Henrywood and Agarwal, Equation (13)

Average Error: 59.7 → 16.4

Time: 31.0s

Precision: binary64

Cost: 43852

Math

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

↓

\[\begin{array}{l} t_0 := \frac{d}{D} \cdot \sqrt{2 \cdot \frac{c0}{w \cdot h}}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_3 := t_1 \cdot \left(t_2 + \sqrt{t_2 \cdot t_2 - M \cdot M}\right)\\ \mathbf{if}\;t_3 \leq -1 \cdot 10^{-175}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \frac{\frac{d}{D}}{\left(h \cdot \frac{w}{c0}\right) \cdot \frac{D}{d}}\right)\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;0.25 \cdot \left(\frac{M}{\frac{d}{D} \cdot \frac{d}{D}} \cdot \left(h \cdot M\right)\right)\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\frac{t_0 \cdot \frac{c0}{\frac{w}{t_0}}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{\frac{D}{d} \cdot \left(h \cdot M\right)}{\frac{\frac{d}{D}}{M}}\\ \end{array} \]

FPCore

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

↓

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ d D) (sqrt (* 2.0 (/ c0 (* w h))))))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_3 (* t_1 (+ t_2 (sqrt (- (* t_2 t_2) (* M M)))))))
   (if (<= t_3 -1e-175)
     (* t_1 (* 2.0 (/ (/ d D) (* (* h (/ w c0)) (/ D d)))))
     (if (<= t_3 0.0)
       (* 0.25 (* (/ M (* (/ d D) (/ d D))) (* h M)))
       (if (<= t_3 INFINITY)
         (/ (* t_0 (/ c0 (/ w t_0))) 2.0)
         (* 0.25 (/ (* (/ D d) (* h M)) (/ (/ d D) 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 = (d / D) * sqrt((2.0 * (c0 / (w * h))));
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_3 = t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))));
	double tmp;
	if (t_3 <= -1e-175) {
		tmp = t_1 * (2.0 * ((d / D) / ((h * (w / c0)) * (D / d))));
	} else if (t_3 <= 0.0) {
		tmp = 0.25 * ((M / ((d / D) * (d / D))) * (h * M));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (t_0 * (c0 / (w / t_0))) / 2.0;
	} else {
		tmp = 0.25 * (((D / d) * (h * M)) / ((d / D) / 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 = (d / D) * Math.sqrt((2.0 * (c0 / (w * h))));
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_3 = t_1 * (t_2 + Math.sqrt(((t_2 * t_2) - (M * M))));
	double tmp;
	if (t_3 <= -1e-175) {
		tmp = t_1 * (2.0 * ((d / D) / ((h * (w / c0)) * (D / d))));
	} else if (t_3 <= 0.0) {
		tmp = 0.25 * ((M / ((d / D) * (d / D))) * (h * M));
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = (t_0 * (c0 / (w / t_0))) / 2.0;
	} else {
		tmp = 0.25 * (((D / d) * (h * M)) / ((d / D) / 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 = (d / D) * math.sqrt((2.0 * (c0 / (w * h))))
	t_1 = c0 / (2.0 * w)
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D))
	t_3 = t_1 * (t_2 + math.sqrt(((t_2 * t_2) - (M * M))))
	tmp = 0
	if t_3 <= -1e-175:
		tmp = t_1 * (2.0 * ((d / D) / ((h * (w / c0)) * (D / d))))
	elif t_3 <= 0.0:
		tmp = 0.25 * ((M / ((d / D) * (d / D))) * (h * M))
	elif t_3 <= math.inf:
		tmp = (t_0 * (c0 / (w / t_0))) / 2.0
	else:
		tmp = 0.25 * (((D / d) * (h * M)) / ((d / D) / 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(d / D) * sqrt(Float64(2.0 * Float64(c0 / Float64(w * h)))))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	t_3 = Float64(t_1 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M)))))
	tmp = 0.0
	if (t_3 <= -1e-175)
		tmp = Float64(t_1 * Float64(2.0 * Float64(Float64(d / D) / Float64(Float64(h * Float64(w / c0)) * Float64(D / d)))));
	elseif (t_3 <= 0.0)
		tmp = Float64(0.25 * Float64(Float64(M / Float64(Float64(d / D) * Float64(d / D))) * Float64(h * M)));
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(t_0 * Float64(c0 / Float64(w / t_0))) / 2.0);
	else
		tmp = Float64(0.25 * Float64(Float64(Float64(D / d) * Float64(h * M)) / Float64(Float64(d / D) / 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 = (d / D) * sqrt((2.0 * (c0 / (w * h))));
	t_1 = c0 / (2.0 * w);
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	t_3 = t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))));
	tmp = 0.0;
	if (t_3 <= -1e-175)
		tmp = t_1 * (2.0 * ((d / D) / ((h * (w / c0)) * (D / d))));
	elseif (t_3 <= 0.0)
		tmp = 0.25 * ((M / ((d / D) * (d / D))) * (h * M));
	elseif (t_3 <= Inf)
		tmp = (t_0 * (c0 / (w / t_0))) / 2.0;
	else
		tmp = 0.25 * (((D / d) * (h * M)) / ((d / D) / 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[(d / D), $MachinePrecision] * N[Sqrt[N[(2.0 * N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1e-175], N[(t$95$1 * N[(2.0 * N[(N[(d / D), $MachinePrecision] / N[(N[(h * N[(w / c0), $MachinePrecision]), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(0.25 * N[(N[(M / N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(t$95$0 * N[(c0 / N[(w / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(0.25 * N[(N[(N[(D / d), $MachinePrecision] * N[(h * M), $MachinePrecision]), $MachinePrecision] / N[(N[(d / D), $MachinePrecision] / M), $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))))) < -1e-175

