Henrywood and Agarwal, Equation (13)

Average Error: 59.3 → 17.2

Time: 30.0s

Precision: binary64

Cost: 30540

Math

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

↓

\[\begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_1 := \frac{c0}{2 \cdot w} \cdot \left(t_0 + \sqrt{t_0 \cdot t_0 - M \cdot M}\right)\\ \mathbf{if}\;t_1 \leq -4 \cdot 10^{-188}:\\ \;\;\;\;d \cdot \left(c0 \cdot \frac{d}{D \cdot \left(D \cdot \left(\left(w \cdot h\right) \cdot \frac{w}{c0}\right)\right)}\right)\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{\left(D \cdot \left(h \cdot \frac{M}{d}\right)\right) \cdot \left(D \cdot 0.25\right)}{\frac{d}{M}}\\ \mathbf{elif}\;t_1 \leq 10^{+119}:\\ \;\;\;\;\frac{c0 \cdot 0.5}{\frac{w}{2 \cdot \left(\frac{d}{D} \cdot \frac{\frac{d}{D}}{h \cdot \frac{w}{c0}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{M \cdot \left(D \cdot 0.25\right)}{\frac{d}{h}}}{\frac{d}{D \cdot M}}\\ \end{array} \]

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))))
        (t_1 (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))))
   (if (<= t_1 -4e-188)
     (* d (* c0 (/ d (* D (* D (* (* w h) (/ w c0)))))))
     (if (<= t_1 0.0)
       (/ (* (* D (* h (/ M d))) (* D 0.25)) (/ d M))
       (if (<= t_1 1e+119)
         (/ (* c0 0.5) (/ w (* 2.0 (* (/ d D) (/ (/ d D) (* h (/ w c0)))))))
         (/ (/ (* M (* D 0.25)) (/ d h)) (/ d (* D M))))))))

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 t_1 = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if (t_1 <= -4e-188) {
		tmp = d * (c0 * (d / (D * (D * ((w * h) * (w / c0))))));
	} else if (t_1 <= 0.0) {
		tmp = ((D * (h * (M / d))) * (D * 0.25)) / (d / M);
	} else if (t_1 <= 1e+119) {
		tmp = (c0 * 0.5) / (w / (2.0 * ((d / D) * ((d / D) / (h * (w / c0))))));
	} else {
		tmp = ((M * (D * 0.25)) / (d / h)) / (d / (D * M));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = (c0 / (2.0d0 * w)) * (((c0 * (d_1 * d_1)) / ((w * h) * (d * d))) + sqrt(((((c0 * (d_1 * d_1)) / ((w * h) * (d * d))) * ((c0 * (d_1 * d_1)) / ((w * h) * (d * d)))) - (m * m))))
end function

↓

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (c0 * (d_1 * d_1)) / ((w * h) * (d * d))
    t_1 = (c0 / (2.0d0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (m * m))))
    if (t_1 <= (-4d-188)) then
        tmp = d_1 * (c0 * (d_1 / (d * (d * ((w * h) * (w / c0))))))
    else if (t_1 <= 0.0d0) then
        tmp = ((d * (h * (m / d_1))) * (d * 0.25d0)) / (d_1 / m)
    else if (t_1 <= 1d+119) then
        tmp = (c0 * 0.5d0) / (w / (2.0d0 * ((d_1 / d) * ((d_1 / d) / (h * (w / c0))))))
    else
        tmp = ((m * (d * 0.25d0)) / (d_1 / h)) / (d_1 / (d * m))
    end if
    code = tmp
end function

Java

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

↓

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_1 = (c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if (t_1 <= -4e-188) {
		tmp = d * (c0 * (d / (D * (D * ((w * h) * (w / c0))))));
	} else if (t_1 <= 0.0) {
		tmp = ((D * (h * (M / d))) * (D * 0.25)) / (d / M);
	} else if (t_1 <= 1e+119) {
		tmp = (c0 * 0.5) / (w / (2.0 * ((d / D) * ((d / D) / (h * (w / c0))))));
	} else {
		tmp = ((M * (D * 0.25)) / (d / h)) / (d / (D * M));
	}
	return tmp;
}

