Henrywood and Agarwal, Equation (13)

Average Error: 92.92% → 23.48%

Time: 35.9s

Precision: binary64

Cost: 30540

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\frac{d}{D}}{M}\\ t_1 := \frac{\frac{\frac{c0 \cdot d}{w}}{D} \cdot \frac{2}{h}}{\frac{D}{d} \cdot \left(2 \cdot \frac{w}{c0}\right)}\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_3 := \frac{c0}{2 \cdot w} \cdot \left(t_2 + \sqrt{t_2 \cdot t_2 - M \cdot M}\right)\\ \mathbf{if}\;t_3 \leq -4 \cdot 10^{-152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;0.25 \cdot \frac{M \cdot \frac{h \cdot D}{d}}{t_0}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{\frac{D}{d} \cdot \left(h \cdot M\right)}{t_0}\\ \end{array} \]

FPCore

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

↓

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (/ d D) M))
        (t_1
         (/ (* (/ (/ (* c0 d) w) D) (/ 2.0 h)) (* (/ D d) (* 2.0 (/ w c0)))))
        (t_2 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_3 (* (/ c0 (* 2.0 w)) (+ t_2 (sqrt (- (* t_2 t_2) (* M M)))))))
   (if (<= t_3 -4e-152)
     t_1
     (if (<= t_3 0.0)
       (* 0.25 (/ (* M (/ (* h D) d)) t_0))
       (if (<= t_3 INFINITY) t_1 (* 0.25 (/ (* (/ D d) (* h M)) t_0)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return (c0 / (2.0 * w)) * (((c0 * (d * d)) / ((w * h) * (D * D))) + sqrt(((((c0 * (d * d)) / ((w * h) * (D * D))) * ((c0 * (d * d)) / ((w * h) * (D * D)))) - (M * M))));
}

↓

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (d / D) / M;
	double t_1 = ((((c0 * d) / w) / D) * (2.0 / h)) / ((D / d) * (2.0 * (w / c0)));
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_3 = (c0 / (2.0 * w)) * (t_2 + sqrt(((t_2 * t_2) - (M * M))));
	double tmp;
	if (t_3 <= -4e-152) {
		tmp = t_1;
	} else if (t_3 <= 0.0) {
		tmp = 0.25 * ((M * ((h * D) / d)) / t_0);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = 0.25 * (((D / d) * (h * M)) / t_0);
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return (c0 / (2.0 * w)) * (((c0 * (d * d)) / ((w * h) * (D * D))) + Math.sqrt(((((c0 * (d * d)) / ((w * h) * (D * D))) * ((c0 * (d * d)) / ((w * h) * (D * D)))) - (M * M))));
}

↓

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (d / D) / M;
	double t_1 = ((((c0 * d) / w) / D) * (2.0 / h)) / ((D / d) * (2.0 * (w / c0)));
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_3 = (c0 / (2.0 * w)) * (t_2 + Math.sqrt(((t_2 * t_2) - (M * M))));
	double tmp;
	if (t_3 <= -4e-152) {
		tmp = t_1;
	} else if (t_3 <= 0.0) {
		tmp = 0.25 * ((M * ((h * D) / d)) / t_0);
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = 0.25 * (((D / d) * (h * M)) / t_0);
	}
	return tmp;
}

Python

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

↓

def code(c0, w, h, D, d, M):
	t_0 = (d / D) / M
	t_1 = ((((c0 * d) / w) / D) * (2.0 / h)) / ((D / d) * (2.0 * (w / c0)))
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D))
	t_3 = (c0 / (2.0 * w)) * (t_2 + math.sqrt(((t_2 * t_2) - (M * M))))
	tmp = 0
	if t_3 <= -4e-152:
		tmp = t_1
	elif t_3 <= 0.0:
		tmp = 0.25 * ((M * ((h * D) / d)) / t_0)
	elif t_3 <= math.inf:
		tmp = t_1
	else:
		tmp = 0.25 * (((D / d) * (h * M)) / t_0)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	return Float64(Float64(c0 / Float64(2.0 * w)) * Float64(Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D))) + sqrt(Float64(Float64(Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D))) * Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))) - Float64(M * M)))))
end