   1. Initial program 50.7

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

   2. Simplified 47.8

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

      [Start] 50.7	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      times-frac [=>] 52.7	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      fma-def [=>] 52.7	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d \cdot d}{D \cdot D}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)} \]
      associate-/r* [=>] 52.7	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\color{blue}{\frac{\frac{c0}{w}}{h}}, \frac{d \cdot d}{D \cdot D}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      times-frac [=>] 52.8	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \color{blue}{\frac{d}{D} \cdot \frac{d}{D}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      difference-of-squares [=>] 52.8	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \sqrt{\color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + M\right) \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M\right)}}\right) \]
      sub-neg [=>] 52.8	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \sqrt{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + M\right) \cdot \color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \left(-M\right)\right)}}\right) \]

   3. Taylor expanded in c0 around inf 43.9

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

   4. Simplified 37.8

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

      [Start] 43.9	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]
      times-frac [=>] 43.6	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{{d}^{2}}{{D}^{2}} \cdot \frac{c0}{w \cdot h}\right)}\right) \]
      *-commutative [=>] 43.6	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{{d}^{2}}{{D}^{2}} \cdot \frac{c0}{\color{blue}{h \cdot w}}\right)\right) \]
      associate-/l/ [<=] 42.8	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{{d}^{2}}{{D}^{2}} \cdot \color{blue}{\frac{\frac{c0}{w}}{h}}\right)\right) \]
      associate-*l/ [=>] 42.3	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{{d}^{2} \cdot \frac{\frac{c0}{w}}{h}}{{D}^{2}}}\right) \]
      unpow2 [=>] 42.3	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\color{blue}{\left(d \cdot d\right)} \cdot \frac{\frac{c0}{w}}{h}}{{D}^{2}}\right) \]
      unpow2 [=>] 42.3	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\left(d \cdot d\right) \cdot \frac{\frac{c0}{w}}{h}}{\color{blue}{D \cdot D}}\right) \]
      *-commutative [<=] 42.3	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\color{blue}{\frac{\frac{c0}{w}}{h} \cdot \left(d \cdot d\right)}}{D \cdot D}\right) \]
      times-frac [=>] 36.7	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{\frac{\frac{c0}{w}}{h}}{D} \cdot \frac{d \cdot d}{D}\right)}\right) \]
      associate-*l/ [<=] 35.0	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\frac{\frac{c0}{w}}{h}}{D} \cdot \color{blue}{\left(\frac{d}{D} \cdot d\right)}\right)\right) \]
      associate-/r/ [<=] 35.0	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\frac{\frac{c0}{w}}{h}}{D} \cdot \color{blue}{\frac{d}{\frac{D}{d}}}\right)\right) \]
      associate-*l/ [=>] 35.7	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{\frac{\frac{c0}{w}}{h} \cdot \frac{d}{\frac{D}{d}}}{D}}\right) \]
      associate-*r/ [<=] 38.2	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{\frac{c0}{w}}{h} \cdot \frac{\frac{d}{\frac{D}{d}}}{D}\right)}\right) \]
      associate-/l/ [=>] 39.0	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\color{blue}{\frac{c0}{h \cdot w}} \cdot \frac{\frac{d}{\frac{D}{d}}}{D}\right)\right) \]
