Henrywood and Agarwal, Equation (12)

Average Error: 26.9 → 13.5

Time: 40.3s

Precision: binary64

Cost: 104464

\[ \begin{array}{c}[M, D] = \mathsf{sort}([M, D])\\ \end{array} \]

Math

\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

↓

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right)\\ t_3 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{+303}:\\ \;\;\;\;\left(t_0 \cdot t_1\right) \cdot \left(-0.125 \cdot {\left(\frac{D}{d} \cdot \frac{M}{\sqrt{\frac{\ell}{h}}}\right)}^{2}\right)\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{-187}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(t_1 \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(\frac{M \cdot \frac{D}{2}}{d}\right)}^{2} \cdot -0.5, 1\right)\right)\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+245}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (*
  (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
  (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))

↓

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h)))
        (t_1 (sqrt (/ d l)))
        (t_2
         (*
          (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
          (+ 1.0 (* (/ h l) (* (pow (/ (* M D) (* d 2.0)) 2.0) -0.5)))))
        (t_3 (fabs (/ d (sqrt (* h l))))))
   (if (<= t_2 -5e+303)
     (* (* t_0 t_1) (* -0.125 (pow (* (/ D d) (/ M (sqrt (/ l h)))) 2.0)))
     (if (<= t_2 -2e-187)
       (*
        (/ 1.0 (sqrt (/ h d)))
        (* t_1 (fma (/ h l) (* (pow (/ (* M (/ D 2.0)) d) 2.0) -0.5) 1.0)))
       (if (<= t_2 0.0)
         t_3
         (if (<= t_2 5e+245)
           (*
            t_1
            (*
             t_0
             (- 1.0 (* 0.5 (/ (* h (pow (* (* 0.5 M) (/ D d)) 2.0)) l)))))
           t_3))))))

C

double code(double d, double h, double l, double M, double D) {
	return (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
}

↓

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h));
	double t_1 = sqrt((d / l));
	double t_2 = (pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 + ((h / l) * (pow(((M * D) / (d * 2.0)), 2.0) * -0.5)));
	double t_3 = fabs((d / sqrt((h * l))));
	double tmp;
	if (t_2 <= -5e+303) {
		tmp = (t_0 * t_1) * (-0.125 * pow(((D / d) * (M / sqrt((l / h)))), 2.0));
	} else if (t_2 <= -2e-187) {
		tmp = (1.0 / sqrt((h / d))) * (t_1 * fma((h / l), (pow(((M * (D / 2.0)) / d), 2.0) * -0.5), 1.0));
	} else if (t_2 <= 0.0) {
		tmp = t_3;
	} else if (t_2 <= 5e+245) {
		tmp = t_1 * (t_0 * (1.0 - (0.5 * ((h * pow(((0.5 * M) * (D / d)), 2.0)) / l))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	return Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
end

↓

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / h))
	t_1 = sqrt(Float64(d / l))
	t_2 = Float64(Float64((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5)) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * -0.5))))
	t_3 = abs(Float64(d / sqrt(Float64(h * l))))
	tmp = 0.0
	if (t_2 <= -5e+303)
		tmp = Float64(Float64(t_0 * t_1) * Float64(-0.125 * (Float64(Float64(D / d) * Float64(M / sqrt(Float64(l / h)))) ^ 2.0)));
	elseif (t_2 <= -2e-187)
		tmp = Float64(Float64(1.0 / sqrt(Float64(h / d))) * Float64(t_1 * fma(Float64(h / l), Float64((Float64(Float64(M * Float64(D / 2.0)) / d) ^ 2.0) * -0.5), 1.0)));
	elseif (t_2 <= 0.0)
		tmp = t_3;
	elseif (t_2 <= 5e+245)
		tmp = Float64(t_1 * Float64(t_0 * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(Float64(0.5 * M) * Float64(D / d)) ^ 2.0)) / l)))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, -5e+303], N[(N[(t$95$0 * t$95$1), $MachinePrecision] * N[(-0.125 * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M / N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -2e-187], N[(N[(1.0 / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M * N[(D / 2.0), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], t$95$3, If[LessEqual[t$95$2, 5e+245], N[(t$95$1 * N[(t$95$0 * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(N[(0.5 * M), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < -4.9999999999999997e303