↓

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(d / D) / M)
	t_1 = Float64(Float64(Float64(Float64(Float64(c0 * d) / w) / D) * Float64(2.0 / h)) / Float64(Float64(D / d) * Float64(2.0 * Float64(w / c0))))
	t_2 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	t_3 = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M)))))
	tmp = 0.0
	if (t_3 <= -4e-152)
		tmp = t_1;
	elseif (t_3 <= 0.0)
		tmp = Float64(0.25 * Float64(Float64(M * Float64(Float64(h * D) / d)) / t_0));
	elseif (t_3 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(0.25 * Float64(Float64(Float64(D / d) * Float64(h * M)) / t_0));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (d / D) / M;
	t_1 = ((((c0 * d) / w) / D) * (2.0 / h)) / ((D / d) * (2.0 * (w / c0)));
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	t_3 = (c0 / (2.0 * w)) * (t_2 + sqrt(((t_2 * t_2) - (M * M))));
	tmp = 0.0;
	if (t_3 <= -4e-152)
		tmp = t_1;
	elseif (t_3 <= 0.0)
		tmp = 0.25 * ((M * ((h * D) / d)) / t_0);
	elseif (t_3 <= Inf)
		tmp = t_1;
	else
		tmp = 0.25 * (((D / d) * (h * M)) / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(d / D), $MachinePrecision] / M), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(c0 * d), $MachinePrecision] / w), $MachinePrecision] / D), $MachinePrecision] * N[(2.0 / h), $MachinePrecision]), $MachinePrecision] / N[(N[(D / d), $MachinePrecision] * N[(2.0 * N[(w / c0), $MachinePrecision]), $MachinePrecision]), $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[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * 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, -4e-152], t$95$1, If[LessEqual[t$95$3, 0.0], N[(0.25 * N[(N[(M * N[(N[(h * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$1, N[(0.25 * N[(N[(N[(D / d), $MachinePrecision] * N[(h * M), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]]]

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))))) < -4.00000000000000026e-152 or 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 76.26

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

   2. Simplified 70.78

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

      [Start] 76.26	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      times-frac [=>] 78.72	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      fma-def [=>] 78.7	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d \cdot d}{D \cdot D}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)} \]
      associate-/r* [=>] 78.71	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\color{blue}{\frac{\frac{c0}{w}}{h}}, \frac{d \cdot d}{D \cdot D}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      times-frac [=>] 78.69	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \color{blue}{\frac{d}{D} \cdot \frac{d}{D}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      difference-of-squares [=>] 78.7	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \sqrt{\color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + M\right) \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M\right)}}\right) \]
      sub-neg [=>] 78.7	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \sqrt{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + M\right) \cdot \color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \left(-M\right)\right)}}\right) \]

