.f64 M)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (fma.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 c0 (*.f64 w h)) (Rewrite<= pow-plus_binary64 (*.f64 (pow.f64 (/.f64 d D) 3) (/.f64 d D)))) (*.f64 M (neg.f64 M)))))): 0 points increase in error, 1 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (fma.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (Rewrite=> unpow3_binary64 (*.f64 (*.f64 (/.f64 d D) (/.f64 d D)) (/.f64 d D))) (/.f64 d D))) (*.f64 M (neg.f64 M)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (fma.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 d d) (*.f64 D D))) (/.f64 d D)) (/.f64 d D))) (*.f64 M (neg.f64 M)))))): 1 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (fma.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 c0 (*.f64 w h)) (Rewrite<= associate-*r*_binary64 (*.f64 (/.f64 (*.f64 d d) (*.f64 D D)) (*.f64 (/.f64 d D) (/.f64 d D))))) (*.f64 M (neg.f64 M)))))): 0 points increase in error, 1 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (fma.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 (*.f64 d d) (*.f64 D D)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 d d) (*.f64 D D))))) (*.f64 M (neg.f64 M)))))): 1 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (fma.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 (*.f64 d d) (*.f64 D D)) (/.f64 (*.f64 d d) (*.f64 D D)))) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 M M))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 (*.f64 d d) (*.f64 D D)) (/.f64 (*.f64 d d) (*.f64 D D))))) (*.f64 M M)))))): 1 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 c0 (*.f64 w h)) (/.f64 c0 (*.f64 w h))) (*.f64 (/.f64 (*.f64 d d) (*.f64 D D)) (/.f64 (*.f64 d d) (*.f64 D D))))) (*.f64 M M))))): 2 points increase in error, 1 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (Rewrite<= swap-sqr_binary64 (*.f64 (*.f64 (/.f64 c0 (*.f64 w h)) (/.f64 (*.f64 d d) (*.f64 D D))) (*.f64 (/.f64 c0 (*.f64 w h)) (/.f64 (*.f64 d d) (*.f64 D D))))) (*.f64 M M))))): 0 points increase in error, 10 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 (/.f64 c0 (*.f64 w h)) (/.f64 (*.f64 d d) (*.f64 D D)))) (*.f64 M M))))): 5 points increase in error, 1 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))))) (*.f64 M M))))): 2 points increase in error, 6 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (*.f64 d d) (/.f64 c0 (*.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)))))): 3 points increase in error, 4 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (*.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 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (Rewrite=> associate-*l/_binary64 (/.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))))): 2 points increase in error, 2 points decrease in error