      *-commutative [<=] 39.0	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{\color{blue}{w \cdot h}} \cdot \frac{\frac{d}{\frac{D}{d}}}{D}\right)\right) \]
      associate-/r/ [=>] 39.0	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{\color{blue}{\frac{d}{D} \cdot d}}{D}\right)\right) \]
      associate-*r/ [<=] 37.8	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w \cdot h} \cdot \color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D}\right)}\right)\right) \]
      unpow2 [<=] 37.8	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w \cdot h} \cdot \color{blue}{{\left(\frac{d}{D}\right)}^{2}}\right)\right) \]

   5. Applied egg-rr 30.7

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

   if -1e-175 < (*.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 29.3

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

   2. Simplified 54.9

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

      [Start] 29.3	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      associate-*l/ [<=] 34.5	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \left(d \cdot d\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      *-commutative [=>] 34.5	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(d \cdot d\right) \cdot \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      fma-def [=>] 42.7	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(d \cdot d, \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)} \]
      associate-*l* [=>] 45.1	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{\color{blue}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      associate-/r* [=>] 45.5	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \color{blue}{\frac{\frac{c0}{w}}{h \cdot \left(D \cdot D\right)}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      associate-*r* [=>] 46.2	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{\color{blue}{\left(h \cdot D\right) \cdot D}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      *-commutative [=>] 46.2	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{\color{blue}{D \cdot \left(h \cdot D\right)}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Taylor expanded in c0 around -inf 29.0

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

   4. Simplified 23.0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{\frac{w \cdot \left(M \cdot \left(M \cdot h\right)\right)}{{\left(\frac{d}{D}\right)}^{2}}}{c0}, c0 \cdot 0\right)} \]
      Proof

      [Start] 29.0	\[ \frac{c0}{2 \cdot w} \cdot \left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0} + -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right) \]
      fma-def [=>] 29.0	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0}, -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right)} \]

   5. Taylor expanded in c0 around 0 24.9

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

   6. Simplified 23.6

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

      [Start] 24.9	\[ 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \]
      unpow2 [=>] 24.9	\[ 0.25 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \]
      unpow2 [=>] 24.9	\[ 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot h\right)}{{d}^{2}} \]
      associate-*r* [<=] 23.6	\[ 0.25 \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot \left(M \cdot h\right)\right)}}{{d}^{2}} \]
      *-commutative [<=] 23.6	\[ 0.25 \cdot \frac{\color{blue}{\left(M \cdot \left(M \cdot h\right)\right) \cdot \left(D \cdot D\right)}}{{d}^{2}} \]
      unpow2 [=>] 23.6	\[ 0.25 \cdot \frac{\left(M \cdot \left(M \cdot h\right)\right) \cdot \left(D \cdot D\right)}{\color{blue}{d \cdot d}} \]