   1. Initial program 63.8

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Simplified 61.6

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)} \]
      Proof
      (*.f64 (*.f64 (sqrt.f64 (/.f64 d h)) (sqrt.f64 (/.f64 d l))) (fma.f64 (pow.f64 (*.f64 D (/.f64 M (*.f64 d 2))) 2) (*.f64 (/.f64 h l) -1/2) 1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (Rewrite<= unpow1/2_binary64 (pow.f64 (/.f64 d h) 1/2)) (sqrt.f64 (/.f64 d l))) (fma.f64 (pow.f64 (*.f64 D (/.f64 M (*.f64 d 2))) 2) (*.f64 (/.f64 h l) -1/2) 1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (Rewrite<= metadata-eval (/.f64 1 2))) (sqrt.f64 (/.f64 d l))) (fma.f64 (pow.f64 (*.f64 D (/.f64 M (*.f64 d 2))) 2) (*.f64 (/.f64 h l) -1/2) 1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (Rewrite<= unpow1/2_binary64 (pow.f64 (/.f64 d l) 1/2))) (fma.f64 (pow.f64 (*.f64 D (/.f64 M (*.f64 d 2))) 2) (*.f64 (/.f64 h l) -1/2) 1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (Rewrite<= metadata-eval (/.f64 1 2)))) (fma.f64 (pow.f64 (*.f64 D (/.f64 M (*.f64 d 2))) 2) (*.f64 (/.f64 h l) -1/2) 1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (fma.f64 (pow.f64 (*.f64 D (/.f64 M (Rewrite<= *-commutative_binary64 (*.f64 2 d)))) 2) (*.f64 (/.f64 h l) -1/2) 1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (fma.f64 (pow.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 M (*.f64 2 d)) D)) 2) (*.f64 (/.f64 h l) -1/2) 1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (fma.f64 (pow.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 M D) (*.f64 2 d))) 2) (*.f64 (/.f64 h l) -1/2) 1)): 6 points increase in error, 11 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (fma.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (*.f64 (/.f64 h l) (Rewrite<= metadata-eval (neg.f64 1/2))) 1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (fma.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (*.f64 (/.f64 h l) (neg.f64 (Rewrite<= metadata-eval (/.f64 1 2)))) 1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (fma.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (/.f64 h l) (/.f64 1 2)))) 1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (fma.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 1 2) (/.f64 h l)))) 1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (neg.f64 (*.f64 (/.f64 1 2) (/.f64 h l)))) 1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (+.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (*.f64 (/.f64 1 2) (/.f64 h l))))) 1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (+.f64 (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 1 2)) (/.f64 h l)))) 1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (+.f64 (neg.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2))) (/.f64 h l))) 1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (+.f64 (Rewrite<= distribute-lft-neg-out_binary64 (*.f64 (neg.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2))) (/.f64 h l))) 1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 (neg.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2))) (/.f64 h l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l))))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in D around inf 59.7

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell}\right)} \]

