Henrywood and Agarwal, Equation (13)

Average Error: 59.4 → 17.7

Time: 49.6s

Precision: binary64

Cost: 52552

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\frac{c0}{h}}{w}\\ t_1 := {\left(\frac{d}{D}\right)}^{2}\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_3 := \frac{c0}{2 \cdot w} \cdot \left(t_2 + \sqrt{t_2 \cdot t_2 - M \cdot M}\right)\\ t_4 := \sqrt[3]{\frac{d}{D}}\\ \mathbf{if}\;t_3 \leq -1 \cdot 10^{-25}:\\ \;\;\;\;\frac{c0}{\frac{2 \cdot w}{\mathsf{fma}\left(t_0, t_1, \mathsf{hypot}\left(t_0 \cdot t_1, M\right)\right)}}\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(0.25, \frac{M}{{t_4}^{4}} \cdot \frac{M}{\frac{{t_4}^{2}}{h}}, 0\right)\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;{\left(\frac{\frac{c0 \cdot d}{\sqrt{h \cdot \left(w \cdot w\right)}}}{D}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.25, \frac{h \cdot M}{\frac{d}{D}} \cdot \frac{M}{\frac{d}{D}}, 0\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (/ c0 h) w))
        (t_1 (pow (/ d D) 2.0))
        (t_2 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_3 (* (/ c0 (* 2.0 w)) (+ t_2 (sqrt (- (* t_2 t_2) (* M M))))))
        (t_4 (cbrt (/ d D))))
   (if (<= t_3 -1e-25)
     (/ c0 (/ (* 2.0 w) (fma t_0 t_1 (hypot (* t_0 t_1) M))))
     (if (<= t_3 0.0)
       (fma 0.25 (* (/ M (pow t_4 4.0)) (/ M (/ (pow t_4 2.0) h))) 0.0)
       (if (<= t_3 INFINITY)
         (pow (/ (/ (* c0 d) (sqrt (* h (* w w)))) D) 2.0)
         (fma 0.25 (* (/ (* h M) (/ d D)) (/ M (/ d D))) 0.0))))))

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 = (c0 / h) / w;
	double t_1 = pow((d / D), 2.0);
	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 t_4 = cbrt((d / D));
	double tmp;
	if (t_3 <= -1e-25) {
		tmp = c0 / ((2.0 * w) / fma(t_0, t_1, hypot((t_0 * t_1), M)));
	} else if (t_3 <= 0.0) {
		tmp = fma(0.25, ((M / pow(t_4, 4.0)) * (M / (pow(t_4, 2.0) / h))), 0.0);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = pow((((c0 * d) / sqrt((h * (w * w)))) / D), 2.0);
	} else {
		tmp = fma(0.25, (((h * M) / (d / D)) * (M / (d / D))), 0.0);
	}
	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(c0 / h) / w)
	t_1 = Float64(d / D) ^ 2.0
	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)))))
	t_4 = cbrt(Float64(d / D))
	tmp = 0.0
	if (t_3 <= -1e-25)
		tmp = Float64(c0 / Float64(Float64(2.0 * w) / fma(t_0, t_1, hypot(Float64(t_0 * t_1), M))));
	elseif (t_3 <= 0.0)
		tmp = fma(0.25, Float64(Float64(M / (t_4 ^ 4.0)) * Float64(M / Float64((t_4 ^ 2.0) / h))), 0.0);
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(Float64(c0 * d) / sqrt(Float64(h * Float64(w * w)))) / D) ^ 2.0;
	else
		tmp = fma(0.25, Float64(Float64(Float64(h * M) / Float64(d / D)) * Float64(M / Float64(d / D))), 0.0);
	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[(N[(c0 / h), $MachinePrecision] / w), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(d / D), $MachinePrecision], 2.0], $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]}, Block[{t$95$4 = N[Power[N[(d / D), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[t$95$3, -1e-25], N[(c0 / N[(N[(2.0 * w), $MachinePrecision] / N[(t$95$0 * t$95$1 + N[Sqrt[N[(t$95$0 * t$95$1), $MachinePrecision] ^ 2 + M ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(0.25 * N[(N[(M / N[Power[t$95$4, 4.0], $MachinePrecision]), $MachinePrecision] * N[(M / N[(N[Power[t$95$4, 2.0], $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Power[N[(N[(N[(c0 * d), $MachinePrecision] / N[Sqrt[N[(h * N[(w * w), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision], 2.0], $MachinePrecision], N[(0.25 * N[(N[(N[(h * M), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision] * N[(M / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.0), $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))))) < -1.00000000000000004e-25

