Henrywood and Agarwal, Equation (13)

Average Error: 59.8 → 20.9

Time: 27.1s

Precision: binary64

Cost: 20880

Math

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

↓

\[\begin{array}{l} t_0 := D \cdot \sqrt{h}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := 0.25 \cdot \left(\frac{h \cdot D}{\frac{d}{M}} \cdot \frac{D}{\frac{d}{M}}\right)\\ \mathbf{if}\;h \leq -7 \cdot 10^{+31}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;h \leq -4.4 \cdot 10^{-33}:\\ \;\;\;\;t_1 \cdot \mathsf{fma}\left(\frac{d}{D}, \frac{d}{D} \cdot \frac{\frac{c0}{h}}{w}, \mathsf{hypot}\left(\frac{\frac{d}{D} \cdot \frac{c0}{w \cdot h}}{\frac{D}{d}}, M\right)\right)\\ \mathbf{elif}\;h \leq 1.3 \cdot 10^{+115}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;h \leq 4.5 \cdot 10^{+146}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot {\left(\frac{d}{t_0} \cdot \sqrt{\frac{c0}{w}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot {\left(\frac{t_0}{\frac{d}{M}}\right)}^{2}\\ \end{array} \]

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 (sqrt h)))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (* 0.25 (* (/ (* h D) (/ d M)) (/ D (/ d M))))))
   (if (<= h -7e+31)
     t_2
     (if (<= h -4.4e-33)
       (*
        t_1
        (fma
         (/ d D)
         (* (/ d D) (/ (/ c0 h) w))
         (hypot (/ (* (/ d D) (/ c0 (* w h))) (/ D d)) M)))
       (if (<= h 1.3e+115)
         t_2
         (if (<= h 4.5e+146)
           (* t_1 (* 2.0 (pow (* (/ d t_0) (sqrt (/ c0 w))) 2.0)))
           (* 0.25 (pow (/ t_0 (/ d M)) 2.0))))))))

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 * sqrt(h);
	double t_1 = c0 / (2.0 * w);
	double t_2 = 0.25 * (((h * D) / (d / M)) * (D / (d / M)));
	double tmp;
	if (h <= -7e+31) {
		tmp = t_2;
	} else if (h <= -4.4e-33) {
		tmp = t_1 * fma((d / D), ((d / D) * ((c0 / h) / w)), hypot((((d / D) * (c0 / (w * h))) / (D / d)), M));
	} else if (h <= 1.3e+115) {
		tmp = t_2;
	} else if (h <= 4.5e+146) {
		tmp = t_1 * (2.0 * pow(((d / t_0) * sqrt((c0 / w))), 2.0));
	} else {
		tmp = 0.25 * pow((t_0 / (d / M)), 2.0);
	}
	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(D * sqrt(h))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(0.25 * Float64(Float64(Float64(h * D) / Float64(d / M)) * Float64(D / Float64(d / M))))
	tmp = 0.0
	if (h <= -7e+31)
		tmp = t_2;
	elseif (h <= -4.4e-33)
		tmp = Float64(t_1 * fma(Float64(d / D), Float64(Float64(d / D) * Float64(Float64(c0 / h) / w)), hypot(Float64(Float64(Float64(d / D) * Float64(c0 / Float64(w * h))) / Float64(D / d)), M)));
	elseif (h <= 1.3e+115)
		tmp = t_2;
	elseif (h <= 4.5e+146)
		tmp = Float64(t_1 * Float64(2.0 * (Float64(Float64(d / t_0) * sqrt(Float64(c0 / w))) ^ 2.0)));
	else
		tmp = Float64(0.25 * (Float64(t_0 / Float64(d / M)) ^ 2.0));
	end
	return 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[(D * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.25 * N[(N[(N[(h * D), $MachinePrecision] / N[(d / M), $MachinePrecision]), $MachinePrecision] * N[(D / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -7e+31], t$95$2, If[LessEqual[h, -4.4e-33], N[(t$95$1 * N[(N[(d / D), $MachinePrecision] * N[(N[(d / D), $MachinePrecision] * N[(N[(c0 / h), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[(N[(d / D), $MachinePrecision] * N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(D / d), $MachinePrecision]), $MachinePrecision] ^ 2 + M ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 1.3e+115], t$95$2, If[LessEqual[h, 4.5e+146], N[(t$95$1 * N[(2.0 * N[Power[N[(N[(d / t$95$0), $MachinePrecision] * N[Sqrt[N[(c0 / w), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[Power[N[(t$95$0 / N[(d / M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if h < -7e31 or -4.40000000000000011e-33 < h < 1.3e115