   7. Applied egg-rr 17.5

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

   if 0.0 < (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 47.2

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

   2. Simplified 44.6

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

      [Start] 47.2	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      times-frac [=>] 49.7	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      fma-def [=>] 49.6	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d \cdot d}{D \cdot D}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)} \]
      associate-/r* [=>] 49.6	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\color{blue}{\frac{\frac{c0}{w}}{h}}, \frac{d \cdot d}{D \cdot D}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      times-frac [=>] 49.6	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \color{blue}{\frac{d}{D} \cdot \frac{d}{D}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      difference-of-squares [=>] 49.6	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \sqrt{\color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + M\right) \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M\right)}}\right) \]
      sub-neg [=>] 49.6	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \sqrt{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + M\right) \cdot \color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \left(-M\right)\right)}}\right) \]

   3. Taylor expanded in c0 around inf 41.4

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

   4. Simplified 35.1

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

      [Start] 41.4	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]
      times-frac [=>] 41.8	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{{d}^{2}}{{D}^{2}} \cdot \frac{c0}{w \cdot h}\right)}\right) \]
      *-commutative [=>] 41.8	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{{d}^{2}}{{D}^{2}} \cdot \frac{c0}{\color{blue}{h \cdot w}}\right)\right) \]
      associate-/l/ [<=] 40.0	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{{d}^{2}}{{D}^{2}} \cdot \color{blue}{\frac{\frac{c0}{w}}{h}}\right)\right) \]
      associate-*l/ [=>] 38.7	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{{d}^{2} \cdot \frac{\frac{c0}{w}}{h}}{{D}^{2}}}\right) \]
      unpow2 [=>] 38.7	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\color{blue}{\left(d \cdot d\right)} \cdot \frac{\frac{c0}{w}}{h}}{{D}^{2}}\right) \]
      unpow2 [=>] 38.7	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\left(d \cdot d\right) \cdot \frac{\frac{c0}{w}}{h}}{\color{blue}{D \cdot D}}\right) \]
      *-commutative [<=] 38.7	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\color{blue}{\frac{\frac{c0}{w}}{h} \cdot \left(d \cdot d\right)}}{D \cdot D}\right) \]
      times-frac [=>] 33.0	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{\frac{\frac{c0}{w}}{h}}{D} \cdot \frac{d \cdot d}{D}\right)}\right) \]
      associate-*l/ [<=] 30.7	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\frac{\frac{c0}{w}}{h}}{D} \cdot \color{blue}{\left(\frac{d}{D} \cdot d\right)}\right)\right) \]
      associate-/r/ [<=] 30.7	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\frac{\frac{c0}{w}}{h}}{D} \cdot \color{blue}{\frac{d}{\frac{D}{d}}}\right)\right) \]
      associate-*l/ [=>] 30.5	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{\frac{\frac{c0}{w}}{h} \cdot \frac{d}{\frac{D}{d}}}{D}}\right) \]
      associate-*r/ [<=] 34.4	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{\frac{c0}{w}}{h} \cdot \frac{\frac{d}{\frac{D}{d}}}{D}\right)}\right) \]
      associate-/l/ [=>] 36.3	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\color{blue}{\frac{c0}{h \cdot w}} \cdot \frac{\frac{d}{\frac{D}{d}}}{D}\right)\right) \]
      *-commutative [<=] 36.3	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{\color{blue}{w \cdot h}} \cdot \frac{\frac{d}{\frac{D}{d}}}{D}\right)\right) \]
      associate-/r/ [=>] 36.3	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{\color{blue}{\frac{d}{D} \cdot d}}{D}\right)\right) \]
      associate-*r/ [<=] 35.1	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w \cdot h} \cdot \color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D}\right)}\right)\right) \]
      unpow2 [<=] 35.1	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w \cdot h} \cdot \color{blue}{{\left(\frac{d}{D}\right)}^{2}}\right)\right) \]