   1. Initial program 53.2

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

   2. Simplified 52.8

      \[\leadsto \color{blue}{\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}, \frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}, -M \cdot M\right)}\right)} \]
      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)))))): 7 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 (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)))))): 1 points increase in error, 1 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)))))): 0 points increase in error, 9 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 45.9

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

   if -1.00000000000000004e-25 < (*.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 26.7

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

   2. Simplified 31.8

      \[\leadsto \color{blue}{\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}, \frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}, -M \cdot M\right)}\right)} \]
      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)))))): 7 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 (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)))))): 1 points increase in error, 1 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)))))): 0 points increase in error, 9 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. Taylor expanded in c0 around -inf 36.5

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

   4. Simplified 30.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.25, \frac{M \cdot \left(M \cdot h\right)}{{\left(\frac{d}{D}\right)}^{2}}, \frac{0}{w} \cdot \left(c0 \cdot c0\right)\right)} \]
      Proof
      (fma.f64 1/4 (/.f64 (*.f64 M (*.f64 M h)) (pow.f64 (/.f64 d D) 2)) (*.f64 (/.f64 0 w) (*.f64 c0 c0))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 M M) h)) (pow.f64 (/.f64 d D) 2)) (*.f64 (/.f64 0 w) (*.f64 c0 c0))): 17 points increase in error, 5 points decrease in error
      (fma.f64 1/4 (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 M 2)) h) (pow.f64 (/.f64 d D) 2)) (*.f64 (/.f64 0 w) (*.f64 c0 c0))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 h (pow.f64 M 2))) (pow.f64 (/.f64 d D) 2)) (*.f64 (/.f64 0 w) (*.f64 c0 c0))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (/.f64 (*.f64 h (pow.f64 M 2)) (Rewrite=> unpow2_binary64 (*.f64 (/.f64 d D) (/.f64 d D)))) (*.f64 (/.f64 0 w) (*.f64 c0 c0))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (/.f64 (*.f64 h (pow.f64 M 2)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 d d) (*.f64 D D)))) (*.f64 (/.f64 0 w) (*.f64 c0 c0))): 29 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (/.f64 (*.f64 h (pow.f64 M 2)) (/.f64 (*.f64 d d) (Rewrite<= unpow2_binary64 (pow.f64 D 2)))) (*.f64 (/.f64 0 w) (*.f64 c0 c0))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 h (pow.f64 M 2)) (pow.f64 D 2)) (*.f64 d d))) (*.f64 (/.f64 0 w) (*.f64 c0 c0))): 3 points increase in error, 5 points decrease in error
      (fma.f64 1/4 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 D 2) (*.f64 h (pow.f64 M 2)))) (*.f64 d d)) (*.f64 (/.f64 0 w) (*.f64 c0 c0))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 M 2) h))) (*.f64 d d)) (*.f64 (/.f64 0 w) (*.f64 c0 c0))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (Rewrite<= unpow2_binary64 (pow.f64 d 2))) (*.f64 (/.f64 0 w) (*.f64 c0 c0))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2)) (*.f64 (/.f64 0 w) (Rewrite<= unpow2_binary64 (pow.f64 c0 2)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2)) (Rewrite<= associate-/r/_binary64 (/.f64 0 (/.f64 w (pow.f64 c0 2))))): 11 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2)) (/.f64 (Rewrite<= metadata-eval (*.f64 -1/2 0)) (/.f64 w (pow.f64 c0 2)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2)) (Rewrite<= associate-*r/_binary64 (*.f64 -1/2 (/.f64 0 (/.f64 w (pow.f64 c0 2)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2)) (*.f64 -1/2 (/.f64 (Rewrite<= mul0-lft_binary64 (*.f64 0 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h))))) (/.f64 w (pow.f64 c0 2))))): 90 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2)) (*.f64 -1/2 (/.f64 (*.f64 (Rewrite<= metadata-eval (+.f64 -1 1)) (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h)))) (/.f64 w (pow.f64 c0 2))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2)) (*.f64 -1/2 (/.f64 (Rewrite<= distribute-rgt1-in_binary64 (+.f64 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h))) (*.f64 -1 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h)))))) (/.f64 w (pow.f64 c0 2))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2)) (*.f64 -1/2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (+.f64 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h))) (*.f64 -1 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h))))) (pow.f64 c0 2)) w)))): 0 points increase in error, 1 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2))) (*.f64 -1/2 (/.f64 (*.f64 (+.f64 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h))) (*.f64 -1 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h))))) (pow.f64 c0 2)) w)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1/2 (/.f64 (*.f64 (+.f64 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h))) (*.f64 -1 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h))))) (pow.f64 c0 2)) w)) (*.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2))))): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in w around 0 22.8

      \[\leadsto \mathsf{fma}\left(0.25, \frac{M \cdot \left(M \cdot h\right)}{{\left(\frac{d}{D}\right)}^{2}}, \color{blue}{0}\right) \]