   4. Simplified 59.8

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{\left(-0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right)\right)} \]
      Proof
      (*.f64 -1/8 (*.f64 (/.f64 (*.f64 D D) (*.f64 d d)) (/.f64 (*.f64 (*.f64 M M) h) l))): 0 points increase in error, 0 points decrease in error
      (*.f64 -1/8 (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 D 2)) (*.f64 d d)) (/.f64 (*.f64 (*.f64 M M) h) l))): 0 points increase in error, 0 points decrease in error
      (*.f64 -1/8 (*.f64 (/.f64 (pow.f64 D 2) (Rewrite<= unpow2_binary64 (pow.f64 d 2))) (/.f64 (*.f64 (*.f64 M M) h) l))): 0 points increase in error, 0 points decrease in error
      (*.f64 -1/8 (*.f64 (/.f64 (pow.f64 D 2) (pow.f64 d 2)) (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 M 2)) h) l))): 0 points increase in error, 0 points decrease in error
      (*.f64 -1/8 (*.f64 (/.f64 (pow.f64 D 2) (pow.f64 d 2)) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 h (pow.f64 M 2))) l))): 0 points increase in error, 0 points decrease in error
      (*.f64 -1/8 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 h (pow.f64 M 2))) (*.f64 (pow.f64 d 2) l)))): 13 points increase in error, 7 points decrease in error

   5. Applied egg-rr 44.3

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-0.125 \cdot \color{blue}{\left(0 + {\left(\frac{M}{\sqrt{\frac{\ell}{h}}} \cdot \frac{D}{d}\right)}^{2}\right)}\right) \]

   6. Simplified 44.3

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-0.125 \cdot \color{blue}{{\left(\frac{M}{\sqrt{\frac{\ell}{h}}} \cdot \frac{D}{d}\right)}^{2}}\right) \]
      Proof
      (pow.f64 (*.f64 (/.f64 M (sqrt.f64 (/.f64 l h))) (/.f64 D d)) 2): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-lft-identity_binary64 (+.f64 0 (pow.f64 (*.f64 (/.f64 M (sqrt.f64 (/.f64 l h))) (/.f64 D d)) 2))): 0 points increase in error, 0 points decrease in error

   if -4.9999999999999997e303 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < -2e-187

   1. Initial program 1.1

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Simplified 2.5

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(\frac{\frac{D}{2} \cdot M}{d}\right)}^{2} \cdot -0.5, 1\right)\right)} \]
      Proof
      (*.f64 (sqrt.f64 (/.f64 d h)) (*.f64 (sqrt.f64 (/.f64 d l)) (fma.f64 (/.f64 h l) (*.f64 (pow.f64 (/.f64 (*.f64 (/.f64 D 2) M) d) 2) -1/2) 1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= unpow1/2_binary64 (pow.f64 (/.f64 d h) 1/2)) (*.f64 (sqrt.f64 (/.f64 d l)) (fma.f64 (/.f64 h l) (*.f64 (pow.f64 (/.f64 (*.f64 (/.f64 D 2) M) d) 2) -1/2) 1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d h) (Rewrite<= metadata-eval (/.f64 1 2))) (*.f64 (sqrt.f64 (/.f64 d l)) (fma.f64 (/.f64 h l) (*.f64 (pow.f64 (/.f64 (*.f64 (/.f64 D 2) M) d) 2) -1/2) 1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (*.f64 (Rewrite<= unpow1/2_binary64 (pow.f64 (/.f64 d l) 1/2)) (fma.f64 (/.f64 h l) (*.f64 (pow.f64 (/.f64 (*.f64 (/.f64 D 2) M) d) 2) -1/2) 1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d l) (Rewrite<= metadata-eval (/.f64 1 2))) (fma.f64 (/.f64 h l) (*.f64 (pow.f64 (/.f64 (*.f64 (/.f64 D 2) M) d) 2) -1/2) 1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (fma.f64 (/.f64 h l) (*.f64 (pow.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (/.f64 D 2) (/.f64 M d))) 2) -1/2) 1))): 10 points increase in error, 6 points decrease in error
      (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (fma.f64 (/.f64 h l) (*.f64 (pow.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 D M) (*.f64 2 d))) 2) -1/2) 1))): 6 points increase in error, 10 points decrease in error
      (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (fma.f64 (/.f64 h l) (*.f64 (pow.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 M D)) (*.f64 2 d)) 2) -1/2) 1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (fma.f64 (/.f64 h l) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (Rewrite<= metadata-eval (neg.f64 1/2))) 1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (fma.f64 (/.f64 h l) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (neg.f64 (Rewrite<= metadata-eval (/.f64 1 2)))) 1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (fma.f64 (/.f64 h l) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 1 2)))) 1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (fma.f64 (/.f64 h l) (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)))) 1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (/.f64 h l) (neg.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)))) 1)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2))) (/.f64 h l))) 1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 (neg.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2))) (/.f64 h l)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l))))): 5 points increase in error, 3 points decrease in error