   1. Initial program 59.8

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

   2. Simplified 62.1

      \[\leadsto \color{blue}{\frac{\frac{c0}{2}}{w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{4}, M \cdot \left(-M\right)\right)}\right)} \]
      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)))))): 2 points increase in error, 4 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)))))): 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)) (*.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, 1 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (fma.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 d d) (*.f64 D D))) (/.f64 d D)) (/.f64 d D))) (*.f64 M (neg.f64 M)))))): 2 points increase in error, 1 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (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)))))): 1 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (fma.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 (*.f64 d d) (*.f64 D D)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 d d) (*.f64 D D))))) (*.f64 M (neg.f64 M)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (fma.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 (*.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))))): 4 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, 5 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 (/.f64 c0 (*.f64 w h)) (/.f64 (*.f64 d d) (*.f64 D D)))) (*.f64 M M))))): 5 points increase in error, 1 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))))) (*.f64 M M))))): 1 points increase in error, 5 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)))))): 1 points increase in error, 6 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (*.f64 d d))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))): 2 points increase in error, 2 points decrease in error

   3. Taylor expanded in c0 around -inf 59.9

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

   4. Simplified 36.0

      \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{\frac{w \cdot \left(h \cdot \left(M \cdot M\right)\right)}{{\left(\frac{d}{D}\right)}^{2}}}{c0}, c0 \cdot 0\right)} \]
      Proof
      (fma.f64 1/2 (/.f64 (/.f64 (*.f64 w (*.f64 h (*.f64 M M))) (pow.f64 (/.f64 d D) 2)) c0) (*.f64 c0 0)): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (/.f64 (*.f64 w (*.f64 h (Rewrite<= unpow2_binary64 (pow.f64 M 2)))) (pow.f64 (/.f64 d D) 2)) c0) (*.f64 c0 0)): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (/.f64 (*.f64 w (*.f64 h (pow.f64 M 2))) (Rewrite=> unpow2_binary64 (*.f64 (/.f64 d D) (/.f64 d D)))) c0) (*.f64 c0 0)): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (/.f64 (*.f64 w (*.f64 h (pow.f64 M 2))) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 d d) (*.f64 D D)))) c0) (*.f64 c0 0)): 45 points increase in error, 7 points decrease in error
      (fma.f64 1/2 (/.f64 (/.f64 (*.f64 w (*.f64 h (pow.f64 M 2))) (/.f64 (*.f64 d d) (Rewrite<= unpow2_binary64 (pow.f64 D 2)))) c0) (*.f64 c0 0)): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 w (*.f64 h (pow.f64 M 2))) (pow.f64 D 2)) (*.f64 d d))) c0) (*.f64 c0 0)): 6 points increase in error, 2 points decrease in error
      (fma.f64 1/2 (/.f64 (/.f64 (*.f64 (*.f64 w (*.f64 h (Rewrite=> unpow2_binary64 (*.f64 M M)))) (pow.f64 D 2)) (*.f64 d d)) c0) (*.f64 c0 0)): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (/.f64 (*.f64 (*.f64 w (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 M M) h))) (pow.f64 D 2)) (*.f64 d d)) c0) (*.f64 c0 0)): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (/.f64 (*.f64 (*.f64 w (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 M 2)) h)) (pow.f64 D 2)) (*.f64 d d)) c0) (*.f64 c0 0)): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 D 2) (*.f64 w (*.f64 (pow.f64 M 2) h)))) (*.f64 d d)) c0) (*.f64 c0 0)): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (Rewrite<= associate-/r*_binary64 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 w (*.f64 (pow.f64 M 2) h))) (*.f64 (*.f64 d d) c0))) (*.f64 c0 0)): 8 points increase in error, 1 points decrease in error
      (fma.f64 1/2 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 w (*.f64 (pow.f64 M 2) h))) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 d 2)) c0)) (*.f64 c0 0)): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 w (*.f64 (pow.f64 M 2) h))) (*.f64 (pow.f64 d 2) c0)) (*.f64 c0 (Rewrite<= metadata-eval (neg.f64 0)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 w (*.f64 (pow.f64 M 2) h))) (*.f64 (pow.f64 d 2) c0)) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 c0 0)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 w (*.f64 (pow.f64 M 2) h))) (*.f64 (pow.f64 d 2) c0)) (neg.f64 (*.f64 c0 (Rewrite<= mul0-lft_binary64 (*.f64 0 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h)))))))): 125 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 w (*.f64 (pow.f64 M 2) h))) (*.f64 (pow.f64 d 2) c0)) (neg.f64 (*.f64 c0 (*.f64 (Rewrite<= metadata-eval (+.f64 -1 1)) (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h))))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 w (*.f64 (pow.f64 M 2) h))) (*.f64 (pow.f64 d 2) c0)) (neg.f64 (*.f64 c0 (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))))))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 w (*.f64 (pow.f64 M 2) h))) (*.f64 (pow.f64 d 2) c0)) (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h))) (*.f64 -1 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h))))) c0)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 w (*.f64 (pow.f64 M 2) h))) (*.f64 (pow.f64 d 2) c0)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 (+.f64 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h))) (*.f64 -1 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h))))) c0)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 1/2 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 w (*.f64 (pow.f64 M 2) h))) (*.f64 (pow.f64 d 2) c0))) (*.f64 -1 (*.f64 (+.f64 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h))) (*.f64 -1 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h))))) c0)))): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in c0 around 0 35.5