   6. Applied egg-rr 18.1

      \[\leadsto \mathsf{fma}\left(0.25, \color{blue}{\frac{M}{{\left(\sqrt[3]{\frac{d}{D}}\right)}^{4}} \cdot \frac{M \cdot h}{{\left(\sqrt[3]{\frac{d}{D}}\right)}^{2}}}, 0\right) \]

   7. Simplified 16.8

      \[\leadsto \mathsf{fma}\left(0.25, \color{blue}{\frac{M}{{\left(\sqrt[3]{\frac{d}{D}}\right)}^{4}} \cdot \frac{M}{\frac{{\left(\sqrt[3]{\frac{d}{D}}\right)}^{2}}{h}}}, 0\right) \]
      Proof
      (*.f64 (/.f64 M (pow.f64 (cbrt.f64 (/.f64 d D)) 4)) (/.f64 M (/.f64 (pow.f64 (cbrt.f64 (/.f64 d D)) 2) h))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 M (pow.f64 (cbrt.f64 (/.f64 d D)) 4)) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 M h) (pow.f64 (cbrt.f64 (/.f64 d D)) 2)))): 23 points increase in error, 14 points decrease in error

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

   1. Initial program 46.6

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

   2. Simplified 53.5

      \[\leadsto \color{blue}{\frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d}{D \cdot D} \cdot d, \sqrt{\mathsf{fma}\left(\frac{c0 \cdot d}{w}, \frac{d}{h \cdot \left(D \cdot D\right)}, M\right) \cdot \left(\frac{d}{\frac{h}{d}} \cdot \frac{c0}{w \cdot \left(D \cdot D\right)} - M\right)}\right)} \]
      Proof
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 d (*.f64 D D)) d) (sqrt.f64 (*.f64 (fma.f64 (/.f64 (*.f64 c0 d) w) (/.f64 d (*.f64 h (*.f64 D D))) M) (-.f64 (*.f64 (/.f64 d (/.f64 h d)) (/.f64 c0 (*.f64 w (*.f64 D D)))) M))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (/.f64 c0 (*.f64 w h)) (Rewrite<= associate-/r/_binary64 (/.f64 d (/.f64 (*.f64 D D) d))) (sqrt.f64 (*.f64 (fma.f64 (/.f64 (*.f64 c0 d) w) (/.f64 d (*.f64 h (*.f64 D D))) M) (-.f64 (*.f64 (/.f64 d (/.f64 h d)) (/.f64 c0 (*.f64 w (*.f64 D D)))) M))))): 2 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (/.f64 c0 (*.f64 w h)) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 d d) (*.f64 D D))) (sqrt.f64 (*.f64 (fma.f64 (/.f64 (*.f64 c0 d) w) (/.f64 d (*.f64 h (*.f64 D D))) M) (-.f64 (*.f64 (/.f64 d (/.f64 h d)) (/.f64 c0 (*.f64 w (*.f64 D D)))) M))))): 1 points increase in error, 1 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (/.f64 c0 (*.f64 w h)) (/.f64 (*.f64 d d) (*.f64 D D)) (sqrt.f64 (*.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (/.f64 (*.f64 c0 d) w) (/.f64 d (*.f64 h (*.f64 D D)))) M)) (-.f64 (*.f64 (/.f64 d (/.f64 h d)) (/.f64 c0 (*.f64 w (*.f64 D D)))) M))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (/.f64 c0 (*.f64 w h)) (/.f64 (*.f64 d d) (*.f64 D D)) (sqrt.f64 (*.f64 (+.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (*.f64 c0 d) d) (*.f64 w (*.f64 h (*.f64 D D))))) M) (-.f64 (*.f64 (/.f64 d (/.f64 h d)) (/.f64 c0 (*.f64 w (*.f64 D D)))) M))))): 3 points increase in error, 1 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (/.f64 c0 (*.f64 w h)) (/.f64 (*.f64 d d) (*.f64 D D)) (sqrt.f64 (*.f64 (+.f64 (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 c0 (*.f64 d d))) (*.f64 w (*.f64 h (*.f64 D D)))) M) (-.f64 (*.f64 (/.f64 d (/.f64 h d)) (/.f64 c0 (*.f64 w (*.f64 D D)))) M))))): 0 points increase in error, 2 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (/.f64 c0 (*.f64 w h)) (/.f64 (*.f64 d d) (*.f64 D D)) (sqrt.f64 (*.f64 (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 w h) (*.f64 D D)))) M) (-.f64 (*.f64 (/.f64 d (/.f64 h d)) (/.f64 c0 (*.f64 w (*.f64 D D)))) M))))): 1 points increase in error, 3 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (/.f64 c0 (*.f64 w h)) (/.f64 (*.f64 d d) (*.f64 D D)) (sqrt.f64 (*.f64 (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) M) (-.f64 (*.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 d d) h)) (/.f64 c0 (*.f64 w (*.f64 D D)))) M))))): 1 points increase in error, 1 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (/.f64 c0 (*.f64 w h)) (/.f64 (*.f64 d d) (*.f64 D D)) (sqrt.f64 (*.f64 (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) M) (-.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (*.f64 d d) c0) (*.f64 h (*.f64 w (*.f64 D D))))) M))))): 1 points increase in error, 3 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (/.f64 c0 (*.f64 w h)) (/.f64 (*.f64 d d) (*.f64 D D)) (sqrt.f64 (*.f64 (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) M) (-.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 c0 (*.f64 d d))) (*.f64 h (*.f64 w (*.f64 D D)))) M))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (/.f64 c0 (*.f64 w h)) (/.f64 (*.f64 d d) (*.f64 D D)) (sqrt.f64 (*.f64 (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) M) (-.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 h w) (*.f64 D D)))) M))))): 1 points increase in error, 2 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (/.f64 c0 (*.f64 w h)) (/.f64 (*.f64 d d) (*.f64 D D)) (sqrt.f64 (*.f64 (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) M) (-.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 w h)) (*.f64 D D))) M))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (/.f64 c0 (*.f64 w h)) (/.f64 (*.f64 d d) (*.f64 D D)) (sqrt.f64 (Rewrite<= difference-of-squares_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
      (*.f64 (/.f64 c0 (*.f64 2 w)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (/.f64 c0 (*.f64 w h)) (/.f64 (*.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)))))): 2 points increase in error, 2 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 (-.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, 11 points decrease in error