   3. Applied egg-rr 2.9

      \[\leadsto \color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(\frac{\frac{D}{2} \cdot M}{d}\right)}^{2} \cdot -0.5, 1\right)\right) \]

   if -2e-187 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < -0.0 or 5.00000000000000034e245 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l))))

   1. Initial program 56.3

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Simplified 56.6

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)} \]
      Proof
      (*.f64 (*.f64 (sqrt.f64 (/.f64 d h)) (sqrt.f64 (/.f64 d l))) (fma.f64 (pow.f64 (*.f64 D (/.f64 M (*.f64 d 2))) 2) (*.f64 (/.f64 h l) -1/2) 1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (Rewrite<= unpow1/2_binary64 (pow.f64 (/.f64 d h) 1/2)) (sqrt.f64 (/.f64 d l))) (fma.f64 (pow.f64 (*.f64 D (/.f64 M (*.f64 d 2))) 2) (*.f64 (/.f64 h l) -1/2) 1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (Rewrite<= metadata-eval (/.f64 1 2))) (sqrt.f64 (/.f64 d l))) (fma.f64 (pow.f64 (*.f64 D (/.f64 M (*.f64 d 2))) 2) (*.f64 (/.f64 h l) -1/2) 1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (Rewrite<= unpow1/2_binary64 (pow.f64 (/.f64 d l) 1/2))) (fma.f64 (pow.f64 (*.f64 D (/.f64 M (*.f64 d 2))) 2) (*.f64 (/.f64 h l) -1/2) 1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (Rewrite<= metadata-eval (/.f64 1 2)))) (fma.f64 (pow.f64 (*.f64 D (/.f64 M (*.f64 d 2))) 2) (*.f64 (/.f64 h l) -1/2) 1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (fma.f64 (pow.f64 (*.f64 D (/.f64 M (Rewrite<= *-commutative_binary64 (*.f64 2 d)))) 2) (*.f64 (/.f64 h l) -1/2) 1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (fma.f64 (pow.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 M (*.f64 2 d)) D)) 2) (*.f64 (/.f64 h l) -1/2) 1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (fma.f64 (pow.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 M D) (*.f64 2 d))) 2) (*.f64 (/.f64 h l) -1/2) 1)): 6 points increase in error, 11 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (fma.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (*.f64 (/.f64 h l) (Rewrite<= metadata-eval (neg.f64 1/2))) 1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (fma.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (*.f64 (/.f64 h l) (neg.f64 (Rewrite<= metadata-eval (/.f64 1 2)))) 1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (fma.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (/.f64 h l) (/.f64 1 2)))) 1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (fma.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 1 2) (/.f64 h l)))) 1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (neg.f64 (*.f64 (/.f64 1 2) (/.f64 h l)))) 1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (+.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (*.f64 (/.f64 1 2) (/.f64 h l))))) 1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (+.f64 (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 1 2)) (/.f64 h l)))) 1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (+.f64 (neg.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2))) (/.f64 h l))) 1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (+.f64 (Rewrite<= distribute-lft-neg-out_binary64 (*.f64 (neg.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2))) (/.f64 h l))) 1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 (neg.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2))) (/.f64 h l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l))))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in d around inf 41.3

