Henrywood and Agarwal, Equation (13)

Average Error: 59.9 → 13.9

Time: 37.0s

Precision: binary64

Cost: 36556

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\frac{c0}{D}}{w}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := M \cdot \frac{D}{d}\\ t_3 := h \cdot t_2\\ t_4 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_5 := t_1 \cdot \left(t_4 + \sqrt{t_4 \cdot t_4 - M \cdot M}\right)\\ \mathbf{if}\;t_5 \leq -1 \cdot 10^{-175}:\\ \;\;\;\;d \cdot \left(\frac{d}{h} \cdot \left(t_0 \cdot t_0\right)\right)\\ \mathbf{elif}\;t_5 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(0.25, \frac{D}{\frac{d}{M}} \cdot t_3, 0\right)\\ \mathbf{elif}\;t_5 \leq \infty:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \frac{c0 \cdot \frac{d}{D}}{\left(w \cdot h\right) \cdot \frac{D}{d}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.25, t_3 \cdot t_2, 0\right)\\ \end{array} \]

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 D) w))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (* M (/ D d)))
        (t_3 (* h t_2))
        (t_4 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_5 (* t_1 (+ t_4 (sqrt (- (* t_4 t_4) (* M M)))))))
   (if (<= t_5 -1e-175)
     (* d (* (/ d h) (* t_0 t_0)))
     (if (<= t_5 0.0)
       (fma 0.25 (* (/ D (/ d M)) t_3) 0.0)
       (if (<= t_5 INFINITY)
         (* t_1 (* 2.0 (/ (* c0 (/ d D)) (* (* w h) (/ D d)))))
         (fma 0.25 (* t_3 t_2) 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 / D) / w;
	double t_1 = c0 / (2.0 * w);
	double t_2 = M * (D / d);
	double t_3 = h * t_2;
	double t_4 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_5 = t_1 * (t_4 + sqrt(((t_4 * t_4) - (M * M))));
	double tmp;
	if (t_5 <= -1e-175) {
		tmp = d * ((d / h) * (t_0 * t_0));
	} else if (t_5 <= 0.0) {
		tmp = fma(0.25, ((D / (d / M)) * t_3), 0.0);
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = t_1 * (2.0 * ((c0 * (d / D)) / ((w * h) * (D / d))));
	} else {
		tmp = fma(0.25, (t_3 * t_2), 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 / D) / w)
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(M * Float64(D / d))
	t_3 = Float64(h * t_2)
	t_4 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	t_5 = Float64(t_1 * Float64(t_4 + sqrt(Float64(Float64(t_4 * t_4) - Float64(M * M)))))
	tmp = 0.0
	if (t_5 <= -1e-175)
		tmp = Float64(d * Float64(Float64(d / h) * Float64(t_0 * t_0)));
	elseif (t_5 <= 0.0)
		tmp = fma(0.25, Float64(Float64(D / Float64(d / M)) * t_3), 0.0);
	elseif (t_5 <= Inf)
		tmp = Float64(t_1 * Float64(2.0 * Float64(Float64(c0 * Float64(d / D)) / Float64(Float64(w * h) * Float64(D / d)))));
	else
		tmp = fma(0.25, Float64(t_3 * t_2), 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 / D), $MachinePrecision] / w), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(h * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$1 * N[(t$95$4 + N[Sqrt[N[(N[(t$95$4 * t$95$4), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, -1e-175], N[(d * N[(N[(d / h), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 0.0], N[(0.25 * N[(N[(D / N[(d / M), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] + 0.0), $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[(t$95$1 * N[(2.0 * N[(N[(c0 * N[(d / D), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(t$95$3 * t$95$2), $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))))) < -1e-175

