Henrywood and Agarwal, Equation (13)

Average Error: 59.4 → 19.0

Time: 31.7s

Precision: binary64

Cost: 17092

Math

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

↓

\[\begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t_0 + \sqrt{t_0 \cdot t_0 - M \cdot M}\right) \leq 4 \cdot 10^{+292}:\\ \;\;\;\;\frac{\frac{c0}{2}}{w} \cdot \left(2 \cdot \left(\frac{\frac{d}{D}}{w \cdot h} \cdot \left(c0 \cdot \frac{d}{D}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.25, D \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(h \cdot \frac{M}{d}\right)\right), 0\right)\\ \end{array} \]

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 d)) (* (* w h) (* D D)))))
   (if (<= (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M))))) 4e+292)
     (* (/ (/ c0 2.0) w) (* 2.0 (* (/ (/ d D) (* w h)) (* c0 (/ d D)))))
     (fma 0.25 (* D (* (* D (/ M d)) (* h (/ M 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 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= 4e+292) {
		tmp = ((c0 / 2.0) / w) * (2.0 * (((d / D) / (w * h)) * (c0 * (d / D))));
	} else {
		tmp = fma(0.25, (D * ((D * (M / d)) * (h * (M / 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 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M))))) <= 4e+292)
		tmp = Float64(Float64(Float64(c0 / 2.0) / w) * Float64(2.0 * Float64(Float64(Float64(d / D) / Float64(w * h)) * Float64(c0 * Float64(d / D)))));
	else
		tmp = fma(0.25, Float64(D * Float64(Float64(D * Float64(M / d)) * Float64(h * Float64(M / 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 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+292], N[(N[(N[(c0 / 2.0), $MachinePrecision] / w), $MachinePrecision] * N[(2.0 * N[(N[(N[(d / D), $MachinePrecision] / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[(c0 * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(D * N[(N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < 4.0000000000000001e292

   1. Initial program 34.1

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

   2. Simplified 52.3

      \[\leadsto \color{blue}{\frac{\frac{c0}{2}}{w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{4}, M \cdot \left(-M\right)\right)}\right)} \]
      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)))))): 1 points increase in error, 2 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (fma.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 c0 (*.f64 w h)) (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, 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 (Rewrite=> unpow3_binary64 (*.f64 (*.f64 (/.f64 d D) (/.f64 d D)) (/.f64 d D))) (/.f64 d D))) (*.f64 M (neg.f64 M)))))): 1 points increase in error, 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)))))): 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<= 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)))))): 1 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (fma.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 (*.f64 d d) (*.f64 D D)) (/.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))))): 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 (Rewrite<= swap-sqr_binary64 (*.f64 (*.f64 (/.f64 c0 (*.f64 w h)) (/.f64 (*.f64 d d) (*.f64 D D))) (*.f64 (/.f64 c0 (*.f64 w h)) (/.f64 (*.f64 d d) (*.f64 D D))))) (*.f64 M M))))): 0 points increase in error, 8 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))))): 1 points increase in error, 2 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.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, 11 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D)))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))): 1 points increase in error, 5 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))))): 1 points increase in error, 3 points decrease in error

   3. Taylor expanded in d around inf 37.9

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

   4. Simplified 39.8

      \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \color{blue}{\left(2 \cdot \left(c0 \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w \cdot h}\right)\right)} \]
      Proof
      (*.f64 2 (*.f64 c0 (/.f64 (*.f64 (/.f64 d D) (/.f64 d D)) (*.f64 w h)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 c0 (/.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 d d) (*.f64 D D))) (*.f64 w h)))): 34 points increase in error, 10 points decrease in error
      (*.f64 2 (*.f64 c0 (/.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 d 2)) (*.f64 D D)) (*.f64 w h)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 c0 (/.f64 (/.f64 (pow.f64 d 2) (Rewrite<= unpow2_binary64 (pow.f64 D 2))) (*.f64 w h)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 c0 (Rewrite<= associate-/r*_binary64 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h)))))): 15 points increase in error, 9 points decrease in error
      (*.f64 2 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 c0 (pow.f64 d 2)) (*.f64 (pow.f64 D 2) (*.f64 w h))))): 11 points increase in error, 7 points decrease in error
      (*.f64 2 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 d 2) c0)) (*.f64 (pow.f64 D 2) (*.f64 w h)))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 30.7

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

   6. Applied egg-rr 31.0

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

   if 4.0000000000000001e292 < (*.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. Applied egg-rr 63.9