   3. Taylor expanded in c0 around inf 67.11

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

   4. Simplified 63.42

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

      [Start] 67.11	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]
      times-frac [=>] 68.42	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{{d}^{2}}{{D}^{2}} \cdot \frac{c0}{w \cdot h}\right)}\right) \]
      associate-*l/ [=>] 67.04	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{{d}^{2} \cdot \frac{c0}{w \cdot h}}{{D}^{2}}}\right) \]
      unpow2 [=>] 67.04	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\color{blue}{\left(d \cdot d\right)} \cdot \frac{c0}{w \cdot h}}{{D}^{2}}\right) \]
      unpow2 [=>] 67.04	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\left(d \cdot d\right) \cdot \frac{c0}{w \cdot h}}{\color{blue}{D \cdot D}}\right) \]
      times-frac [=>] 56.52	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{d \cdot d}{D} \cdot \frac{\frac{c0}{w \cdot h}}{D}\right)}\right) \]
      associate-*r/ [=>] 57.01	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{\frac{d \cdot d}{D} \cdot \frac{c0}{w \cdot h}}{D}}\right) \]
      associate-*l/ [<=] 63.46	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{\frac{d \cdot d}{D}}{D} \cdot \frac{c0}{w \cdot h}\right)}\right) \]
      associate-*l/ [<=] 59.97	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\color{blue}{\frac{d}{D} \cdot d}}{D} \cdot \frac{c0}{w \cdot h}\right)\right) \]
      associate-*r/ [<=] 58.33	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D}\right)} \cdot \frac{c0}{w \cdot h}\right)\right) \]
      unpow2 [<=] 58.33	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\color{blue}{{\left(\frac{d}{D}\right)}^{2}} \cdot \frac{c0}{w \cdot h}\right)\right) \]
      associate-*r/ [=>] 60.5	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{{\left(\frac{d}{D}\right)}^{2} \cdot c0}{w \cdot h}}\right) \]
      associate-/l* [=>] 58.2	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{{\left(\frac{d}{D}\right)}^{2}}{\frac{w \cdot h}{c0}}}\right) \]
      associate-/r/ [=>] 63.42	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h} \cdot c0\right)}\right) \]

   5. Applied egg-rr 57.28

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

   6. Applied egg-rr 24.08

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

   7. Simplified 26.21

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

      [Start] 24.08	\[ \frac{\frac{2}{\frac{h}{\frac{d}{D} \cdot \frac{c0}{w}}}}{\left(2 \cdot \frac{w}{c0}\right) \cdot \frac{D}{d}} \]
      associate-/r/ [=>] 24.08	\[ \frac{\color{blue}{\frac{2}{h} \cdot \left(\frac{d}{D} \cdot \frac{c0}{w}\right)}}{\left(2 \cdot \frac{w}{c0}\right) \cdot \frac{D}{d}} \]
      *-commutative [=>] 24.08	\[ \frac{\color{blue}{\left(\frac{d}{D} \cdot \frac{c0}{w}\right) \cdot \frac{2}{h}}}{\left(2 \cdot \frac{w}{c0}\right) \cdot \frac{D}{d}} \]
      associate-*l/ [=>] 24.96	\[ \frac{\color{blue}{\frac{d \cdot \frac{c0}{w}}{D}} \cdot \frac{2}{h}}{\left(2 \cdot \frac{w}{c0}\right) \cdot \frac{D}{d}} \]
      associate-*r/ [=>] 26.21	\[ \frac{\frac{\color{blue}{\frac{d \cdot c0}{w}}}{D} \cdot \frac{2}{h}}{\left(2 \cdot \frac{w}{c0}\right) \cdot \frac{D}{d}} \]
      *-commutative [=>] 26.21	\[ \frac{\frac{\frac{d \cdot c0}{w}}{D} \cdot \frac{2}{h}}{\color{blue}{\frac{D}{d} \cdot \left(2 \cdot \frac{w}{c0}\right)}} \]

   if -4.00000000000000026e-152 < (*.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 44.26

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

   2. Simplified 50.46

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

      [Start] 44.26	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      times-frac [=>] 62	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      fma-neg [=>] 62	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\color{blue}{\mathsf{fma}\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, -M \cdot M\right)}}\right) \]
      times-frac [=>] 57.15	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\mathsf{fma}\left(\color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}}, \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, -M \cdot M\right)}\right) \]
      times-frac [=>] 50.46	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}, \color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}}, -M \cdot M\right)}\right) \]

   3. Taylor expanded in c0 around -inf 48.12

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

   4. Simplified 48.71

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

      [Start] 48.12	\[ \frac{c0}{2 \cdot w} \cdot \left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0} + -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right) \]
      fma-def [=>] 48.12	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0}, -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right)} \]