   3. Taylor expanded in c0 around -inf 60.5

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

   4. Simplified 41.4

      \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \color{blue}{\mathsf{fma}\left(0.5, \left(\frac{D}{d \cdot d} \cdot D\right) \cdot \left(\frac{w \cdot h}{c0} \cdot \left(M \cdot M\right)\right), 0\right)} \]
      Proof
      (fma.f64 1/2 (*.f64 (*.f64 (/.f64 D (*.f64 d d)) D) (*.f64 (/.f64 (*.f64 w h) c0) (*.f64 M M))) 0): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (*.f64 (*.f64 (/.f64 D (Rewrite<= unpow2_binary64 (pow.f64 d 2))) D) (*.f64 (/.f64 (*.f64 w h) c0) (*.f64 M M))) 0): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (*.f64 (Rewrite<= associate-/r/_binary64 (/.f64 D (/.f64 (pow.f64 d 2) D))) (*.f64 (/.f64 (*.f64 w h) c0) (*.f64 M M))) 0): 1 points increase in error, 5 points decrease in error
      (fma.f64 1/2 (*.f64 (/.f64 D (/.f64 (pow.f64 d 2) D)) (*.f64 (/.f64 (*.f64 w h) c0) (Rewrite<= unpow2_binary64 (pow.f64 M 2)))) 0): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (*.f64 (/.f64 D (/.f64 (pow.f64 d 2) D)) (Rewrite<= associate-/r/_binary64 (/.f64 (*.f64 w h) (/.f64 c0 (pow.f64 M 2))))) 0): 8 points increase in error, 17 points decrease in error
      (fma.f64 1/2 (*.f64 (/.f64 D (/.f64 (pow.f64 d 2) D)) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 w h) (pow.f64 M 2)) c0))) 0): 9 points increase in error, 13 points decrease in error
      (fma.f64 1/2 (*.f64 (/.f64 D (/.f64 (pow.f64 d 2) D)) (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 w (*.f64 h (pow.f64 M 2)))) c0)) 0): 5 points increase in error, 18 points decrease in error
      (fma.f64 1/2 (*.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 D D) (pow.f64 d 2))) (/.f64 (*.f64 w (*.f64 h (pow.f64 M 2))) c0)) 0): 31 points increase in error, 6 points decrease in error
      (fma.f64 1/2 (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 D 2)) (pow.f64 d 2)) (/.f64 (*.f64 w (*.f64 h (pow.f64 M 2))) c0)) 0): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (*.f64 (/.f64 (pow.f64 D 2) (pow.f64 d 2)) (/.f64 (*.f64 w (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 M 2) h))) c0)) 0): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 w (*.f64 (pow.f64 M 2) h))) (*.f64 (pow.f64 d 2) c0))) 0): 16 points increase in error, 11 points decrease in error
      (fma.f64 1/2 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 w (*.f64 (pow.f64 M 2) h))) (*.f64 (pow.f64 d 2) c0)) (Rewrite<= metadata-eval (*.f64 -1 0))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 w (*.f64 (pow.f64 M 2) h))) (*.f64 (pow.f64 d 2) c0)) (*.f64 -1 (Rewrite<= mul0-lft_binary64 (*.f64 0 c0)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 w (*.f64 (pow.f64 M 2) h))) (*.f64 (pow.f64 d 2) c0)) (*.f64 -1 (*.f64 (Rewrite<= mul0-lft_binary64 (*.f64 0 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h))))) c0))): 99 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 w (*.f64 (pow.f64 M 2) h))) (*.f64 (pow.f64 d 2) c0)) (*.f64 -1 (*.f64 (*.f64 (Rewrite<= metadata-eval (+.f64 -1 1)) (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h)))) c0))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 w (*.f64 (pow.f64 M 2) h))) (*.f64 (pow.f64 d 2) c0)) (*.f64 -1 (*.f64 (Rewrite<= distribute-rgt1-in_binary64 (+.f64 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h))) (*.f64 -1 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h)))))) c0))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 1/2 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 w (*.f64 (pow.f64 M 2) h))) (*.f64 (pow.f64 d 2) c0))) (*.f64 -1 (*.f64 (+.f64 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h))) (*.f64 -1 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h))))) c0)))): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in c0 around 0 34.3