   1. Initial program 51.8

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

   2. Simplified 57.9

      \[\leadsto \color{blue}{\frac{\frac{c0}{2}}{w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{4}, M \cdot \left(-M\right)\right)}\right)} \]
      Proof
      (*.f64 (/.f64 (/.f64 c0 2) w) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 w (*.f64 h (*.f64 D D)))) (sqrt.f64 (fma.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 c0 (*.f64 w h)) (pow.f64 (/.f64 d D) 4)) (*.f64 M (neg.f64 M)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 c0 (*.f64 2 w))) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 w (*.f64 h (*.f64 D D)))) (sqrt.f64 (fma.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 c0 (*.f64 w h)) (pow.f64 (/.f64 d D) 4)) (*.f64 M (neg.f64 M)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 w h) (*.f64 D D)))) (sqrt.f64 (fma.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 c0 (*.f64 w h)) (pow.f64 (/.f64 d D) 4)) (*.f64 M (neg.f64 M)))))): 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)))))): 0 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 (Rewrite=> unpow3_binary64 (*.f64 (*.f64 (/.f64 d D) (/.f64 d D)) (/.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 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 d d) (*.f64 D D))) (/.f64 d D)) (/.f64 d D))) (*.f64 M (neg.f64 M)))))): 1 points increase in error, 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)) (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, 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 (*.f64 d d) (*.f64 D D)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 d d) (*.f64 D D))))) (*.f64 M (neg.f64 M)))))): 2 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))))): 3 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))))): 1 points increase in error, 7 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, 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 (*.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, 6 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)))))): 2 points increase in error, 4 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))))): 0 points increase in error, 5 points decrease in error

   3. Taylor expanded in d around inf 47.7

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

   4. Simplified 43.8

      \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \color{blue}{\left(2 \cdot \left(\frac{\frac{\frac{d \cdot d}{D}}{w \cdot D}}{h} \cdot c0\right)\right)} \]
      Proof
      (*.f64 2 (*.f64 (/.f64 (/.f64 (/.f64 (*.f64 d d) D) (*.f64 w D)) h) c0)): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (/.f64 (/.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 d 2)) D) (*.f64 w D)) h) c0)): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (/.f64 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (/.f64 (pow.f64 d 2) D) D) w)) h) c0)): 11 points increase in error, 16 points decrease in error
      (*.f64 2 (*.f64 (/.f64 (/.f64 (Rewrite<= associate-/r*_binary64 (/.f64 (pow.f64 d 2) (*.f64 D D))) w) h) c0)): 25 points increase in error, 8 points decrease in error
      (*.f64 2 (*.f64 (/.f64 (/.f64 (/.f64 (pow.f64 d 2) (Rewrite<= unpow2_binary64 (pow.f64 D 2))) w) h) c0)): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 (/.f64 (pow.f64 d 2) (pow.f64 D 2)) (*.f64 w h))) c0)): 14 points increase in error, 5 points decrease in error
      (*.f64 2 (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h)))) c0)): 9 points increase in error, 13 points decrease in error
      (*.f64 2 (Rewrite<= associate-/r/_binary64 (/.f64 (pow.f64 d 2) (/.f64 (*.f64 (pow.f64 D 2) (*.f64 w h)) c0)))): 13 points increase in error, 6 points decrease in error
      (*.f64 2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 d 2) c0) (*.f64 (pow.f64 D 2) (*.f64 w h))))): 12 points increase in error, 16 points decrease in error