   3. Taylor expanded in c0 around inf 41.3

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

   4. Simplified 42.3

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{\frac{\left(d \cdot d\right) \cdot c0}{D \cdot D}}{w \cdot h}\right)} \]
      Proof
      (*.f64 2 (/.f64 (/.f64 (*.f64 (*.f64 d d) c0) (*.f64 D D)) (*.f64 w h))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 d 2)) c0) (*.f64 D D)) (*.f64 w h))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (/.f64 (*.f64 (pow.f64 d 2) c0) (Rewrite<= unpow2_binary64 (pow.f64 D 2))) (*.f64 w h))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (Rewrite<= associate-/r*_binary64 (/.f64 (*.f64 (pow.f64 d 2) c0) (*.f64 (pow.f64 D 2) (*.f64 w h))))): 13 points increase in error, 7 points decrease in error

   5. Taylor expanded in c0 around 0 54.1

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

   6. Simplified 42.6

      \[\leadsto \color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{c0}{w}\right)} \]
      Proof
      (*.f64 (*.f64 (/.f64 d D) (/.f64 d D)) (*.f64 (/.f64 c0 (*.f64 w h)) (/.f64 c0 w))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 d d) (*.f64 D D))) (*.f64 (/.f64 c0 (*.f64 w h)) (/.f64 c0 w))): 40 points increase in error, 7 points decrease in error
      (*.f64 (/.f64 (*.f64 d d) (*.f64 D D)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 c0 c0) (*.f64 (*.f64 w h) w)))): 14 points increase in error, 5 points decrease in error
      (*.f64 (/.f64 (*.f64 d d) (*.f64 D D)) (/.f64 (*.f64 c0 c0) (Rewrite=> *-commutative_binary64 (*.f64 w (*.f64 w h))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.f64 d d) (*.f64 D D)) (/.f64 (*.f64 c0 c0) (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 w w) h)))): 2 points increase in error, 1 points decrease in error
      (*.f64 (/.f64 (*.f64 d d) (*.f64 D D)) (/.f64 (*.f64 c0 c0) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 w 2)) h))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (*.f64 d d) (*.f64 c0 c0)) (*.f64 (*.f64 D D) (*.f64 (pow.f64 w 2) h)))): 5 points increase in error, 7 points decrease in error
      (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 d 2)) (*.f64 c0 c0)) (*.f64 (*.f64 D D) (*.f64 (pow.f64 w 2) h))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (pow.f64 d 2) (Rewrite<= unpow2_binary64 (pow.f64 c0 2))) (*.f64 (*.f64 D D) (*.f64 (pow.f64 w 2) h))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (pow.f64 d 2) (pow.f64 c0 2)) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 D 2)) (*.f64 (pow.f64 w 2) h))): 0 points increase in error, 0 points decrease in error