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

   6. Simplified 26.6

      \[\leadsto \color{blue}{0.25 \cdot \left(\frac{D \cdot \left(D \cdot h\right)}{d} \cdot \frac{M}{\frac{d}{M}}\right)} \]
      Proof
      (*.f64 1/4 (*.f64 (/.f64 (*.f64 D (*.f64 D h)) d) (/.f64 M (/.f64 d M)))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/4 (*.f64 (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 D D) h)) d) (/.f64 M (/.f64 d M)))): 24 points increase in error, 1 points decrease in error
      (*.f64 1/4 (*.f64 (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 D 2)) h) d) (/.f64 M (/.f64 d M)))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/4 (*.f64 (/.f64 (*.f64 (pow.f64 D 2) h) d) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 M M) d)))): 21 points increase in error, 0 points decrease in error
      (*.f64 1/4 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (*.f64 (pow.f64 D 2) h) (*.f64 M M)) (*.f64 d d)))): 31 points increase in error, 5 points decrease in error
      (*.f64 1/4 (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (pow.f64 D 2) (*.f64 h (*.f64 M M)))) (*.f64 d d))): 9 points increase in error, 9 points decrease in error
      (*.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 h (Rewrite<= unpow2_binary64 (pow.f64 M 2)))) (*.f64 d d))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 h (pow.f64 M 2))) (Rewrite<= unpow2_binary64 (pow.f64 d 2)))): 0 points increase in error, 0 points decrease in error

   7. Applied egg-rr 31.1

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

   8. Applied egg-rr 18.5

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

   if -7e31 < h < -4.40000000000000011e-33

   1. Initial program 59.3

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

   2. Simplified 59.1

      \[\leadsto \color{blue}{\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}, \frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}, -M \cdot M\right)}\right)} \]
      Proof
      (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (*.f64 (/.f64 c0 (*.f64 w h)) (/.f64 (*.f64 d d) (*.f64 D D))) (sqrt.f64 (fma.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))) (neg.f64 (*.f64 M M)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (sqrt.f64 (fma.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))) (neg.f64 (*.f64 M M)))))): 9 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (fma.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))) (neg.f64 (*.f64 M M)))))): 0 points increase in error, 2 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (fma.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)))) (neg.f64 (*.f64 M M)))))): 2 points increase in error, 5 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (Rewrite<= fma-neg_binary64 (-.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