   5. Taylor expanded in c0 around 0 55.1

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

   6. Simplified 32.0

      \[\leadsto \color{blue}{d \cdot \left(\frac{d}{h} \cdot \left(\frac{\frac{c0}{D}}{w} \cdot \frac{\frac{c0}{D}}{w}\right)\right)} \]
      Proof
      (*.f64 d (*.f64 (/.f64 d h) (*.f64 (/.f64 (/.f64 c0 D) w) (/.f64 (/.f64 c0 D) w)))): 0 points increase in error, 0 points decrease in error
      (*.f64 d (*.f64 (/.f64 d h) (*.f64 (Rewrite=> associate-/l/_binary64 (/.f64 c0 (*.f64 w D))) (/.f64 (/.f64 c0 D) w)))): 11 points increase in error, 7 points decrease in error
      (*.f64 d (*.f64 (/.f64 d h) (*.f64 (/.f64 c0 (Rewrite<= *-commutative_binary64 (*.f64 D w))) (/.f64 (/.f64 c0 D) w)))): 0 points increase in error, 0 points decrease in error
      (*.f64 d (*.f64 (/.f64 d h) (*.f64 (/.f64 c0 (*.f64 D w)) (Rewrite=> associate-/l/_binary64 (/.f64 c0 (*.f64 w D)))))): 8 points increase in error, 9 points decrease in error
      (*.f64 d (*.f64 (/.f64 d h) (*.f64 (/.f64 c0 (*.f64 D w)) (/.f64 c0 (Rewrite<= *-commutative_binary64 (*.f64 D w)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 d (*.f64 (/.f64 d h) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 c0 c0) (*.f64 (*.f64 D w) (*.f64 D w)))))): 40 points increase in error, 14 points decrease in error
      (*.f64 d (*.f64 (/.f64 d h) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 c0 2)) (*.f64 (*.f64 D w) (*.f64 D w))))): 0 points increase in error, 0 points decrease in error
      (*.f64 d (*.f64 (/.f64 d h) (/.f64 (pow.f64 c0 2) (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 D D) (*.f64 w w)))))): 28 points increase in error, 4 points decrease in error
      (*.f64 d (*.f64 (/.f64 d h) (/.f64 (pow.f64 c0 2) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 D 2)) (*.f64 w w))))): 0 points increase in error, 0 points decrease in error
      (*.f64 d (*.f64 (/.f64 d h) (/.f64 (pow.f64 c0 2) (*.f64 (pow.f64 D 2) (Rewrite<= unpow2_binary64 (pow.f64 w 2)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 d (Rewrite<= associate-/r/_binary64 (/.f64 d (/.f64 h (/.f64 (pow.f64 c0 2) (*.f64 (pow.f64 D 2) (pow.f64 w 2))))))): 3 points increase in error, 20 points decrease in error
      (*.f64 d (/.f64 d (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 h (*.f64 (pow.f64 D 2) (pow.f64 w 2))) (pow.f64 c0 2))))): 3 points increase in error, 6 points decrease in error
      (*.f64 d (/.f64 d (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (pow.f64 D 2) (pow.f64 w 2)) h)) (pow.f64 c0 2)))): 0 points increase in error, 0 points decrease in error
      (*.f64 d (/.f64 d (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 w 2) h))) (pow.f64 c0 2)))): 5 points increase in error, 6 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 d (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 w 2) h)) (pow.f64 c0 2))) d)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 d (/.f64 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 w 2) h)) (pow.f64 c0 2)) d))): 5 points increase in error, 3 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 d d) (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 w 2) h)) (pow.f64 c0 2)))): 30 points increase in error, 3 points decrease in error
      (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 d 2)) (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 w 2) h)) (pow.f64 c0 2))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 d 2) (pow.f64 c0 2)) (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 w 2) h)))): 3 points increase in error, 4 points decrease in error

   if -1e-175 < (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < 0.0

   1. Initial program 30.4

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

   2. Taylor expanded in c0 around -inf 33.2

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

   3. Simplified 22.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.25, \left(\frac{D}{d \cdot d} \cdot D\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), 0\right)} \]
      Proof
      (fma.f64 1/4 (*.f64 (*.f64 (/.f64 D (*.f64 d d)) D) (*.f64 M (*.f64 M h))) 0): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (*.f64 (*.f64 (/.f64 D (Rewrite<= unpow2_binary64 (pow.f64 d 2))) D) (*.f64 M (*.f64 M h))) 0): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (*.f64 (Rewrite<= associate-/r/_binary64 (/.f64 D (/.f64 (pow.f64 d 2) D))) (*.f64 M (*.f64 M h))) 0): 5 points increase in error, 4 points decrease in error
      (fma.f64 1/4 (*.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 D D) (pow.f64 d 2))) (*.f64 M (*.f64 M h))) 0): 36 points increase in error, 2 points decrease in error
      (fma.f64 1/4 (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 D 2)) (pow.f64 d 2)) (*.f64 M (*.f64 M h))) 0): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (*.f64 (/.f64 (pow.f64 D 2) (pow.f64 d 2)) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 M M) h))) 0): 16 points increase in error, 6 points decrease in error
      (fma.f64 1/4 (*.f64 (/.f64 (pow.f64 D 2) (pow.f64 d 2)) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 M 2)) h)) 0): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (*.f64 (/.f64 (pow.f64 D 2) (pow.f64 d 2)) (Rewrite<= *-commutative_binary64 (*.f64 h (pow.f64 M 2)))) 0): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (Rewrite<= associate-/r/_binary64 (/.f64 (pow.f64 D 2) (/.f64 (pow.f64 d 2) (*.f64 h (pow.f64 M 2))))) 0): 11 points increase in error, 6 points decrease in error
      (fma.f64 1/4 (/.f64 (pow.f64 D 2) (/.f64 (pow.f64 d 2) (Rewrite=> *-commutative_binary64 (*.f64 (pow.f64 M 2) h)))) 0): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2))) 0): 6 points increase in error, 11 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<= div0_binary64 (/.f64 0 (/.f64 w (pow.f64 c0 2))))): 47 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)) (/.f64 (*.f64 -1/2 (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)))): 102 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 -1/2 (*.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 (*.f64 -1/2 (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)) (Rewrite<= associate-*r/_binary64 (*.f64 -1/2 (/.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))))) (/.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