   7. Applied egg-rr 33.8

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

   8. Simplified 35.7

      \[\leadsto \color{blue}{{\left(\frac{\frac{d \cdot c0}{\sqrt{\left(w \cdot w\right) \cdot h}}}{D}\right)}^{2}} \]
      Proof
      (pow.f64 (/.f64 (/.f64 (*.f64 d c0) (sqrt.f64 (*.f64 (*.f64 w w) h))) D) 2): 0 points increase in error, 0 points decrease in error
      (pow.f64 (/.f64 (/.f64 (*.f64 d c0) (sqrt.f64 (Rewrite<= associate-*r*_binary64 (*.f64 w (*.f64 w h))))) D) 2): 7 points increase in error, 7 points decrease in error
      (pow.f64 (/.f64 (Rewrite<= associate-*r/_binary64 (*.f64 d (/.f64 c0 (sqrt.f64 (*.f64 w (*.f64 w h)))))) D) 2): 4 points increase in error, 7 points decrease in error
      (pow.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 d D) (/.f64 c0 (sqrt.f64 (*.f64 w (*.f64 w h)))))) 2): 18 points increase in error, 6 points decrease in error

   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 63.1

      \[\leadsto \color{blue}{\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}, \frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}, -M \cdot M\right)}\right)} \]
      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)))))): 7 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 (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)))))): 1 points increase in error, 1 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)))))): 0 points increase in error, 9 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. Taylor expanded in c0 around -inf 63.5

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

   4. Simplified 34.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.25, \frac{M \cdot \left(M \cdot h\right)}{{\left(\frac{d}{D}\right)}^{2}}, \frac{0}{w} \cdot \left(c0 \cdot c0\right)\right)} \]
      Proof
      (fma.f64 1/4 (/.f64 (*.f64 M (*.f64 M h)) (pow.f64 (/.f64 d D) 2)) (*.f64 (/.f64 0 w) (*.f64 c0 c0))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 M M) h)) (pow.f64 (/.f64 d D) 2)) (*.f64 (/.f64 0 w) (*.f64 c0 c0))): 17 points increase in error, 5 points decrease in error
      (fma.f64 1/4 (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 M 2)) h) (pow.f64 (/.f64 d D) 2)) (*.f64 (/.f64 0 w) (*.f64 c0 c0))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 h (pow.f64 M 2))) (pow.f64 (/.f64 d D) 2)) (*.f64 (/.f64 0 w) (*.f64 c0 c0))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (/.f64 (*.f64 h (pow.f64 M 2)) (Rewrite=> unpow2_binary64 (*.f64 (/.f64 d D) (/.f64 d D)))) (*.f64 (/.f64 0 w) (*.f64 c0 c0))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (/.f64 (*.f64 h (pow.f64 M 2)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 d d) (*.f64 D D)))) (*.f64 (/.f64 0 w) (*.f64 c0 c0))): 29 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (/.f64 (*.f64 h (pow.f64 M 2)) (/.f64 (*.f64 d d) (Rewrite<= unpow2_binary64 (pow.f64 D 2)))) (*.f64 (/.f64 0 w) (*.f64 c0 c0))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 h (pow.f64 M 2)) (pow.f64 D 2)) (*.f64 d d))) (*.f64 (/.f64 0 w) (*.f64 c0 c0))): 3 points increase in error, 5 points decrease in error
      (fma.f64 1/4 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 D 2) (*.f64 h (pow.f64 M 2)))) (*.f64 d d)) (*.f64 (/.f64 0 w) (*.f64 c0 c0))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 M 2) h))) (*.f64 d d)) (*.f64 (/.f64 0 w) (*.f64 c0 c0))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (Rewrite<= unpow2_binary64 (pow.f64 d 2))) (*.f64 (/.f64 0 w) (*.f64 c0 c0))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2)) (*.f64 (/.f64 0 w) (Rewrite<= unpow2_binary64 (pow.f64 c0 2)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2)) (Rewrite<= associate-/r/_binary64 (/.f64 0 (/.f64 w (pow.f64 c0 2))))): 11 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2)) (/.f64 (Rewrite<= metadata-eval (*.f64 -1/2 0)) (/.f64 w (pow.f64 c0 2)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2)) (Rewrite<= associate-*r/_binary64 (*.f64 -1/2 (/.f64 0 (/.f64 w (pow.f64 c0 2)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2)) (*.f64 -1/2 (/.f64 (Rewrite<= mul0-lft_binary64 (*.f64 0 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h))))) (/.f64 w (pow.f64 c0 2))))): 90 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2)) (*.f64 -1/2 (/.f64 (*.f64 (Rewrite<= metadata-eval (+.f64 -1 1)) (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h)))) (/.f64 w (pow.f64 c0 2))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2)) (*.f64 -1/2 (/.f64 (Rewrite<= distribute-rgt1-in_binary64 (+.f64 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h))) (*.f64 -1 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h)))))) (/.f64 w (pow.f64 c0 2))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2)) (*.f64 -1/2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (+.f64 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h))) (*.f64 -1 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h))))) (pow.f64 c0 2)) w)))): 0 points increase in error, 1 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2))) (*.f64 -1/2 (/.f64 (*.f64 (+.f64 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h))) (*.f64 -1 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h))))) (pow.f64 c0 2)) w)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1/2 (/.f64 (*.f64 (+.f64 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h))) (*.f64 -1 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h))))) (pow.f64 c0 2)) w)) (*.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2))))): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in w around 0 22.0