   3. Applied egg-rr 54.0

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

   4. Applied egg-rr 50.5

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

   if 1.3e115 < h < 4.50000000000000026e146

   1. Initial program 61.6

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

   2. Taylor expanded in c0 around inf 61.5

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

   3. Simplified 55.5

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d \cdot \frac{d}{D}}{D \cdot h}\right)\right)} \]
      Proof
      (*.f64 2 (*.f64 (/.f64 c0 w) (/.f64 (*.f64 d (/.f64 d D)) (*.f64 D h)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (/.f64 c0 w) (/.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 d d) D)) (*.f64 D h)))): 17 points increase in error, 12 points decrease in error
      (*.f64 2 (*.f64 (/.f64 c0 w) (/.f64 (/.f64 (*.f64 d d) D) (Rewrite<= *-commutative_binary64 (*.f64 h D))))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 c0 (/.f64 (*.f64 d d) D)) (*.f64 w (*.f64 h D))))): 15 points increase in error, 26 points decrease in error
      (*.f64 2 (/.f64 (*.f64 c0 (/.f64 (*.f64 d d) D)) (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 w h) D)))): 12 points increase in error, 15 points decrease in error
      (*.f64 2 (Rewrite=> times-frac_binary64 (*.f64 (/.f64 c0 (*.f64 w h)) (/.f64 (/.f64 (*.f64 d d) D) D)))): 24 points increase in error, 21 points decrease in error
      (*.f64 2 (*.f64 (/.f64 c0 (*.f64 w h)) (Rewrite<= associate-/r*_binary64 (/.f64 (*.f64 d d) (*.f64 D D))))): 24 points increase in error, 8 points decrease in error
      (*.f64 2 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))))): 13 points increase in error, 14 points decrease in error
      (*.f64 2 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 d d) c0)) (*.f64 (*.f64 w h) (*.f64 D D)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 d 2)) c0) (*.f64 (*.f64 w h) (*.f64 D D)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (*.f64 (pow.f64 d 2) c0) (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 D D) (*.f64 w h))))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (*.f64 (pow.f64 d 2) c0) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 D 2)) (*.f64 w h)))): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 52.4

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

   5. Simplified 51.5

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{{\left(\frac{d}{\sqrt{h} \cdot D} \cdot \sqrt{\frac{c0}{w}}\right)}^{2}}\right) \]
      Proof
      (pow.f64 (*.f64 (/.f64 d (*.f64 (sqrt.f64 h) D)) (sqrt.f64 (/.f64 c0 w))) 2): 0 points increase in error, 0 points decrease in error
      (pow.f64 (*.f64 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 d D) (sqrt.f64 h))) (sqrt.f64 (/.f64 c0 w))) 2): 7 points increase in error, 5 points decrease in error

   if 4.50000000000000026e146 < h

   1. Initial program 59.2

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

   2. Simplified 62.5

      \[\leadsto \color{blue}{\frac{\frac{c0}{2}}{w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{4}, M \cdot \left(-M\right)\right)}\right)} \]
      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)))))): 2 points increase in error, 4 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)))))): 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)) (*.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, 1 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (fma.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 d d) (*.f64 D D))) (/.f64 d D)) (/.f64 d D))) (*.f64 M (neg.f64 M)))))): 2 points increase in error, 1 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (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)))))): 1 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (fma.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 (*.f64 d d) (*.f64 D D)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 d d) (*.f64 D D))))) (*.f64 M (neg.f64 M)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (fma.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 (*.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))))): 4 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, 5 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 (/.f64 c0 (*.f64 w h)) (/.f64 (*.f64 d d) (*.f64 D D)))) (*.f64 M M))))): 5 points increase in error, 1 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))))) (*.f64 M M))))): 1 points increase in error, 5 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)))))): 1 points increase in error, 6 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (*.f64 d d))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))): 2 points increase in error, 2 points decrease in error