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

   4. Simplified 41.3

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
      Proof
      (*.f64 d (sqrt.f64 (/.f64 (/.f64 1 l) h))): 0 points increase in error, 0 points decrease in error
      (*.f64 d (sqrt.f64 (Rewrite<= associate-/r*_binary64 (/.f64 1 (*.f64 l h))))): 23 points increase in error, 13 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (sqrt.f64 (/.f64 1 (*.f64 l h))) d)): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 51.6

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h} \cdot \left(d \cdot d\right)}} \]

   6. Simplified 24.6

      \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{\ell \cdot h}}\right|} \]
      Proof
      (fabs.f64 (/.f64 d (sqrt.f64 (*.f64 l h)))): 0 points increase in error, 0 points decrease in error
      (fabs.f64 (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 d)) (sqrt.f64 (*.f64 l h)))): 0 points increase in error, 0 points decrease in error
      (fabs.f64 (/.f64 (*.f64 1 d) (Rewrite<= unpow1/2_binary64 (pow.f64 (*.f64 l h) 1/2)))): 0 points increase in error, 0 points decrease in error
      (fabs.f64 (/.f64 (*.f64 1 d) (Rewrite<= exp-to-pow_binary64 (exp.f64 (*.f64 (log.f64 (*.f64 l h)) 1/2))))): 143 points increase in error, 27 points decrease in error
      (fabs.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (exp.f64 (*.f64 (log.f64 (*.f64 l h)) 1/2))) d))): 23 points increase in error, 16 points decrease in error
      (fabs.f64 (*.f64 (Rewrite<= exp-neg_binary64 (exp.f64 (neg.f64 (*.f64 (log.f64 (*.f64 l h)) 1/2)))) d)): 18 points increase in error, 27 points decrease in error
      (fabs.f64 (*.f64 (exp.f64 (Rewrite<= distribute-lft-neg-out_binary64 (*.f64 (neg.f64 (log.f64 (*.f64 l h))) 1/2))) d)): 0 points increase in error, 0 points decrease in error
      (fabs.f64 (*.f64 (Rewrite=> exp-prod_binary64 (pow.f64 (exp.f64 (neg.f64 (log.f64 (*.f64 l h)))) 1/2)) d)): 17 points increase in error, 11 points decrease in error
      (fabs.f64 (*.f64 (pow.f64 (Rewrite=> exp-neg_binary64 (/.f64 1 (exp.f64 (log.f64 (*.f64 l h))))) 1/2) d)): 6 points increase in error, 5 points decrease in error
      (fabs.f64 (*.f64 (pow.f64 (/.f64 1 (Rewrite=> rem-exp-log_binary64 (*.f64 l h))) 1/2) d)): 28 points increase in error, 138 points decrease in error
      (fabs.f64 (*.f64 (pow.f64 (/.f64 1 (*.f64 l h)) (Rewrite<= metadata-eval (*.f64 2 1/4))) d)): 0 points increase in error, 0 points decrease in error
      (fabs.f64 (*.f64 (Rewrite<= pow-sqr_binary64 (*.f64 (pow.f64 (/.f64 1 (*.f64 l h)) 1/4) (pow.f64 (/.f64 1 (*.f64 l h)) 1/4))) d)): 43 points increase in error, 19 points decrease in error
      (fabs.f64 (*.f64 (Rewrite=> pow-sqr_binary64 (pow.f64 (/.f64 1 (*.f64 l h)) (*.f64 2 1/4))) d)): 19 points increase in error, 43 points decrease in error
      (fabs.f64 (*.f64 (pow.f64 (/.f64 1 (*.f64 l h)) (Rewrite=> metadata-eval 1/2)) d)): 0 points increase in error, 0 points decrease in error
      (fabs.f64 (*.f64 (Rewrite=> unpow1/2_binary64 (sqrt.f64 (/.f64 1 (*.f64 l h)))) d)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= rem-sqrt-square_binary64 (sqrt.f64 (*.f64 (*.f64 (sqrt.f64 (/.f64 1 (*.f64 l h))) d) (*.f64 (sqrt.f64 (/.f64 1 (*.f64 l h))) d)))): 64 points increase in error, 22 points decrease in error
      (sqrt.f64 (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 (sqrt.f64 (/.f64 1 (*.f64 l h))) (sqrt.f64 (/.f64 1 (*.f64 l h)))) (*.f64 d d)))): 29 points increase in error, 6 points decrease in error
      (sqrt.f64 (*.f64 (Rewrite=> rem-square-sqrt_binary64 (/.f64 1 (*.f64 l h))) (*.f64 d d))): 2 points increase in error, 12 points decrease in error