      \[\leadsto \mathsf{fma}\left(0.25, \frac{M \cdot \left(M \cdot h\right)}{{\left(\frac{d}{D}\right)}^{2}}, \color{blue}{0}\right) \]

   6. Applied egg-rr 14.5

      \[\leadsto \mathsf{fma}\left(0.25, \color{blue}{\frac{M \cdot h}{\frac{d}{D}} \cdot \frac{M}{\frac{d}{D}}}, 0\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 17.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq -1 \cdot 10^{-25}:\\ \;\;\;\;\frac{c0}{\frac{2 \cdot w}{\mathsf{fma}\left(\frac{\frac{c0}{h}}{w}, {\left(\frac{d}{D}\right)}^{2}, \mathsf{hypot}\left(\frac{\frac{c0}{h}}{w} \cdot {\left(\frac{d}{D}\right)}^{2}, M\right)\right)}}\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 0:\\ \;\;\;\;\mathsf{fma}\left(0.25, \frac{M}{{\left(\sqrt[3]{\frac{d}{D}}\right)}^{4}} \cdot \frac{M}{\frac{{\left(\sqrt[3]{\frac{d}{D}}\right)}^{2}}{h}}, 0\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;{\left(\frac{\frac{c0 \cdot d}{\sqrt{h \cdot \left(w \cdot w\right)}}}{D}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.25, \frac{h \cdot M}{\frac{d}{D}} \cdot \frac{M}{\frac{d}{D}}, 0\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	17.5
Cost	42764

\[\begin{array}{l} t_0 := \mathsf{fma}\left(0.25, \frac{h \cdot M}{\frac{d}{D}} \cdot \frac{M}{\frac{d}{D}}, 0\right)\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_2 := \frac{c0}{2 \cdot w} \cdot \left(t_1 + \sqrt{t_1 \cdot t_1 - M \cdot M}\right)\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{c0}{2}}{w} \cdot \left({\left(\frac{d}{D}\right)}^{2} \cdot \left(2 \cdot \frac{c0}{w \cdot h}\right)\right)\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;{\left(\frac{\frac{c0 \cdot d}{\sqrt{h \cdot \left(w \cdot w\right)}}}{D}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	17.5
Cost	42764

\[\begin{array}{l} t_0 := c0 \cdot \frac{0.5}{w}\\ t_1 := {\left(\frac{d}{D}\right)}^{2}\\ t_2 := \frac{c0}{w \cdot h}\\ t_3 := \mathsf{fma}\left(0.25, \frac{h \cdot M}{\frac{d}{D}} \cdot \frac{M}{\frac{d}{D}}, 0\right)\\ t_4 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_5 := \frac{c0}{2 \cdot w} \cdot \left(t_4 + \sqrt{t_4 \cdot t_4 - M \cdot M}\right)\\ \mathbf{if}\;t_5 \leq -1 \cdot 10^{-25}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot t_2\right) + t_0 \cdot \mathsf{fma}\left(t_2, t_1, M\right)\\ \mathbf{elif}\;t_5 \leq 0:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_5 \leq \infty:\\ \;\;\;\;{\left(\frac{\frac{c0 \cdot d}{\sqrt{h \cdot \left(w \cdot w\right)}}}{D}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 3
Error	17.8
Cost	42764