   5. Taylor expanded in c0 around 0 40.76

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

   6. Simplified 36.68

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

      [Start] 40.76	\[ 0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}} \]
      *-commutative [=>] 40.76	\[ 0.25 \cdot \frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{{d}^{2}} \]
      associate-/l* [=>] 41.78	\[ 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{{M}^{2} \cdot h}}} \]
      unpow2 [=>] 41.78	\[ 0.25 \cdot \frac{{D}^{2}}{\frac{{d}^{2}}{\color{blue}{\left(M \cdot M\right)} \cdot h}} \]
      *-commutative [<=] 41.78	\[ 0.25 \cdot \frac{{D}^{2}}{\frac{{d}^{2}}{\color{blue}{h \cdot \left(M \cdot M\right)}}} \]
      associate-/r/ [=>] 41.11	\[ 0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)} \]
      unpow2 [=>] 41.11	\[ 0.25 \cdot \left(\frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]
      unpow2 [=>] 41.11	\[ 0.25 \cdot \left(\frac{D \cdot D}{\color{blue}{d \cdot d}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]
      times-frac [=>] 36.68	\[ 0.25 \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

   7. Applied egg-rr 23.51

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

   8. Applied egg-rr 22.21

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

   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 100

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

   2. Simplified 98.91

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

      [Start] 100	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      times-frac [=>] 100	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      fma-neg [=>] 100	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\color{blue}{\mathsf{fma}\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, -M \cdot M\right)}}\right) \]
      times-frac [=>] 100	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\mathsf{fma}\left(\color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}}, \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, -M \cdot M\right)}\right) \]
      times-frac [=>] 98.91	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}, \color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}}, -M \cdot M\right)}\right) \]

   3. Taylor expanded in c0 around -inf 98.35

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

   4. Simplified 89.96

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

      [Start] 98.35	\[ \frac{c0}{2 \cdot w} \cdot \left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0} + -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right) \]
      fma-def [=>] 98.35	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0}, -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right)} \]

   5. Taylor expanded in c0 around 0 52.7

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

   6. Simplified 40.22

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

      [Start] 52.7	\[ 0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}} \]
      *-commutative [=>] 52.7	\[ 0.25 \cdot \frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{{d}^{2}} \]
      associate-/l* [=>] 52.96	\[ 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{{M}^{2} \cdot h}}} \]
      unpow2 [=>] 52.96	\[ 0.25 \cdot \frac{{D}^{2}}{\frac{{d}^{2}}{\color{blue}{\left(M \cdot M\right)} \cdot h}} \]
      *-commutative [<=] 52.96	\[ 0.25 \cdot \frac{{D}^{2}}{\frac{{d}^{2}}{\color{blue}{h \cdot \left(M \cdot M\right)}}} \]
      associate-/r/ [=>] 52.94	\[ 0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)} \]
      unpow2 [=>] 52.94	\[ 0.25 \cdot \left(\frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]
      unpow2 [=>] 52.94	\[ 0.25 \cdot \left(\frac{D \cdot D}{\color{blue}{d \cdot d}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]
      times-frac [=>] 40.22	\[ 0.25 \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

   7. Applied egg-rr 23.15

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

4. Final simplification 23.48

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq -4 \cdot 10^{-152}:\\ \;\;\;\;\frac{\frac{\frac{c0 \cdot d}{w}}{D} \cdot \frac{2}{h}}{\frac{D}{d} \cdot \left(2 \cdot \frac{w}{c0}\right)}\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 0:\\ \;\;\;\;0.25 \cdot \frac{M \cdot \frac{h \cdot D}{d}}{\frac{\frac{d}{D}}{M}}\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{\frac{\frac{c0 \cdot d}{w}}{D} \cdot \frac{2}{h}}{\frac{D}{d} \cdot \left(2 \cdot \frac{w}{c0}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{\frac{D}{d} \cdot \left(h \cdot M\right)}{\frac{\frac{d}{D}}{M}}\\ \end{array} \]