Python

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

↓

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	t_1 = (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))
	tmp = 0
	if t_1 <= -4e-188:
		tmp = d * (c0 * (d / (D * (D * ((w * h) * (w / c0))))))
	elif t_1 <= 0.0:
		tmp = ((D * (h * (M / d))) * (D * 0.25)) / (d / M)
	elif t_1 <= 1e+119:
		tmp = (c0 * 0.5) / (w / (2.0 * ((d / D) * ((d / D) / (h * (w / c0))))))
	else:
		tmp = ((M * (D * 0.25)) / (d / h)) / (d / (D * M))
	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)))
	t_1 = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))))
	tmp = 0.0
	if (t_1 <= -4e-188)
		tmp = Float64(d * Float64(c0 * Float64(d / Float64(D * Float64(D * Float64(Float64(w * h) * Float64(w / c0)))))));
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(Float64(D * Float64(h * Float64(M / d))) * Float64(D * 0.25)) / Float64(d / M));
	elseif (t_1 <= 1e+119)
		tmp = Float64(Float64(c0 * 0.5) / Float64(w / Float64(2.0 * Float64(Float64(d / D) * Float64(Float64(d / D) / Float64(h * Float64(w / c0)))))));
	else
		tmp = Float64(Float64(Float64(M * Float64(D * 0.25)) / Float64(d / h)) / Float64(d / Float64(D * M)));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	t_1 = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	tmp = 0.0;
	if (t_1 <= -4e-188)
		tmp = d * (c0 * (d / (D * (D * ((w * h) * (w / c0))))));
	elseif (t_1 <= 0.0)
		tmp = ((D * (h * (M / d))) * (D * 0.25)) / (d / M);
	elseif (t_1 <= 1e+119)
		tmp = (c0 * 0.5) / (w / (2.0 * ((d / D) * ((d / D) / (h * (w / c0))))));
	else
		tmp = ((M * (D * 0.25)) / (d / h)) / (d / (D * M));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -4e-188], N[(d * N[(c0 * N[(d / N[(D * N[(D * N[(N[(w * h), $MachinePrecision] * N[(w / c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(N[(D * N[(h * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D * 0.25), $MachinePrecision]), $MachinePrecision] / N[(d / M), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+119], N[(N[(c0 * 0.5), $MachinePrecision] / N[(w / N[(2.0 * N[(N[(d / D), $MachinePrecision] * N[(N[(d / D), $MachinePrecision] / N[(h * N[(w / c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(M * N[(D * 0.25), $MachinePrecision]), $MachinePrecision] / N[(d / h), $MachinePrecision]), $MachinePrecision] / N[(d / N[(D * M), $MachinePrecision]), $MachinePrecision]), $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))))) < -3.9999999999999998e-188

   1. Initial program 51.6

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

   2. Simplified 56.8

      \[\leadsto \color{blue}{\frac{\frac{c0}{2}}{w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{4}, M \cdot \left(-M\right)\right)}\right)} \]
      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, 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)))))): 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 (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)))))): 1 points increase in error, 3 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)))))): 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 (/.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, 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)) (/.f64 (*.f64 d d) (*.f64 D D)))) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 M M))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 (*.f64 d d) (*.f64 D D)) (/.f64 (*.f64 d d) (*.f64 D D))))) (*.f64 M M)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 c0 (*.f64 w h)) (/.f64 c0 (*.f64 w h))) (*.f64 (/.f64 (*.f64 d d) (*.f64 D D)) (/.f64 (*.f64 d d) (*.f64 D D))))) (*.f64 M M))))): 4 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (Rewrite<= swap-sqr_binary64 (*.f64 (*.f64 (/.f64 c0 (*.f64 w h)) (/.f64 (*.f64 d d) (*.f64 D D))) (*.f64 (/.f64 c0 (*.f64 w h)) (/.f64 (*.f64 d d) (*.f64 D D))))) (*.f64 M M))))): 2 points increase in error, 9 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))))): 6 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))))): 2 points increase in error, 5 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D)))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))): 1 points increase in error, 9 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))))): 3 points increase in error, 7 points decrease in error

   3. Taylor expanded in d around inf 43.8

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

   4. Simplified 42.5

      \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \color{blue}{\left(2 \cdot \left(c0 \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w \cdot h}\right)\right)} \]
      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)))): 33 points increase in error, 16 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)))))): 8 points increase in error, 6 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))))): 16 points increase in error, 11 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 43.1