\[\begin{array}{l} t_0 := \frac{\frac{c0}{h}}{w}\\ t_1 := {\left(\frac{d}{D}\right)}^{2}\\ t_2 := \mathsf{fma}\left(0.25, \frac{h \cdot M}{\frac{d}{D}} \cdot \frac{M}{\frac{d}{D}}, 0\right)\\ t_3 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_4 := \frac{c0}{2 \cdot w} \cdot \left(t_3 + \sqrt{t_3 \cdot t_3 - M \cdot M}\right)\\ \mathbf{if}\;t_4 \leq -1 \cdot 10^{-25}:\\ \;\;\;\;\frac{c0}{\frac{2 \cdot w}{\mathsf{fma}\left(t_0, t_1, \mathsf{hypot}\left(t_0 \cdot t_1, M\right)\right)}}\\ \mathbf{elif}\;t_4 \leq 0:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;{\left(\frac{\frac{c0 \cdot d}{\sqrt{h \cdot \left(w \cdot w\right)}}}{D}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	18.0
Cost	36556

\[\begin{array}{l} t_0 := \mathsf{fma}\left(0.25, \frac{h \cdot M}{\frac{d}{D}} \cdot \frac{M}{\frac{d}{D}}, 0\right)\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_2 := \frac{c0}{2 \cdot w} \cdot \left(t_1 + \sqrt{t_1 \cdot t_1 - M \cdot M}\right)\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{c0}{2}}{w} \cdot \left({\left(\frac{d}{D}\right)}^{2} \cdot \left(2 \cdot \frac{c0}{w \cdot h}\right)\right)\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{-23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\frac{c0 \cdot \frac{\frac{c0}{w}}{h}}{\frac{w}{d}} \cdot \frac{\frac{-d}{D}}{-D}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	25.5
Cost	7960

\[\begin{array}{l} t_0 := 0.25 \cdot \frac{D \cdot D}{\frac{d}{h \cdot M} \cdot \frac{d}{M}}\\ \mathbf{if}\;D \leq -1.55 \cdot 10^{+92}:\\ \;\;\;\;\frac{\left(M \cdot M\right) \cdot -0.5}{\frac{\frac{d}{D} \cdot \frac{d \cdot -2}{D}}{h}}\\ \mathbf{elif}\;D \leq -1 \cdot 10^{-121}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;D \leq -3.25 \cdot 10^{-239}:\\ \;\;\;\;0.25 \cdot \left(\frac{D}{\frac{d}{h}} \cdot \frac{D}{\frac{d}{M \cdot M}}\right)\\ \mathbf{elif}\;D \leq 3.7 \cdot 10^{+53}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;D \leq 1.95 \cdot 10^{+96}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(c0 \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w \cdot h}\right)\right)\\ \mathbf{elif}\;D \leq 1.9 \cdot 10^{+133}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(h \cdot \left(\left(M \cdot M\right) \cdot {\left(\frac{D}{d}\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 6
Error	21.4
Cost	7688

\[\begin{array}{l} t_0 := 0.25 \cdot \left(M \cdot \left({\left(\frac{D}{d}\right)}^{2} \cdot \left(h \cdot M\right)\right)\right)\\ \mathbf{if}\;D \cdot D \leq 5 \cdot 10^{-309}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;D \cdot D \leq 10^{+179}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot D}{\frac{d}{h \cdot M} \cdot \frac{d}{M}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	24.0
Cost	7624

\[\begin{array}{l} \mathbf{if}\;D \leq 3.7 \cdot 10^{+53}:\\ \;\;\;\;0.25 \cdot \left(M \cdot \left({\left(\frac{D}{d}\right)}^{2} \cdot \left(h \cdot M\right)\right)\right)\\ \mathbf{elif}\;D \leq 1.95 \cdot 10^{+96}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(c0 \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w \cdot h}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.25, \frac{D}{d} \cdot \frac{M \cdot \left(h \cdot M\right)}{\frac{d}{D}}, 0\right)\\ \end{array} \]