   3. Taylor expanded in c0 around -inf 60.2

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

   4. Simplified 40.9

      \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{\frac{w \cdot \left(h \cdot \left(M \cdot M\right)\right)}{{\left(\frac{d}{D}\right)}^{2}}}{c0}, c0 \cdot 0\right)} \]
      Proof
      (fma.f64 1/2 (/.f64 (/.f64 (*.f64 w (*.f64 h (*.f64 M M))) (pow.f64 (/.f64 d D) 2)) c0) (*.f64 c0 0)): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (/.f64 (*.f64 w (*.f64 h (Rewrite<= unpow2_binary64 (pow.f64 M 2)))) (pow.f64 (/.f64 d D) 2)) c0) (*.f64 c0 0)): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (/.f64 (*.f64 w (*.f64 h (pow.f64 M 2))) (Rewrite=> unpow2_binary64 (*.f64 (/.f64 d D) (/.f64 d D)))) c0) (*.f64 c0 0)): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (/.f64 (*.f64 w (*.f64 h (pow.f64 M 2))) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 d d) (*.f64 D D)))) c0) (*.f64 c0 0)): 45 points increase in error, 7 points decrease in error
      (fma.f64 1/2 (/.f64 (/.f64 (*.f64 w (*.f64 h (pow.f64 M 2))) (/.f64 (*.f64 d d) (Rewrite<= unpow2_binary64 (pow.f64 D 2)))) c0) (*.f64 c0 0)): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 w (*.f64 h (pow.f64 M 2))) (pow.f64 D 2)) (*.f64 d d))) c0) (*.f64 c0 0)): 6 points increase in error, 2 points decrease in error
      (fma.f64 1/2 (/.f64 (/.f64 (*.f64 (*.f64 w (*.f64 h (Rewrite=> unpow2_binary64 (*.f64 M M)))) (pow.f64 D 2)) (*.f64 d d)) c0) (*.f64 c0 0)): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (/.f64 (*.f64 (*.f64 w (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 M M) h))) (pow.f64 D 2)) (*.f64 d d)) c0) (*.f64 c0 0)): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (/.f64 (*.f64 (*.f64 w (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 M 2)) h)) (pow.f64 D 2)) (*.f64 d d)) c0) (*.f64 c0 0)): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 D 2) (*.f64 w (*.f64 (pow.f64 M 2) h)))) (*.f64 d d)) c0) (*.f64 c0 0)): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (Rewrite<= associate-/r*_binary64 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 w (*.f64 (pow.f64 M 2) h))) (*.f64 (*.f64 d d) c0))) (*.f64 c0 0)): 8 points increase in error, 1 points decrease in error
      (fma.f64 1/2 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 w (*.f64 (pow.f64 M 2) h))) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 d 2)) c0)) (*.f64 c0 0)): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 w (*.f64 (pow.f64 M 2) h))) (*.f64 (pow.f64 d 2) c0)) (*.f64 c0 (Rewrite<= metadata-eval (neg.f64 0)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 w (*.f64 (pow.f64 M 2) h))) (*.f64 (pow.f64 d 2) c0)) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 c0 0)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 w (*.f64 (pow.f64 M 2) h))) (*.f64 (pow.f64 d 2) c0)) (neg.f64 (*.f64 c0 (Rewrite<= mul0-lft_binary64 (*.f64 0 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h)))))))): 125 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 w (*.f64 (pow.f64 M 2) h))) (*.f64 (pow.f64 d 2) c0)) (neg.f64 (*.f64 c0 (*.f64 (Rewrite<= metadata-eval (+.f64 -1 1)) (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h))))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 w (*.f64 (pow.f64 M 2) h))) (*.f64 (pow.f64 d 2) c0)) (neg.f64 (*.f64 c0 (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))))))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 w (*.f64 (pow.f64 M 2) h))) (*.f64 (pow.f64 d 2) c0)) (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h))) (*.f64 -1 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h))))) c0)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 w (*.f64 (pow.f64 M 2) h))) (*.f64 (pow.f64 d 2) c0)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 (+.f64 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h))) (*.f64 -1 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h))))) c0)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 1/2 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 w (*.f64 (pow.f64 M 2) h))) (*.f64 (pow.f64 d 2) c0))) (*.f64 -1 (*.f64 (+.f64 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h))) (*.f64 -1 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h))))) c0)))): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in c0 around 0 38.1