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

   6. Taylor expanded in d around 0 46.2

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

   7. Simplified 45.1

      \[\leadsto \frac{\left(2 \cdot \frac{c0}{w \cdot 2}\right) \cdot \left(\color{blue}{\left(d \cdot \frac{d}{D \cdot D}\right)} \cdot c0\right)}{w \cdot h} \]
      Proof
      (*.f64 d (/.f64 d (*.f64 D D))): 0 points increase in error, 0 points decrease in error
      (*.f64 d (/.f64 d (Rewrite<= unpow2_binary64 (pow.f64 D 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 d (pow.f64 D 2)) d)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 d d) (pow.f64 D 2))): 33 points increase in error, 9 points decrease in error
      (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 d 2)) (pow.f64 D 2)): 0 points increase in error, 0 points decrease in error

   8. Taylor expanded in c0 around 0 53.7

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

   9. Simplified 34.4

      \[\leadsto \color{blue}{d \cdot \left(\frac{d}{D \cdot \left(D \cdot \left(\frac{w}{c0} \cdot \left(w \cdot h\right)\right)\right)} \cdot c0\right)} \]
      Proof
      (*.f64 d (*.f64 (/.f64 d (*.f64 D (*.f64 D (*.f64 (/.f64 w c0) (*.f64 w h))))) c0)): 0 points increase in error, 0 points decrease in error
      (*.f64 d (*.f64 (/.f64 d (*.f64 D (*.f64 D (*.f64 (/.f64 w c0) (Rewrite<= *-commutative_binary64 (*.f64 h w)))))) c0)): 0 points increase in error, 0 points decrease in error
      (*.f64 d (*.f64 (/.f64 d (*.f64 D (*.f64 D (Rewrite<= associate-/r/_binary64 (/.f64 w (/.f64 c0 (*.f64 h w))))))) c0)): 8 points increase in error, 2 points decrease in error
      (*.f64 d (*.f64 (/.f64 d (*.f64 D (*.f64 D (/.f64 w (/.f64 c0 (Rewrite=> *-commutative_binary64 (*.f64 w h))))))) c0)): 0 points increase in error, 0 points decrease in error
      (*.f64 d (*.f64 (/.f64 d (*.f64 D (*.f64 D (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 w (*.f64 w h)) c0))))) c0)): 7 points increase in error, 6 points decrease in error
      (*.f64 d (*.f64 (/.f64 d (*.f64 D (*.f64 D (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 w w) h)) c0)))) c0)): 5 points increase in error, 11 points decrease in error
      (*.f64 d (*.f64 (/.f64 d (*.f64 D (*.f64 D (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 w 2)) h) c0)))) c0)): 0 points increase in error, 0 points decrease in error
      (*.f64 d (*.f64 (/.f64 d (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 D D) (/.f64 (*.f64 (pow.f64 w 2) h) c0)))) c0)): 35 points increase in error, 1 points decrease in error
      (*.f64 d (*.f64 (/.f64 d (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 D 2)) (/.f64 (*.f64 (pow.f64 w 2) h) c0))) c0)): 0 points increase in error, 0 points decrease in error
      (*.f64 d (*.f64 (/.f64 d (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 w 2) h)) c0))) c0)): 1 points increase in error, 4 points decrease in error
      (*.f64 d (Rewrite<= associate-/r/_binary64 (/.f64 d (/.f64 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 w 2) h)) c0) c0)))): 4 points increase in error, 3 points decrease in error
      (*.f64 d (/.f64 d (Rewrite<= associate-/r*_binary64 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 w 2) h)) (*.f64 c0 c0))))): 17 points increase in error, 1 points decrease in error
      (*.f64 d (/.f64 d (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 w 2) h)) (Rewrite<= unpow2_binary64 (pow.f64 c0 2))))): 0 points increase in error, 0 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-*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, 2 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)))): 4 points increase in error, 0 points decrease in error

   if -3.9999999999999998e-188 < (*.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 27.1