   5. Applied egg-rr 38.1

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

   6. Applied egg-rr 23.3

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

   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 63.7

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

      [Start] 64.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      associate-*l/ [<=] 64.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \left(d \cdot d\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      *-commutative [=>] 64.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(d \cdot d\right) \cdot \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      fma-def [=>] 64.0	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(d \cdot d, \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)} \]
      associate-*l* [=>] 64.0	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{\color{blue}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      associate-/r* [=>] 64.0	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \color{blue}{\frac{\frac{c0}{w}}{h \cdot \left(D \cdot D\right)}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      associate-*r* [=>] 64.0	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{\color{blue}{\left(h \cdot D\right) \cdot D}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      *-commutative [=>] 64.0	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{\color{blue}{D \cdot \left(h \cdot D\right)}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Taylor expanded in c0 around -inf 62.7

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

   4. Simplified 32.9

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{\frac{w \cdot \left(M \cdot \left(M \cdot h\right)\right)}{{\left(\frac{d}{D}\right)}^{2}}}{c0}, c0 \cdot 0\right)} \]
      Proof

      [Start] 62.7	\[ \frac{c0}{2 \cdot w} \cdot \left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0} + -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right) \]
      fma-def [=>] 62.7	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0}, -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right)} \]

   5. Taylor expanded in c0 around 0 34.0

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

   6. Simplified 23.7

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

      [Start] 34.0	\[ 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \]
      *-commutative [<=] 34.0	\[ 0.25 \cdot \frac{{D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{{d}^{2}} \]
      associate-/l* [=>] 34.0	\[ 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]
      unpow2 [=>] 34.0	\[ 0.25 \cdot \frac{\color{blue}{D \cdot D}}{\frac{{d}^{2}}{h \cdot {M}^{2}}} \]
      unpow2 [=>] 34.0	\[ 0.25 \cdot \frac{D \cdot D}{\frac{\color{blue}{d \cdot d}}{h \cdot {M}^{2}}} \]
      *-commutative [=>] 34.0	\[ 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{\color{blue}{{M}^{2} \cdot h}}} \]
      unpow2 [=>] 34.0	\[ 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{\color{blue}{\left(M \cdot M\right)} \cdot h}} \]
      associate-*r* [<=] 32.0	\[ 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{\color{blue}{M \cdot \left(M \cdot h\right)}}} \]
      associate-/r/ [=>] 31.9	\[ 0.25 \cdot \color{blue}{\left(\frac{D \cdot D}{d \cdot d} \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)} \]
      associate-/r* [=>] 28.6	\[ 0.25 \cdot \left(\color{blue}{\frac{\frac{D \cdot D}{d}}{d}} \cdot \left(M \cdot \left(M \cdot h\right)\right)\right) \]
      associate-*r/ [<=] 23.7	\[ 0.25 \cdot \left(\frac{\color{blue}{D \cdot \frac{D}{d}}}{d} \cdot \left(M \cdot \left(M \cdot h\right)\right)\right) \]

   7. Applied egg-rr 22.2

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

   8. Applied egg-rr 14.7

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

4. Final simplification 16.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq -1 \cdot 10^{-175}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{d}{D}}{\left(h \cdot \frac{w}{c0}\right) \cdot \frac{D}{d}}\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 0:\\ \;\;\;\;0.25 \cdot \left(\frac{M}{\frac{d}{D} \cdot \frac{d}{D}} \cdot \left(h \cdot M\right)\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{\left(\frac{d}{D} \cdot \sqrt{2 \cdot \frac{c0}{w \cdot h}}\right) \cdot \frac{c0}{\frac{w}{\frac{d}{D} \cdot \sqrt{2 \cdot \frac{c0}{w \cdot h}}}}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{\frac{D}{d} \cdot \left(h \cdot M\right)}{\frac{\frac{d}{D}}{M}}\\ \end{array} \]

Alternatives

Alternative 1
Error	16.9
Cost	43148

\[\begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_2 := t_0 \cdot \left(t_1 + \sqrt{t_1 \cdot t_1 - M \cdot M}\right)\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{-175}:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \frac{\frac{d}{D}}{\left(h \cdot \frac{w}{c0}\right) \cdot \frac{D}{d}}\right)\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;0.25 \cdot \left(\frac{M}{\frac{d}{D} \cdot \frac{d}{D}} \cdot \left(h \cdot M\right)\right)\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\frac{\frac{c0 \cdot {\left(\frac{d}{D} \cdot \sqrt{2 \cdot \frac{c0}{w \cdot h}}\right)}^{2}}{w}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{\frac{D}{d} \cdot \left(h \cdot M\right)}{\frac{\frac{d}{D}}{M}}\\ \end{array} \]