   4. Taylor expanded in D around 0 26.7

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

   5. Simplified 13.3

      \[\leadsto \mathsf{fma}\left(0.25, \color{blue}{\frac{D}{\frac{d}{M}} \cdot \frac{D}{\frac{\frac{d}{M}}{h}}}, 0\right) \]
      Proof
      (*.f64 (/.f64 D (/.f64 d M)) (/.f64 D (/.f64 (/.f64 d M) h))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 D (/.f64 d M)) (/.f64 D (Rewrite<= associate-/r*_binary64 (/.f64 d (*.f64 M h))))): 23 points increase in error, 8 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 D D) (*.f64 (/.f64 d M) (/.f64 d (*.f64 M h))))): 59 points increase in error, 14 points decrease in error
      (/.f64 (*.f64 D D) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 d d) (*.f64 M (*.f64 M h))))): 42 points increase in error, 4 points decrease in error
      (/.f64 (*.f64 D D) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 d 2)) (*.f64 M (*.f64 M h)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 D D) (/.f64 (pow.f64 d 2) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 M M) h)))): 16 points increase in error, 1 points decrease in error
      (/.f64 (*.f64 D D) (/.f64 (pow.f64 d 2) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 M 2)) h))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 D D) (/.f64 (pow.f64 d 2) (Rewrite<= *-commutative_binary64 (*.f64 h (pow.f64 M 2))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 D 2)) (/.f64 (pow.f64 d 2) (*.f64 h (pow.f64 M 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 h (pow.f64 M 2))) (pow.f64 d 2))): 6 points increase in error, 11 points decrease in error

   6. Applied egg-rr 11.9

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

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

   1. Initial program 48.8

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

   2. Taylor expanded in c0 around inf 42.2

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

   3. Simplified 41.3

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \left(\frac{\frac{d}{D} \cdot \frac{d}{D}}{w \cdot h} \cdot c0\right)\right)} \]
      Proof
      (*.f64 2 (*.f64 (/.f64 (*.f64 (/.f64 d D) (/.f64 d D)) (*.f64 w h)) c0)): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (/.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 d d) (*.f64 D D))) (*.f64 w h)) c0)): 48 points increase in error, 10 points decrease in error
      (*.f64 2 (*.f64 (/.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 d 2)) (*.f64 D D)) (*.f64 w h)) c0)): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (/.f64 (/.f64 (pow.f64 d 2) (Rewrite<= unpow2_binary64 (pow.f64 D 2))) (*.f64 w h)) c0)): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h)))) c0)): 9 points increase in error, 13 points decrease in error
      (*.f64 2 (Rewrite<= associate-/r/_binary64 (/.f64 (pow.f64 d 2) (/.f64 (*.f64 (pow.f64 D 2) (*.f64 w h)) c0)))): 13 points increase in error, 6 points decrease in error
      (*.f64 2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 d 2) c0) (*.f64 (pow.f64 D 2) (*.f64 w h))))): 12 points increase in error, 16 points decrease in error