Alternatives

Alternative 1
Error	32.39%
Cost	2384

\[\begin{array}{l} t_0 := \frac{\frac{d}{D}}{M}\\ t_1 := \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{d \cdot \frac{d}{D}}{D \cdot \frac{w \cdot h}{c0}}\right)\\ t_2 := 0.25 \cdot \frac{\frac{D}{d} \cdot \left(h \cdot M\right)}{t_0}\\ \mathbf{if}\;d \cdot d \leq 4 \cdot 10^{-250}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \cdot d \leq 10^{-216}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \cdot d \leq 10^{-110}:\\ \;\;\;\;0.25 \cdot \frac{M \cdot \frac{h \cdot D}{d}}{t_0}\\ \mathbf{elif}\;d \cdot d \leq 2 \cdot 10^{-63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \cdot d \leq 10^{+200}:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot M\right) \cdot \frac{\frac{h \cdot \left(D \cdot M\right)}{d}}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	32.37%
Cost	2384

\[\begin{array}{l} t_0 := \frac{\frac{d}{D}}{M}\\ t_1 := \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{d}{D}}{\frac{D}{d} \cdot \frac{w \cdot h}{c0}}\right)\\ t_2 := 0.25 \cdot \frac{\frac{D}{d} \cdot \left(h \cdot M\right)}{t_0}\\ \mathbf{if}\;d \cdot d \leq 4 \cdot 10^{-250}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \cdot d \leq 10^{-216}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \cdot d \leq 10^{-110}:\\ \;\;\;\;0.25 \cdot \frac{M \cdot \frac{h \cdot D}{d}}{t_0}\\ \mathbf{elif}\;d \cdot d \leq 2 \cdot 10^{-63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \cdot d \leq 10^{+200}:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot M\right) \cdot \frac{\frac{h \cdot \left(D \cdot M\right)}{d}}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	31.97%
Cost	2384

\[\begin{array}{l} t_0 := \frac{\frac{d}{D}}{M}\\ t_1 := \frac{2}{\left(\frac{D}{d} \cdot \left(2 \cdot \frac{w}{c0}\right)\right) \cdot \frac{h}{\frac{d}{D} \cdot \frac{c0}{w}}}\\ t_2 := 0.25 \cdot \frac{\frac{D}{d} \cdot \left(h \cdot M\right)}{t_0}\\ \mathbf{if}\;d \cdot d \leq 4 \cdot 10^{-250}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \cdot d \leq 4 \cdot 10^{-213}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \cdot d \leq 10^{-110}:\\ \;\;\;\;0.25 \cdot \frac{M \cdot \frac{h \cdot D}{d}}{t_0}\\ \mathbf{elif}\;d \cdot d \leq 2 \cdot 10^{-63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \cdot d \leq 10^{+200}:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot M\right) \cdot \frac{\frac{h \cdot \left(D \cdot M\right)}{d}}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	45.23%
Cost	1225

\[\begin{array}{l} \mathbf{if}\;d \leq -7.6 \cdot 10^{-93} \lor \neg \left(d \leq 4.1 \cdot 10^{-162}\right):\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \left(h \cdot \frac{M}{\frac{d \cdot d}{M}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5
Error	44.2%
Cost	1225

\[\begin{array}{l} \mathbf{if}\;M \leq -5 \cdot 10^{+107} \lor \neg \left(M \leq 1.9 \cdot 10^{+25}\right):\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \left(h \cdot \frac{M}{\frac{d \cdot d}{M}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \left(\frac{h}{d} \cdot \frac{M \cdot M}{d}\right)\right)\right)\\ \end{array} \]

Alternative 6
Error	41.65%
Cost	1225

\[\begin{array}{l} \mathbf{if}\;M \leq -5.4 \cdot 10^{+118} \lor \neg \left(M \leq 2 \cdot 10^{+19}\right):\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \frac{h \cdot M}{d \cdot \frac{d}{M}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \left(\frac{h}{d} \cdot \frac{M \cdot M}{d}\right)\right)\right)\\ \end{array} \]

Alternative 7
Error	40.61%
Cost	1225

\[\begin{array}{l} \mathbf{if}\;M \leq -1.9 \cdot 10^{+119} \lor \neg \left(M \leq 2.6 \cdot 10^{+226}\right):\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \frac{h \cdot M}{d \cdot \frac{d}{M}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \frac{M \cdot \frac{M}{d}}{\frac{d}{h}}\right)\right)\\ \end{array} \]

