Henrywood and Agarwal, Equation (13)

Average Error: 59.7 → 17.5

Time: 30.8s

Precision: binary64

Cost: 30540

Math

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

↓

\[\begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_1 := \frac{c0}{2 \cdot w} \cdot \left(t_0 + \sqrt{t_0 \cdot t_0 - M \cdot M}\right)\\ t_2 := \frac{\frac{c0}{2}}{w}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-292}:\\ \;\;\;\;t_2 \cdot \left(2 \cdot \frac{c0 \cdot \frac{d}{D}}{\left(w \cdot h\right) \cdot \frac{D}{d}}\right)\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;0.25 \cdot \frac{M}{\frac{\frac{d}{M}}{h} \cdot \frac{d}{D \cdot D}}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;t_2 \cdot \left(2 \cdot \frac{\left(c0 \cdot \frac{d}{w \cdot D}\right) \cdot \frac{d}{h}}{D}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{M}{\frac{d}{D}} \cdot \frac{D}{\frac{d}{h \cdot M}}\right)\\ \end{array} \]

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 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_1 (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M))))))
        (t_2 (/ (/ c0 2.0) w)))
   (if (<= t_1 -2e-292)
     (* t_2 (* 2.0 (/ (* c0 (/ d D)) (* (* w h) (/ D d)))))
     (if (<= t_1 0.0)
       (* 0.25 (/ M (* (/ (/ d M) h) (/ d (* D D)))))
       (if (<= t_1 INFINITY)
         (* t_2 (* 2.0 (/ (* (* c0 (/ d (* w D))) (/ d h)) D)))
         (* 0.25 (* (/ M (/ d D)) (/ D (/ d (* h M))))))))))

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 t_1 = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	double t_2 = (c0 / 2.0) / w;
	double tmp;
	if (t_1 <= -2e-292) {
		tmp = t_2 * (2.0 * ((c0 * (d / D)) / ((w * h) * (D / d))));
	} else if (t_1 <= 0.0) {
		tmp = 0.25 * (M / (((d / M) / h) * (d / (D * D))));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2 * (2.0 * (((c0 * (d / (w * D))) * (d / h)) / D));
	} else {
		tmp = 0.25 * ((M / (d / D)) * (D / (d / (h * M))));
	}
	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 t_1 = (c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
	double t_2 = (c0 / 2.0) / w;
	double tmp;
	if (t_1 <= -2e-292) {
		tmp = t_2 * (2.0 * ((c0 * (d / D)) / ((w * h) * (D / d))));
	} else if (t_1 <= 0.0) {
		tmp = 0.25 * (M / (((d / M) / h) * (d / (D * D))));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_2 * (2.0 * (((c0 * (d / (w * D))) * (d / h)) / D));
	} else {
		tmp = 0.25 * ((M / (d / D)) * (D / (d / (h * M))));
	}
	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))
	t_1 = (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))
	t_2 = (c0 / 2.0) / w
	tmp = 0
	if t_1 <= -2e-292:
		tmp = t_2 * (2.0 * ((c0 * (d / D)) / ((w * h) * (D / d))))
	elif t_1 <= 0.0:
		tmp = 0.25 * (M / (((d / M) / h) * (d / (D * D))))
	elif t_1 <= math.inf:
		tmp = t_2 * (2.0 * (((c0 * (d / (w * D))) * (d / h)) / D))
	else:
		tmp = 0.25 * ((M / (d / D)) * (D / (d / (h * M))))
	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)))
	t_1 = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))))
	t_2 = Float64(Float64(c0 / 2.0) / w)
	tmp = 0.0
	if (t_1 <= -2e-292)
		tmp = Float64(t_2 * Float64(2.0 * Float64(Float64(c0 * Float64(d / D)) / Float64(Float64(w * h) * Float64(D / d)))));
	elseif (t_1 <= 0.0)
		tmp = Float64(0.25 * Float64(M / Float64(Float64(Float64(d / M) / h) * Float64(d / Float64(D * D)))));
	elseif (t_1 <= Inf)
		tmp = Float64(t_2 * Float64(2.0 * Float64(Float64(Float64(c0 * Float64(d / Float64(w * D))) * Float64(d / h)) / D)));
	else
		tmp = Float64(0.25 * Float64(Float64(M / Float64(d / D)) * Float64(D / Float64(d / Float64(h * M)))));
	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));
	t_1 = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	t_2 = (c0 / 2.0) / w;
	tmp = 0.0;
	if (t_1 <= -2e-292)
		tmp = t_2 * (2.0 * ((c0 * (d / D)) / ((w * h) * (D / d))));
	elseif (t_1 <= 0.0)
		tmp = 0.25 * (M / (((d / M) / h) * (d / (D * D))));
	elseif (t_1 <= Inf)
		tmp = t_2 * (2.0 * (((c0 * (d / (w * D))) * (d / h)) / D));
	else
		tmp = 0.25 * ((M / (d / D)) * (D / (d / (h * M))));
	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]}, Block[{t$95$1 = 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]}, Block[{t$95$2 = N[(N[(c0 / 2.0), $MachinePrecision] / w), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-292], N[(t$95$2 * N[(2.0 * N[(N[(c0 * N[(d / D), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(0.25 * N[(M / N[(N[(N[(d / M), $MachinePrecision] / h), $MachinePrecision] * N[(d / N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(t$95$2 * N[(2.0 * N[(N[(N[(c0 * N[(d / N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(M / N[(d / D), $MachinePrecision]), $MachinePrecision] * N[(D / N[(d / N[(h * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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))))) < -2.0000000000000001e-292