   4. Applied egg-rr 28.4

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

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

   1. Initial program 64.0

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

   2. Taylor expanded in c0 around -inf 63.4

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

   3. Simplified 28.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.25, \left(\frac{D}{d \cdot d} \cdot D\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), 0\right)} \]
      Proof
      (fma.f64 1/4 (*.f64 (*.f64 (/.f64 D (*.f64 d d)) D) (*.f64 M (*.f64 M h))) 0): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (*.f64 (*.f64 (/.f64 D (Rewrite<= unpow2_binary64 (pow.f64 d 2))) D) (*.f64 M (*.f64 M h))) 0): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (*.f64 (Rewrite<= associate-/r/_binary64 (/.f64 D (/.f64 (pow.f64 d 2) D))) (*.f64 M (*.f64 M h))) 0): 5 points increase in error, 4 points decrease in error
      (fma.f64 1/4 (*.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 D D) (pow.f64 d 2))) (*.f64 M (*.f64 M h))) 0): 36 points increase in error, 2 points decrease in error
      (fma.f64 1/4 (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 D 2)) (pow.f64 d 2)) (*.f64 M (*.f64 M h))) 0): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (*.f64 (/.f64 (pow.f64 D 2) (pow.f64 d 2)) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 M M) h))) 0): 16 points increase in error, 6 points decrease in error
      (fma.f64 1/4 (*.f64 (/.f64 (pow.f64 D 2) (pow.f64 d 2)) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 M 2)) h)) 0): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (*.f64 (/.f64 (pow.f64 D 2) (pow.f64 d 2)) (Rewrite<= *-commutative_binary64 (*.f64 h (pow.f64 M 2)))) 0): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (Rewrite<= associate-/r/_binary64 (/.f64 (pow.f64 D 2) (/.f64 (pow.f64 d 2) (*.f64 h (pow.f64 M 2))))) 0): 11 points increase in error, 6 points decrease in error
      (fma.f64 1/4 (/.f64 (pow.f64 D 2) (/.f64 (pow.f64 d 2) (Rewrite=> *-commutative_binary64 (*.f64 (pow.f64 M 2) h)))) 0): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2))) 0): 6 points increase in error, 11 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<= div0_binary64 (/.f64 0 (/.f64 w (pow.f64 c0 2))))): 47 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)) (/.f64 (*.f64 -1/2 (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)))): 102 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 -1/2 (*.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 (*.f64 -1/2 (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)) (Rewrite<= associate-*r/_binary64 (*.f64 -1/2 (/.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))))) (/.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

   4. Taylor expanded in D around 0 33.8

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

   5. Simplified 13.8

      \[\leadsto \mathsf{fma}\left(0.25, \color{blue}{\frac{D}{\frac{d}{M}} \cdot \frac{D}{\frac{\frac{d}{M}}{h}}}, 0\right) \]
      Proof
      (*.f64 (/.f64 D (/.f64 d M)) (/.f64 D (/.f64 (/.f64 d M) h))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 D (/.f64 d M)) (/.f64 D (Rewrite<= associate-/r*_binary64 (/.f64 d (*.f64 M h))))): 23 points increase in error, 8 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 D D) (*.f64 (/.f64 d M) (/.f64 d (*.f64 M h))))): 59 points increase in error, 14 points decrease in error
      (/.f64 (*.f64 D D) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 d d) (*.f64 M (*.f64 M h))))): 42 points increase in error, 4 points decrease in error
      (/.f64 (*.f64 D D) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 d 2)) (*.f64 M (*.f64 M h)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 D D) (/.f64 (pow.f64 d 2) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 M M) h)))): 16 points increase in error, 1 points decrease in error
      (/.f64 (*.f64 D D) (/.f64 (pow.f64 d 2) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 M 2)) h))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 D D) (/.f64 (pow.f64 d 2) (Rewrite<= *-commutative_binary64 (*.f64 h (pow.f64 M 2))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 D 2)) (/.f64 (pow.f64 d 2) (*.f64 h (pow.f64 M 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 h (pow.f64 M 2))) (pow.f64 d 2))): 6 points increase in error, 11 points decrease in error

   6. Applied egg-rr 13.2

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

   7. Applied egg-rr 11.6

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

4. Final simplification 13.9

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

Alternatives

Alternative 1
Error	19.0
Cost	7624

\[\begin{array}{l} t_0 := M \cdot \frac{D}{d}\\ t_1 := \mathsf{fma}\left(0.25, \left(h \cdot t_0\right) \cdot t_0, 0\right)\\ \mathbf{if}\;D \leq 1.5 \cdot 10^{+76}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;D \leq 1.9 \cdot 10^{+285}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot \frac{d}{D}}{\frac{w \cdot h}{\frac{d}{D}}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	16.9
Cost	7624

\[\begin{array}{l} t_0 := M \cdot \frac{D}{d}\\ \mathbf{if}\;w \leq -1.65 \cdot 10^{+142}:\\ \;\;\;\;\mathsf{fma}\left(0.25, \frac{D}{\frac{d}{M} \cdot \frac{d}{D \cdot \left(h \cdot M\right)}}, 0\right)\\ \mathbf{elif}\;w \leq -2.7 \cdot 10^{+128}:\\ \;\;\;\;d \cdot \frac{\frac{d}{w} \cdot \frac{c0}{\frac{D}{\frac{c0}{h}}}}{w \cdot D}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.25, \left(h \cdot t_0\right) \cdot t_0, 0\right)\\ \end{array} \]