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

   6. Simplified 23.1

      \[\leadsto \color{blue}{h \cdot \left(\left(0.25 \cdot \left(D \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{M}{d} \cdot M\right)\right)} \]
      Proof
      (*.f64 h (*.f64 (*.f64 1/4 (*.f64 D (/.f64 D d))) (*.f64 (/.f64 M d) M))): 0 points increase in error, 0 points decrease in error
      (*.f64 h (*.f64 (*.f64 1/4 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 D D) d))) (*.f64 (/.f64 M d) M))): 32 points increase in error, 4 points decrease in error
      (*.f64 h (*.f64 (*.f64 1/4 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 D 2)) d)) (*.f64 (/.f64 M d) M))): 0 points increase in error, 0 points decrease in error
      (*.f64 h (*.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 1/4 (pow.f64 D 2)) d)) (*.f64 (/.f64 M d) M))): 1 points increase in error, 1 points decrease in error
      (*.f64 h (*.f64 (/.f64 (*.f64 1/4 (pow.f64 D 2)) d) (Rewrite<= associate-/r/_binary64 (/.f64 M (/.f64 d M))))): 9 points increase in error, 5 points decrease in error
      (*.f64 h (*.f64 (/.f64 (*.f64 1/4 (pow.f64 D 2)) d) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 M M) d)))): 26 points increase in error, 5 points decrease in error
      (*.f64 h (*.f64 (/.f64 (*.f64 1/4 (pow.f64 D 2)) d) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 M 2)) d))): 0 points increase in error, 0 points decrease in error
      (*.f64 h (Rewrite<= associate-/r/_binary64 (/.f64 (*.f64 1/4 (pow.f64 D 2)) (/.f64 d (/.f64 (pow.f64 M 2) d))))): 14 points increase in error, 3 points decrease in error
      (*.f64 h (/.f64 (*.f64 1/4 (pow.f64 D 2)) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 d d) (pow.f64 M 2))))): 21 points increase in error, 0 points decrease in error
      (*.f64 h (/.f64 (*.f64 1/4 (pow.f64 D 2)) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 d 2)) (pow.f64 M 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (*.f64 1/4 (pow.f64 D 2)) (/.f64 (pow.f64 d 2) (pow.f64 M 2))) h)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 (*.f64 1/4 (pow.f64 D 2)) (/.f64 (/.f64 (pow.f64 d 2) (pow.f64 M 2)) h))): 6 points increase in error, 8 points decrease in error
      (/.f64 (*.f64 1/4 (pow.f64 D 2)) (Rewrite<= associate-/r*_binary64 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 M 2) h)))): 9 points increase in error, 6 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 1/4 (/.f64 (pow.f64 D 2) (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 M 2) h))))): 0 points increase in error, 1 points decrease in error
      (*.f64 1/4 (/.f64 (pow.f64 D 2) (/.f64 (pow.f64 d 2) (Rewrite=> *-commutative_binary64 (*.f64 h (pow.f64 M 2)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/4 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 h (pow.f64 M 2))) (pow.f64 d 2)))): 10 points increase in error, 8 points decrease in error

   7. Taylor expanded in h around 0 34.3

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

   8. Simplified 16.9

      \[\leadsto \color{blue}{0.25 \cdot \left(D \cdot \frac{\frac{D}{d} \cdot M}{\frac{\frac{d}{M}}{h}}\right)} \]
      Proof
      (*.f64 1/4 (*.f64 D (/.f64 (*.f64 (/.f64 D d) M) (/.f64 (/.f64 d M) h)))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/4 (*.f64 D (/.f64 (Rewrite<= associate-/r/_binary64 (/.f64 D (/.f64 d M))) (/.f64 (/.f64 d M) h)))): 12 points increase in error, 20 points decrease in error
      (*.f64 1/4 (*.f64 D (/.f64 (/.f64 D (/.f64 d M)) (Rewrite<= associate-/r*_binary64 (/.f64 d (*.f64 M h)))))): 23 points increase in error, 7 points decrease in error
      (*.f64 1/4 (*.f64 D (Rewrite<= associate-/r*_binary64 (/.f64 D (*.f64 (/.f64 d M) (/.f64 d (*.f64 M h))))))): 28 points increase in error, 5 points decrease in error
      (*.f64 1/4 (*.f64 D (/.f64 D (Rewrite<= times-frac_binary64 (/.f64 (*.f64 d d) (*.f64 M (*.f64 M h))))))): 48 points increase in error, 6 points decrease in error
      (*.f64 1/4 (*.f64 D (/.f64 D (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 d 2)) (*.f64 M (*.f64 M h)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/4 (*.f64 D (/.f64 D (/.f64 (pow.f64 d 2) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 M M) h)))))): 14 points increase in error, 2 points decrease in error
      (*.f64 1/4 (*.f64 D (/.f64 D (/.f64 (pow.f64 d 2) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 M 2)) h))))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/4 (*.f64 D (/.f64 D (/.f64 (pow.f64 d 2) (Rewrite<= *-commutative_binary64 (*.f64 h (pow.f64 M 2))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/4 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 D (/.f64 (pow.f64 d 2) (*.f64 h (pow.f64 M 2)))) D))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/4 (Rewrite<= associate-/r/_binary64 (/.f64 D (/.f64 (/.f64 (pow.f64 d 2) (*.f64 h (pow.f64 M 2))) D)))): 1 points increase in error, 8 points decrease in error
      (*.f64 1/4 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 D D) (/.f64 (pow.f64 d 2) (*.f64 h (pow.f64 M 2)))))): 44 points increase in error, 1 points decrease in error
      (*.f64 1/4 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 D 2)) (/.f64 (pow.f64 d 2) (*.f64 h (pow.f64 M 2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/4 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 h (pow.f64 M 2))) (pow.f64 d 2)))): 10 points increase in error, 8 points decrease in error