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

   3. Taylor expanded in c0 around -inf 63.1

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

   4. Simplified 31.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.25, \frac{D}{\frac{\frac{d \cdot d}{h}}{M \cdot M}} \cdot D, 0\right)} \]
      Proof
      (fma.f64 1/4 (*.f64 (/.f64 D (/.f64 (/.f64 (*.f64 d d) h) (*.f64 M M))) D) 0): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (*.f64 (/.f64 D (/.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 d 2)) h) (*.f64 M M))) D) 0): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (*.f64 (/.f64 D (/.f64 (/.f64 (pow.f64 d 2) h) (Rewrite<= unpow2_binary64 (pow.f64 M 2)))) D) 0): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (*.f64 (/.f64 D (Rewrite<= associate-/r*_binary64 (/.f64 (pow.f64 d 2) (*.f64 h (pow.f64 M 2))))) D) 0): 6 points increase in error, 12 points decrease in error
      (fma.f64 1/4 (Rewrite<= associate-/r/_binary64 (/.f64 D (/.f64 (/.f64 (pow.f64 d 2) (*.f64 h (pow.f64 M 2))) D))) 0): 5 points increase in error, 1 points decrease in error
      (fma.f64 1/4 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 D D) (/.f64 (pow.f64 d 2) (*.f64 h (pow.f64 M 2))))) 0): 33 points increase in error, 1 points decrease in error
      (fma.f64 1/4 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 D 2)) (/.f64 (pow.f64 d 2) (*.f64 h (pow.f64 M 2)))) 0): 0 points increase in error, 0 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, 5 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))))): 45 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)))): 88 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, 0 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 D around 0 30.9

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

   6. Simplified 19.6

      \[\leadsto \mathsf{fma}\left(0.25, \color{blue}{\left(M \cdot \left(\left(\frac{D}{d} \cdot \frac{M}{d}\right) \cdot h\right)\right)} \cdot D, 0\right) \]
      Proof
      (*.f64 M (*.f64 (*.f64 (/.f64 D d) (/.f64 M d)) h)): 0 points increase in error, 0 points decrease in error
      (*.f64 M (*.f64 (Rewrite<= associate-/r/_binary64 (/.f64 D (/.f64 d (/.f64 M d)))) h)): 22 points increase in error, 12 points decrease in error
      (*.f64 M (*.f64 (/.f64 D (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 d d) M))) h)): 32 points increase in error, 16 points decrease in error
      (*.f64 M (*.f64 (/.f64 D (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 d 2)) M)) h)): 0 points increase in error, 0 points decrease in error
      (*.f64 M (*.f64 (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 D (pow.f64 d 2)) M)) h)): 13 points increase in error, 18 points decrease in error
      (*.f64 M (Rewrite<= associate-*r*_binary64 (*.f64 (/.f64 D (pow.f64 d 2)) (*.f64 M h)))): 20 points increase in error, 5 points decrease in error
      (*.f64 M (*.f64 (/.f64 D (pow.f64 d 2)) (Rewrite=> *-commutative_binary64 (*.f64 h M)))): 0 points increase in error, 0 points decrease in error
      (*.f64 M (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 h M) (/.f64 D (pow.f64 d 2))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 M (*.f64 h M)) (/.f64 D (pow.f64 d 2)))): 21 points increase in error, 19 points decrease in error
      (*.f64 (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 h M) M)) (/.f64 D (pow.f64 d 2))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= associate-*r*_binary64 (*.f64 h (*.f64 M M))) (/.f64 D (pow.f64 d 2))): 20 points increase in error, 6 points decrease in error
      (*.f64 (*.f64 h (Rewrite<= unpow2_binary64 (pow.f64 M 2))) (/.f64 D (pow.f64 d 2))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 M 2) h)) (/.f64 D (pow.f64 d 2))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (*.f64 (pow.f64 M 2) h) D) (pow.f64 d 2))): 11 points increase in error, 18 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 D (*.f64 (pow.f64 M 2) h))) (pow.f64 d 2)): 0 points increase in error, 0 points decrease in error

   7. Applied egg-rr 16.8

      \[\leadsto \mathsf{fma}\left(0.25, \color{blue}{0 + D \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(\frac{M}{d} \cdot h\right)\right)}, 0\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 19.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 4 \cdot 10^{+292}:\\ \;\;\;\;\frac{\frac{c0}{2}}{w} \cdot \left(2 \cdot \left(\frac{\frac{d}{D}}{w \cdot h} \cdot \left(c0 \cdot \frac{d}{D}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.25, D \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(h \cdot \frac{M}{d}\right)\right), 0\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	22.8
Cost	7888

\[\begin{array}{l} \mathbf{if}\;M \leq 1.7163436322408752 \cdot 10^{-243}:\\ \;\;\;\;h \cdot \frac{M \cdot \left(D \cdot 0.25\right)}{\frac{d}{D} \cdot \frac{d}{M}}\\ \mathbf{elif}\;M \leq 2.59560543389879 \cdot 10^{-205}:\\ \;\;\;\;\frac{\frac{c0}{2}}{w} \cdot \left(2 \cdot \left(c0 \cdot \frac{\frac{d}{D}}{w \cdot \left(h \cdot \frac{D}{d}\right)}\right)\right)\\ \mathbf{elif}\;M \leq 8.704610907386055 \cdot 10^{-144}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \frac{M \cdot \frac{D}{d}}{\frac{\frac{d}{M}}{h}}\right)\\ \mathbf{elif}\;M \leq 4.7 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(0.25, \left(M \cdot M\right) \cdot \left(\left(D \cdot \frac{D}{d}\right) \cdot \frac{h}{d}\right), 0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.25, D \cdot \left(M \cdot \frac{D \cdot \left(h \cdot \frac{M}{d}\right)}{d}\right), 0\right)\\ \end{array} \]