Alternative 8
Error	39.64%
Cost	1224

\[\begin{array}{l} \mathbf{if}\;d \leq -1.05 \cdot 10^{-183}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \frac{h \cdot M}{d \cdot \frac{d}{M}}\right)\right)\\ \mathbf{elif}\;d \leq 9.5 \cdot 10^{-248}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \frac{M \cdot \frac{M}{d}}{\frac{d}{h}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \frac{h \cdot D}{\frac{d}{M} \cdot \frac{d}{M}}\right)\\ \end{array} \]

Alternative 9
Error	34.04%
Cost	1224

\[\begin{array}{l} \mathbf{if}\;d \leq 2 \cdot 10^{+104}:\\ \;\;\;\;0.25 \cdot \left(\frac{D}{d} \cdot \left(\left(h \cdot M\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{+215}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \frac{h \cdot D}{\frac{d}{M} \cdot \frac{d}{M}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \frac{M \cdot \frac{M}{d}}{\frac{d}{h}}\right)\right)\\ \end{array} \]

Alternative 10
Error	34.25%
Cost	1224

\[\begin{array}{l} t_0 := D \cdot \frac{M}{d}\\ \mathbf{if}\;d \leq 9 \cdot 10^{+27}:\\ \;\;\;\;0.25 \cdot \left(\frac{D}{d} \cdot \left(\left(h \cdot M\right) \cdot t_0\right)\right)\\ \mathbf{elif}\;d \leq 10^{+215}:\\ \;\;\;\;0.25 \cdot \left(\frac{h \cdot D}{d} \cdot \left(M \cdot t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \frac{M \cdot \frac{M}{d}}{\frac{d}{h}}\right)\right)\\ \end{array} \]

Alternative 11
Error	32.73%
Cost	1224

\[\begin{array}{l} \mathbf{if}\;d \leq -1.6 \cdot 10^{+26}:\\ \;\;\;\;0.25 \cdot \left(\frac{D}{d} \cdot \left(\left(h \cdot M\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{+101}:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot M\right) \cdot \frac{\frac{h \cdot \left(D \cdot M\right)}{d}}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \frac{h \cdot D}{\frac{d}{M} \cdot \frac{d}{M}}\right)\\ \end{array} \]

Alternative 12
Error	32.83%
Cost	1224

\[\begin{array}{l} \mathbf{if}\;d \leq -6 \cdot 10^{+25}:\\ \;\;\;\;0.25 \cdot \left(\frac{D}{d} \cdot \left(\left(h \cdot M\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\\ \mathbf{elif}\;d \leq 10^{+195}:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot M\right) \cdot \frac{\frac{h \cdot \left(D \cdot M\right)}{d}}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \frac{M}{\frac{d}{h}}}{\frac{\frac{d}{D}}{M}}\\ \end{array} \]

Alternative 13
Error	30.75%
Cost	1224

\[\begin{array}{l} t_0 := \frac{\frac{d}{D}}{M}\\ \mathbf{if}\;d \leq 2.2 \cdot 10^{-120}:\\ \;\;\;\;0.25 \cdot \frac{\frac{D}{d} \cdot \left(h \cdot M\right)}{t_0}\\ \mathbf{elif}\;d \leq 10^{+195}:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot M\right) \cdot \frac{\frac{h \cdot \left(D \cdot M\right)}{d}}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \frac{M}{\frac{d}{h}}}{t_0}\\ \end{array} \]

Alternative 14
Error	37.5%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;d \leq 2.95 \cdot 10^{-120}:\\ \;\;\;\;0.25 \cdot \left(M \cdot \left(\left(h \cdot M\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \frac{h \cdot D}{\frac{d}{M} \cdot \frac{d}{M}}\right)\\ \end{array} \]

Alternative 15
Error	51.23%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023115 
(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))))))