   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 46.3

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

   2. Taylor expanded in c0 around inf 41.3

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

   3. Simplified 38.4

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \left(c0 \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w \cdot h}\right)\right)} \]
      Proof
      (*.f64 2 (*.f64 c0 (/.f64 (*.f64 (/.f64 d D) (/.f64 d D)) (*.f64 w h)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 c0 (/.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 d d) (*.f64 D D))) (*.f64 w h)))): 42 points increase in error, 15 points decrease in error
      (*.f64 2 (*.f64 c0 (/.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 d 2)) (*.f64 D D)) (*.f64 w h)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 c0 (/.f64 (/.f64 (pow.f64 d 2) (Rewrite<= unpow2_binary64 (pow.f64 D 2))) (*.f64 w h)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 c0 (Rewrite<= associate-/r*_binary64 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h)))))): 6 points increase in error, 8 points decrease in error
      (*.f64 2 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 c0 (pow.f64 d 2)) (*.f64 (pow.f64 D 2) (*.f64 w h))))): 14 points increase in error, 7 points decrease in error
      (*.f64 2 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 d 2) c0)) (*.f64 (pow.f64 D 2) (*.f64 w h)))): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 54.5

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(c0 \cdot \frac{\color{blue}{{\left({\left({\left(\frac{d}{D}\right)}^{2}\right)}^{3}\right)}^{0.3333333333333333}}}{w \cdot h}\right)\right) \]

   5. Applied egg-rr 35.1

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

   6. Applied egg-rr 30.1

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{\frac{c0 \cdot \frac{d}{D}}{w} \cdot d}{D \cdot h}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 18.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq -2 \cdot 10^{-22}:\\ \;\;\;\;\frac{c0 \cdot d}{w \cdot D} \cdot \frac{d \cdot \frac{c0}{h}}{w \cdot D}\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 0:\\ \;\;\;\;0.25 \cdot \left(D \cdot \frac{M \cdot \frac{D}{d}}{\frac{\frac{d}{M}}{h}}\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{d \cdot \frac{c0 \cdot \frac{d}{D}}{w}}{h \cdot D}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \frac{M \cdot \frac{D}{d}}{\frac{\frac{d}{M}}{h}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	21.4
Cost	1352

\[\begin{array}{l} t_0 := 0.25 \cdot \left(D \cdot \frac{M \cdot \frac{D}{d}}{\frac{\frac{d}{M}}{h}}\right)\\ \mathbf{if}\;c0 \leq 2.9 \cdot 10^{-144}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c0 \leq 1.18 \cdot 10^{-125}:\\ \;\;\;\;\frac{c0 \cdot d}{w \cdot D} \cdot \frac{d \cdot \frac{c0}{h}}{w \cdot D}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	21.4
Cost	1352

\[\begin{array}{l} t_0 := 0.25 \cdot \left(D \cdot \frac{M \cdot \frac{D}{d}}{\frac{\frac{d}{M}}{h}}\right)\\ \mathbf{if}\;c0 \leq 2.9 \cdot 10^{-144}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c0 \leq 1.18 \cdot 10^{-125}:\\ \;\;\;\;\frac{c0 \cdot \left(d \cdot \frac{c0}{h}\right)}{w \cdot D} \cdot \frac{d}{w \cdot D}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	28.1
Cost	1220

\[\begin{array}{l} \mathbf{if}\;D \cdot D \leq 7 \cdot 10^{+255}:\\ \;\;\;\;h \cdot \left(\left(M \cdot \frac{M}{d}\right) \cdot \left(0.25 \cdot \frac{D \cdot D}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 4
Error	26.7
Cost	960

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

Alternative 5
Error	21.1
Cost	960

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

Alternative 6
Error	32.7
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022300 
(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))))))