   1. Initial program 49.8

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

   2. Simplified 56.9

      \[\leadsto \color{blue}{\frac{\frac{c0}{2}}{w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{4}, M \cdot \left(-M\right)\right)}\right)} \]
      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, 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)) (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, 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 (Rewrite=> unpow3_binary64 (*.f64 (*.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)) (*.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, 2 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, 2 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)))))): 3 points increase in error, 2 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)))))): 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 (-.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))))): 9 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<= 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, 15 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))))): 6 points increase in error, 2 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))))): 1 points increase in error, 8 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)))))): 7 points increase in error, 2 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))))): 1 points increase in error, 7 points decrease in error

   3. Taylor expanded in d around inf 42.6

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

   4. Simplified 40.2

      \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \color{blue}{\left(2 \cdot \left(c0 \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w \cdot h}\right)\right)} \]
      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)))): 41 points increase in error, 19 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)))))): 12 points increase in error, 13 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))))): 17 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

   5. Applied egg-rr 27.2

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

   if -2.0000000000000001e-292 < (*.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 29.9

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

   2. Applied egg-rr 48.0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(\frac{\frac{c0}{h}}{w}, \frac{d}{D} \cdot \frac{d}{D}, \sqrt{{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - M \cdot M}\right)}\right)}^{3}} \]

   3. Taylor expanded in c0 around -inf 31.9

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

   4. Simplified 22.8

      \[\leadsto \color{blue}{0.25 \cdot \left(\frac{D \cdot D}{d} \cdot \frac{M \cdot \left(M \cdot h\right)}{d}\right)} \]
      Proof
      (*.f64 1/4 (*.f64 (/.f64 (*.f64 D D) d) (/.f64 (*.f64 M (*.f64 M h)) d))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/4 (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 D 2)) d) (/.f64 (*.f64 M (*.f64 M h)) d))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/4 (*.f64 (/.f64 (pow.f64 D 2) d) (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 M M) h)) d))): 18 points increase in error, 6 points decrease in error
      (*.f64 1/4 (*.f64 (/.f64 (pow.f64 D 2) d) (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 M 2)) h) d))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/4 (*.f64 (/.f64 (pow.f64 D 2) d) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 h (pow.f64 M 2))) d))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/4 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 h (pow.f64 M 2))) (*.f64 d d)))): 30 points increase in error, 7 points decrease in error
      (*.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (Rewrite=> *-commutative_binary64 (*.f64 (pow.f64 M 2) h))) (*.f64 d d))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (Rewrite<= unpow2_binary64 (pow.f64 d 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-lft-identity_binary64 (+.f64 0 (*.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= metadata-eval (*.f64 -1/2 0)) (*.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1/2 (Rewrite<= div0_binary64 (/.f64 0 (/.f64 w (pow.f64 c0 2))))) (*.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2)))): 50 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1/2 (/.f64 (Rewrite<= mul0-lft_binary64 (*.f64 0 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h))))) (/.f64 w (pow.f64 c0 2)))) (*.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2)))): 95 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1/2 (/.f64 (*.f64 (Rewrite<= metadata-eval (+.f64 -1 1)) (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h)))) (/.f64 w (pow.f64 c0 2)))) (*.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1/2 (/.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)))))) (/.f64 w (pow.f64 c0 2)))) (*.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1/2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.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))))) (pow.f64 c0 2)) w))) (*.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2)))): 0 points increase in error, 2 points decrease in error