Alternative 2
Error	16.6
Cost	30540

\[\begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := t_0 \cdot \left(2 \cdot \frac{\frac{d}{D}}{\left(h \cdot \frac{w}{c0}\right) \cdot \frac{D}{d}}\right)\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_3 := t_0 \cdot \left(t_2 + \sqrt{t_2 \cdot t_2 - M \cdot M}\right)\\ \mathbf{if}\;t_3 \leq -1 \cdot 10^{-175}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;0.25 \cdot \left(\frac{M}{\frac{d}{D} \cdot \frac{d}{D}} \cdot \left(h \cdot M\right)\right)\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{\frac{D}{d} \cdot \left(h \cdot M\right)}{\frac{\frac{d}{D}}{M}}\\ \end{array} \]

Alternative 3
Error	18.3
Cost	1740

\[\begin{array}{l} \mathbf{if}\;d \cdot d \leq 4 \cdot 10^{-204}:\\ \;\;\;\;0.25 \cdot \left(\frac{D}{d} \cdot \left(\left(h \cdot M\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)\right)\\ \mathbf{elif}\;d \cdot d \leq 5 \cdot 10^{+265}:\\ \;\;\;\;0.25 \cdot \frac{h \cdot \left(D \cdot M\right)}{d \cdot \frac{d}{D \cdot M}}\\ \mathbf{elif}\;d \cdot d \leq 5 \cdot 10^{+288}:\\ \;\;\;\;0.25 \cdot \left(\frac{D \cdot M}{d} \cdot \left(D \cdot \frac{M}{\frac{d}{h}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \frac{M \cdot \left(h \cdot D\right)}{d}\right)\\ \end{array} \]

Alternative 4
Error	20.2
Cost	1224

\[\begin{array}{l} \mathbf{if}\;D \leq -1.4 \cdot 10^{-247}:\\ \;\;\;\;0.25 \cdot \left(\frac{D}{d} \cdot \left(\left(h \cdot M\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)\right)\\ \mathbf{elif}\;D \leq 10^{-177}:\\ \;\;\;\;0.25 \cdot \left(\frac{D \cdot M}{d} \cdot \left(D \cdot \frac{M}{\frac{d}{h}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{M}{d} \cdot \left(D \cdot \frac{M \cdot \left(h \cdot D\right)}{d}\right)\right)\\ \end{array} \]

Alternative 5
Error	19.8
Cost	1092

\[\begin{array}{l} \mathbf{if}\;d \leq 2 \cdot 10^{+145}:\\ \;\;\;\;0.25 \cdot \left(\frac{D}{d} \cdot \left(\left(h \cdot M\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \frac{M \cdot \left(h \cdot D\right)}{d}\right)\\ \end{array} \]

Alternative 6
Error	19.2
Cost	1092

\[\begin{array}{l} \mathbf{if}\;d \leq 1.8 \cdot 10^{+145}:\\ \;\;\;\;0.25 \cdot \frac{M \cdot \frac{D}{d}}{\frac{\frac{d}{D}}{h \cdot M}}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \frac{M \cdot \left(h \cdot D\right)}{d}\right)\\ \end{array} \]

Alternative 7
Error	19.2
Cost	1092

\[\begin{array}{l} \mathbf{if}\;d \leq 1.8 \cdot 10^{+144}:\\ \;\;\;\;0.25 \cdot \frac{\frac{D}{d} \cdot \left(h \cdot M\right)}{\frac{\frac{d}{D}}{M}}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \frac{M \cdot \left(h \cdot D\right)}{d}\right)\\ \end{array} \]

Alternative 8
Error	19.9
Cost	960

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

Alternative 9
Error	32.1
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023047 
(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))))))