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

   2. Taylor expanded in c0 around -inf 32.0

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

   3. Simplified 19.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.25, \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d} \cdot \frac{h}{d}, 0\right)} \]
      Proof
      (fma.f64 1/4 (*.f64 (/.f64 (*.f64 (*.f64 D M) (*.f64 D M)) d) (/.f64 h d)) 0): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (*.f64 (/.f64 (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 D D) (*.f64 M M))) d) (/.f64 h d)) 0): 52 points increase in error, 5 points decrease in error
      (fma.f64 1/4 (*.f64 (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 D 2)) (*.f64 M M)) d) (/.f64 h d)) 0): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (*.f64 (/.f64 (*.f64 (pow.f64 D 2) (Rewrite<= unpow2_binary64 (pow.f64 M 2))) d) (/.f64 h d)) 0): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (*.f64 (pow.f64 D 2) (pow.f64 M 2)) h) (*.f64 d d))) 0): 24 points increase in error, 3 points decrease in error
      (fma.f64 1/4 (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h))) (*.f64 d d)) 0): 5 points increase in error, 4 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))) 0): 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<= div0_binary64 (/.f64 0 (/.f64 w (pow.f64 c0 2))))): 52 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)))): 95 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, 2 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. Applied egg-rr 14.4

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

   5. Applied egg-rr 12.1

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

   6. Applied egg-rr 13.0

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

   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))))) < 9.99999999999999944e118

   1. Initial program 9.8

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

   2. Simplified 34.2

      \[\leadsto \color{blue}{\frac{\frac{c0}{2}}{w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{4}, M \cdot \left(-M\right)\right)}\right)} \]
      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, 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)))))): 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 (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)))))): 1 points increase in error, 3 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)))))): 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 (/.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, 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)) (/.f64 (*.f64 d d) (*.f64 D D)))) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 M M))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 (*.f64 d d) (*.f64 D D)) (/.f64 (*.f64 d d) (*.f64 D D))))) (*.f64 M M)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 c0 (*.f64 w h)) (/.f64 c0 (*.f64 w h))) (*.f64 (/.f64 (*.f64 d d) (*.f64 D D)) (/.f64 (*.f64 d d) (*.f64 D D))))) (*.f64 M M))))): 4 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (Rewrite<= swap-sqr_binary64 (*.f64 (*.f64 (/.f64 c0 (*.f64 w h)) (/.f64 (*.f64 d d) (*.f64 D D))) (*.f64 (/.f64 c0 (*.f64 w h)) (/.f64 (*.f64 d d) (*.f64 D D))))) (*.f64 M M))))): 2 points increase in error, 9 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))))): 6 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))))): 2 points increase in error, 5 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D)))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))): 1 points increase in error, 9 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))))): 3 points increase in error, 7 points decrease in error

   3. Taylor expanded in d around inf 6.5

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

   4. Simplified 15.8

      \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \color{blue}{\left(2 \cdot \left(c0 \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w \cdot h}\right)\right)} \]
      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)))): 33 points increase in error, 16 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)))))): 8 points increase in error, 6 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))))): 16 points increase in error, 11 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. Taylor expanded in c0 around 0 6.5

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

   6. Simplified 18.4

      \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \left(2 \cdot \color{blue}{\left(\frac{d}{D} \cdot \frac{\frac{d \cdot \frac{c0}{h}}{w}}{D}\right)}\right) \]
      Proof
      (*.f64 (/.f64 d D) (/.f64 (/.f64 (*.f64 d (/.f64 c0 h)) w) D)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 d D) (/.f64 (Rewrite<= associate-*r/_binary64 (*.f64 d (/.f64 (/.f64 c0 h) w))) D)): 16 points increase in error, 19 points decrease in error
      (*.f64 (/.f64 d D) (/.f64 (*.f64 d (Rewrite=> associate-/l/_binary64 (/.f64 c0 (*.f64 w h)))) D)): 15 points increase in error, 20 points decrease in error
      (*.f64 (/.f64 d D) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 c0 (*.f64 w h)) d)) D)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 d D) (Rewrite<= associate-*r/_binary64 (*.f64 (/.f64 c0 (*.f64 w h)) (/.f64 d D)))): 10 points increase in error, 20 points decrease in error
      (*.f64 (/.f64 d D) (Rewrite=> *-commutative_binary64 (*.f64 (/.f64 d D) (/.f64 c0 (*.f64 w h))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 d D) (/.f64 d D)) (/.f64 c0 (*.f64 w h)))): 29 points increase in error, 16 points decrease in error
      (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 d d) (*.f64 D D))) (/.f64 c0 (*.f64 w h))): 41 points increase in error, 17 points decrease in error
      (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 d 2)) (*.f64 D D)) (/.f64 c0 (*.f64 w h))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 d 2) (Rewrite<= unpow2_binary64 (pow.f64 D 2))) (/.f64 c0 (*.f64 w h))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (pow.f64 d 2) c0) (*.f64 (pow.f64 D 2) (*.f64 w h)))): 20 points increase in error, 17 points decrease in error