   if -0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < 5.00000000000000034e245

   1. Initial program 1.0

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Simplified 1.2

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} \]
      Proof
      (*.f64 (sqrt.f64 (/.f64 d l)) (*.f64 (sqrt.f64 (/.f64 d h)) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= unpow1/2_binary64 (pow.f64 (/.f64 d l) 1/2)) (*.f64 (sqrt.f64 (/.f64 d h)) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d l) (Rewrite<= metadata-eval (/.f64 1 2))) (*.f64 (sqrt.f64 (/.f64 d h)) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (*.f64 (Rewrite<= unpow1/2_binary64 (pow.f64 (/.f64 d h) 1/2)) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d h) (Rewrite<= metadata-eval (/.f64 1 2))) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (-.f64 1 (*.f64 (Rewrite<= metadata-eval (/.f64 1 2)) (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (-.f64 1 (*.f64 (/.f64 1 2) (*.f64 (pow.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 M D) (*.f64 2 d))) 2) (/.f64 h l)))))): 7 points increase in error, 8 points decrease in error
      (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (-.f64 1 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (pow.f64 (/.f64 d h) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l))))): 4 points increase in error, 2 points decrease in error
      (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2)))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 1.2

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 13.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \leq -5 \cdot 10^{+303}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-0.125 \cdot {\left(\frac{D}{d} \cdot \frac{M}{\sqrt{\frac{\ell}{h}}}\right)}^{2}\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \leq -2 \cdot 10^{-187}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(\frac{M \cdot \frac{D}{2}}{d}\right)}^{2} \cdot -0.5, 1\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \leq 0:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \leq 5 \cdot 10^{+245}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]

Alternatives

Alternative 1
Error	13.3
Cost	104464

\[\begin{array}{l} t_0 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ t_1 := {\left(\frac{d}{\ell}\right)}^{0.5}\\ t_2 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot t_1\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right)\\ \mathbf{if}\;t_2 \leq -2 \cdot 10^{+297}:\\ \;\;\;\;\left(t_1 \cdot \frac{1}{\sqrt{\frac{h}{d}}}\right) \cdot \left(1 + 0.125 \cdot \left(M \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{-d} \cdot \frac{h}{\frac{\ell}{M}}\right)\right)\right)\right)\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{-187}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+245}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	13.4
Cost	104464

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right)\\ t_2 := \sqrt{\frac{d}{h}}\\ t_3 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+297}:\\ \;\;\;\;\left(t_2 \cdot t_0\right) \cdot \left(-0.125 \cdot {\left(\frac{D}{d} \cdot \frac{M}{\sqrt{\frac{\ell}{h}}}\right)}^{2}\right)\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-187}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+245}:\\ \;\;\;\;t_0 \cdot \left(t_2 \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 3
Error	13.5
Cost	104464