Alternative 2
Error	22.8
Cost	7756

\[\begin{array}{l} \mathbf{if}\;M \leq 1.7163436322408752 \cdot 10^{-243}:\\ \;\;\;\;h \cdot \frac{M \cdot \left(D \cdot 0.25\right)}{\frac{d}{D} \cdot \frac{d}{M}}\\ \mathbf{elif}\;M \leq 2.59560543389879 \cdot 10^{-205}:\\ \;\;\;\;\frac{\frac{c0}{2}}{w} \cdot \left(2 \cdot \left(c0 \cdot \frac{\frac{d}{D}}{w \cdot \left(h \cdot \frac{D}{d}\right)}\right)\right)\\ \mathbf{elif}\;M \leq 1.4 \cdot 10^{+26}:\\ \;\;\;\;h \cdot \frac{\left(D \cdot 0.25\right) \cdot \left(M \cdot \frac{M}{d}\right)}{\frac{d}{D}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.25, D \cdot \left(M \cdot \frac{D \cdot \left(h \cdot \frac{M}{d}\right)}{d}\right), 0\right)\\ \end{array} \]

Alternative 3
Error	22.4
Cost	1608

\[\begin{array}{l} \mathbf{if}\;M \leq 1.7163436322408752 \cdot 10^{-243}:\\ \;\;\;\;h \cdot \frac{M \cdot \left(D \cdot 0.25\right)}{\frac{d}{D} \cdot \frac{d}{M}}\\ \mathbf{elif}\;M \leq 2.59560543389879 \cdot 10^{-205}:\\ \;\;\;\;\frac{\frac{c0}{2}}{w} \cdot \left(2 \cdot \left(d \cdot \frac{\frac{c0}{w} \cdot \frac{\frac{d}{D}}{h}}{D}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \frac{M \cdot \frac{D}{d}}{\frac{\frac{d}{M}}{h}}\right)\\ \end{array} \]

Alternative 4
Error	22.5
Cost	1608

\[\begin{array}{l} \mathbf{if}\;M \leq 1.7163436322408752 \cdot 10^{-243}:\\ \;\;\;\;h \cdot \frac{M \cdot \left(D \cdot 0.25\right)}{\frac{d}{D} \cdot \frac{d}{M}}\\ \mathbf{elif}\;M \leq 2.59560543389879 \cdot 10^{-205}:\\ \;\;\;\;\frac{\frac{c0}{2}}{w} \cdot \left(2 \cdot \left(c0 \cdot \frac{\frac{d}{D}}{w \cdot \left(h \cdot \frac{D}{d}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \frac{M \cdot \frac{D}{d}}{\frac{\frac{d}{M}}{h}}\right)\\ \end{array} \]

Alternative 5
Error	24.3
Cost	1488

\[\begin{array}{l} t_0 := \frac{h \cdot \left(\frac{D \cdot D}{d} \cdot \left(M \cdot 0.25\right)\right)}{\frac{d}{M}}\\ t_1 := h \cdot \frac{D \cdot 0.25}{\frac{d}{D} \cdot \frac{d}{M \cdot M}}\\ \mathbf{if}\;D \leq -4.6 \cdot 10^{+123}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;D \leq 3.1 \cdot 10^{-186}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;D \leq 9.2 \cdot 10^{-138}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;D \leq 4 \cdot 10^{+138}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6
Error	25.8
Cost	1224

\[\begin{array}{l} \mathbf{if}\;M \leq -2.45 \cdot 10^{+153}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 1.05 \cdot 10^{+141}:\\ \;\;\;\;h \cdot \frac{D \cdot 0.25}{\frac{d}{D} \cdot \frac{d}{M \cdot M}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7
Error	20.8
Cost	1220

\[\begin{array}{l} \mathbf{if}\;D \cdot D \leq 2 \cdot 10^{+74}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \frac{M \cdot \frac{D}{d}}{\frac{\frac{d}{M}}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;h \cdot \frac{\left(D \cdot 0.25\right) \cdot \left(M \cdot \frac{M}{d}\right)}{\frac{d}{D}}\\ \end{array} \]

Alternative 8
Error	20.5
Cost	960

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

Alternative 9
Error	32.1
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022294 
(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))))))