   5. Applied egg-rr 16.9

      \[\leadsto 0.25 \cdot \color{blue}{\frac{M}{\frac{\frac{d}{M}}{h} \cdot \frac{d}{D \cdot D}}} \]

   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 49.6

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

   2. Simplified 57.1

      \[\leadsto \color{blue}{\frac{\frac{c0}{2}}{w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{4}, M \cdot \left(-M\right)\right)}\right)} \]
      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, 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)) (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, 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 (Rewrite=> unpow3_binary64 (*.f64 (*.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)) (*.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, 2 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, 2 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)))))): 3 points increase in error, 2 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)))))): 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 (-.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))))): 9 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<= 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, 15 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))))): 6 points increase in error, 2 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))))): 1 points increase in error, 8 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)))))): 7 points increase in error, 2 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))))): 1 points increase in error, 7 points decrease in error

   3. Taylor expanded in d around inf 42.3

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

   4. Simplified 41.1

      \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \color{blue}{\left(2 \cdot \left(c0 \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w \cdot h}\right)\right)} \]
      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)))): 41 points increase in error, 19 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)))))): 12 points increase in error, 13 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))))): 17 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

   5. Applied egg-rr 38.1

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

   6. Applied egg-rr 32.0

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

   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. Applied egg-rr 59.5

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(\frac{\frac{c0}{h}}{w}, \frac{d}{D} \cdot \frac{d}{D}, \sqrt{{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - M \cdot M}\right)}\right)}^{3}} \]

   3. Taylor expanded in c0 around -inf 63.5

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

   4. Simplified 27.2

      \[\leadsto \color{blue}{0.25 \cdot \left(\frac{D \cdot D}{d} \cdot \frac{M \cdot \left(M \cdot h\right)}{d}\right)} \]
      Proof
      (*.f64 1/4 (*.f64 (/.f64 (*.f64 D D) d) (/.f64 (*.f64 M (*.f64 M h)) d))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/4 (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 D 2)) d) (/.f64 (*.f64 M (*.f64 M h)) d))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/4 (*.f64 (/.f64 (pow.f64 D 2) d) (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 M M) h)) d))): 18 points increase in error, 6 points decrease in error
      (*.f64 1/4 (*.f64 (/.f64 (pow.f64 D 2) d) (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 M 2)) h) d))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/4 (*.f64 (/.f64 (pow.f64 D 2) d) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 h (pow.f64 M 2))) d))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/4 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 h (pow.f64 M 2))) (*.f64 d d)))): 30 points increase in error, 7 points decrease in error
      (*.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (Rewrite=> *-commutative_binary64 (*.f64 (pow.f64 M 2) h))) (*.f64 d d))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (Rewrite<= unpow2_binary64 (pow.f64 d 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-lft-identity_binary64 (+.f64 0 (*.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= metadata-eval (*.f64 -1/2 0)) (*.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1/2 (Rewrite<= div0_binary64 (/.f64 0 (/.f64 w (pow.f64 c0 2))))) (*.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2)))): 50 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1/2 (/.f64 (Rewrite<= mul0-lft_binary64 (*.f64 0 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h))))) (/.f64 w (pow.f64 c0 2)))) (*.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2)))): 95 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1/2 (/.f64 (*.f64 (Rewrite<= metadata-eval (+.f64 -1 1)) (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h)))) (/.f64 w (pow.f64 c0 2)))) (*.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1/2 (/.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)))))) (/.f64 w (pow.f64 c0 2)))) (*.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1/2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.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))))) (pow.f64 c0 2)) w))) (*.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2)))): 0 points increase in error, 2 points decrease in error