Alternative 3
Error	16.8
Cost	7624

\[\begin{array}{l} t_0 := M \cdot \frac{D}{d}\\ \mathbf{if}\;w \leq -1.65 \cdot 10^{+142}:\\ \;\;\;\;\mathsf{fma}\left(0.25, \frac{D}{\frac{d}{M}} \cdot \frac{D}{\frac{d}{h \cdot M}}, 0\right)\\ \mathbf{elif}\;w \leq -2.7 \cdot 10^{+128}:\\ \;\;\;\;d \cdot \frac{\frac{d}{w} \cdot \frac{c0}{\frac{D}{\frac{c0}{h}}}}{w \cdot D}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.25, \left(h \cdot t_0\right) \cdot t_0, 0\right)\\ \end{array} \]

Alternative 4
Error	20.0
Cost	7492

\[\begin{array}{l} \mathbf{if}\;D \leq 1.5 \cdot 10^{+76}:\\ \;\;\;\;\mathsf{fma}\left(0.25, \frac{D}{\frac{d}{M}} \cdot \frac{h}{\frac{d}{D \cdot M}}, 0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot \frac{d}{D}}{\frac{w \cdot h}{\frac{d}{D}}}\right)\\ \end{array} \]

Alternative 5
Error	33.6
Cost	1872

\[\begin{array}{l} t_0 := \frac{\frac{c0}{D}}{w}\\ \mathbf{if}\;d \leq -9 \cdot 10^{+56}:\\ \;\;\;\;0\\ \mathbf{elif}\;d \leq -7.1 \cdot 10^{-20}:\\ \;\;\;\;d \cdot \left(\frac{d}{h} \cdot \left(t_0 \cdot t_0\right)\right)\\ \mathbf{elif}\;d \leq -3.1 \cdot 10^{-186}:\\ \;\;\;\;0\\ \mathbf{elif}\;d \leq 5 \cdot 10^{-24}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot \frac{d}{D}}{\frac{w \cdot h}{\frac{d}{D}}}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6
Error	33.6
Cost	1872

\[\begin{array}{l} t_0 := \frac{\frac{c0}{D}}{w}\\ \mathbf{if}\;d \leq -9 \cdot 10^{+56}:\\ \;\;\;\;0\\ \mathbf{elif}\;d \leq -7.1 \cdot 10^{-20}:\\ \;\;\;\;d \cdot \left(\frac{d}{h} \cdot \left(t_0 \cdot t_0\right)\right)\\ \mathbf{elif}\;d \leq -3.1 \cdot 10^{-186}:\\ \;\;\;\;0\\ \mathbf{elif}\;d \leq 5 \cdot 10^{-24}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot \frac{d}{D}}{\left(w \cdot h\right) \cdot \frac{D}{d}}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7
Error	33.7
Cost	1616

\[\begin{array}{l} t_0 := \frac{\frac{c0}{D}}{w}\\ t_1 := d \cdot \left(\frac{d}{h} \cdot \left(t_0 \cdot t_0\right)\right)\\ \mathbf{if}\;d \leq -9 \cdot 10^{+56}:\\ \;\;\;\;0\\ \mathbf{elif}\;d \leq -7.1 \cdot 10^{-20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 5.4 \cdot 10^{-164}:\\ \;\;\;\;0\\ \mathbf{elif}\;d \leq 5 \cdot 10^{-24}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8
Error	33.1
Cost	1352

\[\begin{array}{l} t_0 := \frac{d}{w \cdot D}\\ \mathbf{if}\;d \leq -2.9 \cdot 10^{-223}:\\ \;\;\;\;0\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{-66}:\\ \;\;\;\;\left(t_0 \cdot t_0\right) \cdot \frac{c0 \cdot c0}{h}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9
Error	31.7
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022313 
(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))))))