   7. Applied egg-rr 11.0

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

   if 9.99999999999999944e118 < (*.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 63.6

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

   2. Taylor expanded in c0 around -inf 62.9

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

   3. Simplified 21.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.25, \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d} \cdot \frac{h}{d}, 0\right)} \]
      Proof
      (fma.f64 1/4 (*.f64 (/.f64 (*.f64 (*.f64 D M) (*.f64 D M)) d) (/.f64 h d)) 0): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (*.f64 (/.f64 (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 D D) (*.f64 M M))) d) (/.f64 h d)) 0): 52 points increase in error, 5 points decrease in error
      (fma.f64 1/4 (*.f64 (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 D 2)) (*.f64 M M)) d) (/.f64 h d)) 0): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (*.f64 (/.f64 (*.f64 (pow.f64 D 2) (Rewrite<= unpow2_binary64 (pow.f64 M 2))) d) (/.f64 h d)) 0): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/4 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (*.f64 (pow.f64 D 2) (pow.f64 M 2)) h) (*.f64 d d))) 0): 24 points increase in error, 3 points decrease in error
      (fma.f64 1/4 (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h))) (*.f64 d d)) 0): 5 points increase in error, 4 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))) 0): 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<= div0_binary64 (/.f64 0 (/.f64 w (pow.f64 c0 2))))): 52 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)))): 95 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, 2 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 34.7

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

   5. Simplified 22.7

      \[\leadsto \color{blue}{\frac{D \cdot M}{\frac{\frac{\frac{d}{M} \cdot d}{h}}{D}} \cdot 0.25} \]
      Proof
      (*.f64 (/.f64 (*.f64 D M) (/.f64 (/.f64 (*.f64 (/.f64 d M) d) h) D)) 1/4): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.f64 D M) (/.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (/.f64 d M) (/.f64 d h))) D)) 1/4): 7 points increase in error, 16 points decrease in error
      (*.f64 (/.f64 (*.f64 D M) (/.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 d (/.f64 d h)) M)) D)) 1/4): 17 points increase in error, 7 points decrease in error
      (*.f64 (/.f64 (*.f64 D M) (/.f64 (/.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 d d) h)) M) D)) 1/4): 24 points increase in error, 3 points decrease in error
      (*.f64 (/.f64 (*.f64 D M) (/.f64 (/.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 d 2)) h) M) D)) 1/4): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.f64 D M) (Rewrite<= associate-/r*_binary64 (/.f64 (/.f64 (pow.f64 d 2) h) (*.f64 M D)))) 1/4): 4 points increase in error, 7 points decrease in error
      (*.f64 (/.f64 (*.f64 D M) (/.f64 (/.f64 (pow.f64 d 2) h) (Rewrite=> *-commutative_binary64 (*.f64 D M)))) 1/4): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 D M) (*.f64 D M)) (/.f64 (pow.f64 d 2) h))) 1/4): 25 points increase in error, 8 points decrease in error
      (*.f64 (/.f64 (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 D D) (*.f64 M M))) (/.f64 (pow.f64 d 2) h)) 1/4): 52 points increase in error, 1 points decrease in error
      (*.f64 (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 D 2)) (*.f64 M M)) (/.f64 (pow.f64 d 2) h)) 1/4): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.f64 (pow.f64 D 2) (Rewrite<= unpow2_binary64 (pow.f64 M 2))) (/.f64 (pow.f64 d 2) h)) 1/4): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite=> associate-/l*_binary64 (/.f64 (pow.f64 D 2) (/.f64 (/.f64 (pow.f64 d 2) h) (pow.f64 M 2)))) 1/4): 3 points increase in error, 6 points decrease in error
      (*.f64 (/.f64 (pow.f64 D 2) (Rewrite<= associate-/r*_binary64 (/.f64 (pow.f64 d 2) (*.f64 h (pow.f64 M 2))))) 1/4): 7 points increase in error, 8 points decrease in error
      (*.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 h (pow.f64 M 2))) (pow.f64 d 2))) 1/4): 5 points increase in error, 4 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 h (pow.f64 M 2))) (pow.f64 d 2)))): 0 points increase in error, 0 points decrease in error