Alternative 8
Error	19.0
Cost	7624

\[\begin{array}{l} t_0 := \mathsf{fma}\left(0.25, \frac{h \cdot M}{\frac{d}{D}} \cdot \frac{M}{\frac{d}{D}}, 0\right)\\ \mathbf{if}\;d \leq 9.6 \cdot 10^{-42}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{c0}{w} \cdot \left(d \cdot \frac{c0}{h}\right)}{D} \cdot \frac{\frac{d}{D}}{w}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	25.5
Cost	2004

\[\begin{array}{l} t_0 := \frac{\left(M \cdot M\right) \cdot -0.5}{\frac{\frac{d}{D} \cdot \frac{d \cdot -2}{D}}{h}}\\ t_1 := 0.25 \cdot \frac{D \cdot D}{\frac{d}{h \cdot M} \cdot \frac{d}{M}}\\ \mathbf{if}\;D \leq -3 \cdot 10^{+90}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;D \leq -1 \cdot 10^{-121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;D \leq -1.7 \cdot 10^{-239}:\\ \;\;\;\;0.25 \cdot \left(\frac{D}{\frac{d}{h}} \cdot \frac{D}{\frac{d}{M \cdot M}}\right)\\ \mathbf{elif}\;D \leq 3.7 \cdot 10^{+53}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;D \leq 1.95 \cdot 10^{+96}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(c0 \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w \cdot h}\right)\right)\\ \mathbf{elif}\;D \leq 1.1 \cdot 10^{+134}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	25.4
Cost	1880

\[\begin{array}{l} t_0 := \frac{\left(M \cdot M\right) \cdot -0.5}{\frac{\frac{d}{D} \cdot \frac{d \cdot -2}{D}}{h}}\\ t_1 := 0.25 \cdot \frac{D \cdot D}{\frac{d}{h \cdot M} \cdot \frac{d}{M}}\\ \mathbf{if}\;D \leq -1.2 \cdot 10^{+92}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;D \leq -1 \cdot 10^{-121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;D \leq -8.2 \cdot 10^{-239}:\\ \;\;\;\;0.25 \cdot \left(\frac{D}{\frac{d}{h}} \cdot \frac{D}{\frac{d}{M \cdot M}}\right)\\ \mathbf{elif}\;D \leq 3.7 \cdot 10^{+53}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;D \leq 1.95 \cdot 10^{+96}:\\ \;\;\;\;\frac{c0 \cdot \frac{\frac{c0}{w}}{h}}{\frac{w}{d}} \cdot \frac{\frac{-d}{D}}{-D}\\ \mathbf{elif}\;D \leq 2.3 \cdot 10^{+137}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	24.6
Cost	1616

\[\begin{array}{l} t_0 := \frac{\left(M \cdot M\right) \cdot -0.5}{\frac{\frac{d}{D} \cdot \frac{d \cdot -2}{D}}{h}}\\ t_1 := 0.25 \cdot \frac{D \cdot D}{\frac{d}{h \cdot M} \cdot \frac{d}{M}}\\ \mathbf{if}\;D \leq -8.4 \cdot 10^{+91}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;D \leq -1.5 \cdot 10^{-121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;D \leq -1.5 \cdot 10^{-254}:\\ \;\;\;\;0.25 \cdot \left(\frac{D}{\frac{d}{h}} \cdot \frac{D}{\frac{d}{M \cdot M}}\right)\\ \mathbf{elif}\;D \leq 1.25 \cdot 10^{+136}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	25.5
Cost	1488

\[\begin{array}{l} t_0 := 0.25 \cdot \frac{D \cdot D}{\frac{d}{h \cdot M} \cdot \frac{d}{M}}\\ t_1 := 0.25 \cdot \left(\frac{D}{\frac{d}{h}} \cdot \frac{D}{\frac{d}{M \cdot M}}\right)\\ \mathbf{if}\;D \leq -4 \cdot 10^{+90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;D \leq -1 \cdot 10^{-121}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;D \leq -5 \cdot 10^{-253}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;D \leq 6.5 \cdot 10^{+134}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	26.1
Cost	1488

\[\begin{array}{l} t_0 := 0.25 \cdot \left(\frac{D}{\frac{d}{h}} \cdot \frac{D}{\frac{d}{M \cdot M}}\right)\\ t_1 := 0.25 \cdot \frac{D \cdot D}{\frac{d}{h \cdot M} \cdot \frac{d}{M}}\\ \mathbf{if}\;D \leq -3.5 \cdot 10^{+90}:\\ \;\;\;\;0.25 \cdot \left(\frac{D \cdot \left(h \cdot D\right)}{d} \cdot \frac{M \cdot M}{d}\right)\\ \mathbf{elif}\;D \leq -1 \cdot 10^{-121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;D \leq -3.1 \cdot 10^{-239}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;D \leq 2.3 \cdot 10^{+132}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 14
Error	26.0
Cost	1220

\[\begin{array}{l} \mathbf{if}\;D \cdot D \leq 10^{+278}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot D}{\frac{d}{h \cdot M} \cdot \frac{d}{M}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 15
Error	31.7
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022296 
(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))))))