   5. Applied egg-rr 17.7

      \[\leadsto 0.25 \cdot \color{blue}{\frac{M \cdot D}{\frac{\frac{d}{M}}{h} \cdot \frac{d}{D}}} \]

   6. Applied egg-rr 15.6

      \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{M}{\frac{d}{D}} \cdot \frac{D}{\frac{d}{M \cdot h}}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 17.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq -2 \cdot 10^{-292}:\\ \;\;\;\;\frac{\frac{c0}{2}}{w} \cdot \left(2 \cdot \frac{c0 \cdot \frac{d}{D}}{\left(w \cdot h\right) \cdot \frac{D}{d}}\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 0:\\ \;\;\;\;0.25 \cdot \frac{M}{\frac{\frac{d}{M}}{h} \cdot \frac{d}{D \cdot D}}\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{\frac{c0}{2}}{w} \cdot \left(2 \cdot \frac{\left(c0 \cdot \frac{d}{w \cdot D}\right) \cdot \frac{d}{h}}{D}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{M}{\frac{d}{D}} \cdot \frac{D}{\frac{d}{h \cdot M}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	21.2
Cost	1608

\[\begin{array}{l} t_0 := 0.25 \cdot \left(\frac{M}{\frac{d}{D}} \cdot \frac{D}{\frac{d}{h \cdot M}}\right)\\ \mathbf{if}\;d \leq -8.5 \cdot 10^{-73}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -3 \cdot 10^{-150}:\\ \;\;\;\;\frac{\frac{d}{D} \cdot \left(2 \cdot \frac{c0}{2 \cdot w}\right)}{\frac{w}{\frac{c0}{h \cdot \frac{D}{d}}}}\\ \mathbf{elif}\;d \leq 10^{-75}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 6.2 \cdot 10^{+26}:\\ \;\;\;\;0.25 \cdot \left(h \cdot \frac{D \cdot M}{\frac{d}{D} \cdot \frac{d}{M}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \frac{D \cdot M}{\frac{\frac{d}{M}}{h}}}{d}\\ \end{array} \]

Alternative 2
Error	20.3
Cost	1224

\[\begin{array}{l} \mathbf{if}\;d \leq 10^{-75}:\\ \;\;\;\;0.25 \cdot \left(\frac{M}{\frac{d}{D}} \cdot \frac{D}{\frac{d}{h \cdot M}}\right)\\ \mathbf{elif}\;d \leq 6.2 \cdot 10^{+26}:\\ \;\;\;\;0.25 \cdot \left(h \cdot \frac{D \cdot M}{\frac{d}{D} \cdot \frac{d}{M}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \frac{D \cdot M}{\frac{\frac{d}{M}}{h}}}{d}\\ \end{array} \]

Alternative 3
Error	21.5
Cost	1092

\[\begin{array}{l} \mathbf{if}\;D \leq 2.25 \cdot 10^{-209}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot M}{\frac{\frac{\frac{d}{h}}{M}}{\frac{D}{d}}}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \frac{D \cdot M}{\frac{\frac{d}{M}}{h}}}{d}\\ \end{array} \]

Alternative 4
Error	21.3
Cost	960

\[0.25 \cdot \frac{D \cdot M}{\frac{\frac{\frac{d}{h}}{M}}{\frac{D}{d}}} \]

Alternative 5
Error	20.6
Cost	960

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

Alternative 6
Error	31.9
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022291 
(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))))))