   6. Taylor expanded in d around 0 25.2

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

   7. Simplified 18.8

      \[\leadsto \frac{D \cdot M}{\color{blue}{\frac{\frac{d}{h}}{\frac{M}{\frac{d}{D}}}}} \cdot 0.25 \]
      Proof
      (/.f64 (/.f64 d h) (/.f64 M (/.f64 d D))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 (/.f64 d h) M) (/.f64 d D))): 20 points increase in error, 23 points decrease in error
      (*.f64 (Rewrite=> associate-/l/_binary64 (/.f64 d (*.f64 M h))) (/.f64 d D)): 25 points increase in error, 17 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 d d) (*.f64 (*.f64 M h) D))): 42 points increase in error, 16 points decrease in error
      (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 d 2)) (*.f64 (*.f64 M h) D)): 0 points increase in error, 0 points decrease in error
      (/.f64 (pow.f64 d 2) (Rewrite<= *-commutative_binary64 (*.f64 D (*.f64 M h)))): 0 points increase in error, 0 points decrease in error

   8. Applied egg-rr 16.5

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

4. Final simplification 17.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq -4 \cdot 10^{-188}:\\ \;\;\;\;d \cdot \left(c0 \cdot \frac{d}{D \cdot \left(D \cdot \left(\left(w \cdot h\right) \cdot \frac{w}{c0}\right)\right)}\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 0:\\ \;\;\;\;\frac{\left(D \cdot \left(h \cdot \frac{M}{d}\right)\right) \cdot \left(D \cdot 0.25\right)}{\frac{d}{M}}\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 10^{+119}:\\ \;\;\;\;\frac{c0 \cdot 0.5}{\frac{w}{2 \cdot \left(\frac{d}{D} \cdot \frac{\frac{d}{D}}{h \cdot \frac{w}{c0}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{M \cdot \left(D \cdot 0.25\right)}{\frac{d}{h}}}{\frac{d}{D \cdot M}}\\ \end{array} \]

Alternatives

Alternative 1
Error	19.0
Cost	1224

\[\begin{array}{l} t_0 := \frac{\frac{M \cdot \left(D \cdot 0.25\right)}{\frac{d}{h}}}{\frac{d}{D \cdot M}}\\ \mathbf{if}\;M \leq 2.2377834845586857 \cdot 10^{-300}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;M \leq 7.8 \cdot 10^{-162}:\\ \;\;\;\;\frac{\left(D \cdot \left(h \cdot \frac{M}{d}\right)\right) \cdot \left(D \cdot 0.25\right)}{\frac{d}{M}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	17.6
Cost	1224

\[\begin{array}{l} t_0 := \frac{\frac{M \cdot \left(D \cdot 0.25\right)}{\frac{d}{h}}}{\frac{d}{D \cdot M}}\\ \mathbf{if}\;h \leq -5 \cdot 10^{-23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;h \leq 4 \cdot 10^{+220}:\\ \;\;\;\;\left(0.25 \cdot \left(h \cdot \frac{D}{\frac{d}{M}}\right)\right) \cdot \left(M \cdot \frac{D}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	24.4
Cost	960

\[h \cdot \frac{D}{\frac{\frac{d}{M} \cdot \frac{d}{M}}{D \cdot 0.25}} \]

Alternative 4
Error	19.1
Cost	960

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

Alternative 5
Error	16.5
Cost	960

\[\begin{array}{l} t_0 := \frac{D}{\frac{d}{M}}\\ t_0 \cdot \left(0.25 \cdot \left(h \cdot t_0\right)\right) \end{array} \]

Alternative 6
Error	31.5
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022309 
(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))))))