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

   6. Simplified 30.0

      \[\leadsto \color{blue}{0.25 \cdot \left(\frac{D \cdot \left(D \cdot h\right)}{d} \cdot \frac{M}{\frac{d}{M}}\right)} \]
      Proof
      (*.f64 1/4 (*.f64 (/.f64 (*.f64 D (*.f64 D h)) d) (/.f64 M (/.f64 d M)))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/4 (*.f64 (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 D D) h)) d) (/.f64 M (/.f64 d M)))): 24 points increase in error, 1 points decrease in error
      (*.f64 1/4 (*.f64 (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 D 2)) h) d) (/.f64 M (/.f64 d M)))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/4 (*.f64 (/.f64 (*.f64 (pow.f64 D 2) h) d) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 M M) d)))): 21 points increase in error, 0 points decrease in error
      (*.f64 1/4 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (*.f64 (pow.f64 D 2) h) (*.f64 M M)) (*.f64 d d)))): 31 points increase in error, 5 points decrease in error
      (*.f64 1/4 (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (pow.f64 D 2) (*.f64 h (*.f64 M M)))) (*.f64 d d))): 9 points increase in error, 9 points decrease in error
      (*.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 h (Rewrite<= unpow2_binary64 (pow.f64 M 2)))) (*.f64 d d))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 h (pow.f64 M 2))) (Rewrite<= unpow2_binary64 (pow.f64 d 2)))): 0 points increase in error, 0 points decrease in error

   7. Applied egg-rr 34.7

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

   8. Applied egg-rr 19.9

      \[\leadsto 0.25 \cdot \color{blue}{{\left(\frac{D \cdot \sqrt{h}}{\frac{d}{M}}\right)}^{2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 20.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -7 \cdot 10^{+31}:\\ \;\;\;\;0.25 \cdot \left(\frac{h \cdot D}{\frac{d}{M}} \cdot \frac{D}{\frac{d}{M}}\right)\\ \mathbf{elif}\;h \leq -4.4 \cdot 10^{-33}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{d}{D}, \frac{d}{D} \cdot \frac{\frac{c0}{h}}{w}, \mathsf{hypot}\left(\frac{\frac{d}{D} \cdot \frac{c0}{w \cdot h}}{\frac{D}{d}}, M\right)\right)\\ \mathbf{elif}\;h \leq 1.3 \cdot 10^{+115}:\\ \;\;\;\;0.25 \cdot \left(\frac{h \cdot D}{\frac{d}{M}} \cdot \frac{D}{\frac{d}{M}}\right)\\ \mathbf{elif}\;h \leq 4.5 \cdot 10^{+146}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot {\left(\frac{d}{D \cdot \sqrt{h}} \cdot \sqrt{\frac{c0}{w}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot {\left(\frac{D \cdot \sqrt{h}}{\frac{d}{M}}\right)}^{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	18.3
Cost	30212

\[\begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \sqrt[3]{\frac{d}{M}}\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\\ \mathbf{if}\;t_0 \cdot \left(t_2 + \sqrt{t_2 \cdot t_2 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \frac{\left(c0 \cdot d\right) \cdot \frac{d}{D}}{w \cdot \left(h \cdot D\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{D}{t_1} \cdot \frac{D \cdot \left(h \cdot \frac{M}{d}\right)}{{t_1}^{2}}\right)\\ \end{array} \]

Alternative 2
Error	18.6
Cost	11076

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

Alternative 3
Error	23.2
Cost	1488

\[\begin{array}{l} t_0 := 0.25 \cdot \left(\frac{D}{\frac{d}{h}} \cdot \left(D \cdot \frac{M}{\frac{d}{M}}\right)\right)\\ t_1 := 0.25 \cdot \left(\frac{D \cdot D}{\frac{d}{M}} \cdot \frac{h}{\frac{d}{M}}\right)\\ \mathbf{if}\;M \leq -4.6 \cdot 10^{+193}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;M \leq -2.4 \cdot 10^{-143}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;M \leq 4.2 \cdot 10^{-109}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 6.5 \cdot 10^{+27}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	25.0
Cost	1224

\[\begin{array}{l} t_0 := 0.25 \cdot \left(\frac{D}{\frac{d}{h}} \cdot \left(D \cdot \frac{M}{\frac{d}{M}}\right)\right)\\ \mathbf{if}\;M \leq -1 \cdot 10^{-139}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;M \leq 4.2 \cdot 10^{-109}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	18.9
Cost	960

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

Alternative 6
Error	31.6
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022329 
(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))))))