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right)\\ t_3 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{+303}:\\ \;\;\;\;\left(t_0 \cdot t_1\right) \cdot \left(-0.125 \cdot {\left(\frac{D}{d} \cdot \frac{M}{\sqrt{\frac{\ell}{h}}}\right)}^{2}\right)\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{-187}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(\frac{M \cdot \frac{D}{2}}{d}\right)}^{2} \cdot -0.5, 1\right)\right)\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+245}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 4
Error	13.7
Cost	96396

\[\begin{array}{l} t_0 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_3 := \left(t_2 \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right)\\ t_4 := \frac{D}{\frac{d \cdot 2}{M}} \cdot \sqrt{0.5}\\ \mathbf{if}\;t_3 \leq -2 \cdot 10^{-187}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot t_1\right) \cdot \left(1 - {\left(\frac{1}{\sqrt{\frac{\ell}{h}}} \cdot t_4\right)}^{2}\right)\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_3 \leq 5 \cdot 10^{+245}:\\ \;\;\;\;\left(t_2 \cdot t_1\right) \cdot \left(1 - {\left(t_4 \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	13.7
Cost	83532

\[\begin{array}{l} t_0 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ t_3 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right)\\ \mathbf{if}\;t_3 \leq -2 \cdot 10^{-187}:\\ \;\;\;\;\left(t_1 \cdot t_2\right) \cdot \left(1 - {\left(\frac{1}{\sqrt{\frac{\ell}{h}}} \cdot \left(\frac{D}{\frac{d \cdot 2}{M}} \cdot \sqrt{0.5}\right)\right)}^{2}\right)\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_3 \leq 5 \cdot 10^{+245}:\\ \;\;\;\;t_2 \cdot \left(t_1 \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	20.0
Cost	21004

\[\begin{array}{l} t_0 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{if}\;d \leq -4.8 \cdot 10^{+112}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -5.8 \cdot 10^{-211}:\\ \;\;\;\;\left({\left(\frac{d}{\ell}\right)}^{0.5} \cdot \frac{1}{\sqrt{\frac{h}{d}}}\right) \cdot \left(1 + 0.125 \cdot \left(M \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{-d} \cdot \frac{h}{\frac{\ell}{M}}\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{-221}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(M \cdot \frac{0.5 \cdot D}{d}\right)}^{2}\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 7
Error	20.6
Cost	21004

\[\begin{array}{l} \mathbf{if}\;d \leq -4.5 \cdot 10^{+112}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{elif}\;d \leq -2.1 \cdot 10^{-298}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d \cdot \frac{2}{M}}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-251}:\\ \;\;\;\;\frac{D}{\frac{\frac{d}{M}}{M \cdot D}} \cdot \left(-0.125 \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(M \cdot \frac{0.5 \cdot D}{d}\right)}^{2}\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 8
Error	21.3
Cost	15312

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{\ell}\right)}^{0.5} \cdot \frac{1}{\sqrt{\frac{h}{d}}}\right) \cdot \left(1 + 0.125 \cdot \left(M \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{-d} \cdot \frac{h}{\frac{\ell}{M}}\right)\right)\right)\right)\\ t_1 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{if}\;d \leq -5.2 \cdot 10^{+112}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -5.8 \cdot 10^{-211}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 5.2 \cdot 10^{-221}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 5.1 \cdot 10^{+42}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 9
Error	21.8
Cost	14920

\[\begin{array}{l} t_0 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{if}\;d \leq -1.65 \cdot 10^{+111}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -2.3 \cdot 10^{-123}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + 0.5 \cdot \left(\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\ell} \cdot \frac{h}{d \cdot d}\right) \cdot -0.25\right)\right)\right)\\ \mathbf{elif}\;d \leq 10^{-234}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 10
Error	24.6
Cost	14600

\[\begin{array}{l} \mathbf{if}\;d \leq -1.3 \cdot 10^{+90}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{elif}\;d \leq -2.1 \cdot 10^{-298}:\\ \;\;\;\;\sqrt{\frac{d}{\ell \cdot \frac{h}{d}}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D}{\frac{d}{\frac{M}{2}}}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 11
Error	24.3
Cost	14600

\[\begin{array}{l} t_0 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{if}\;d \leq -1.45 \cdot 10^{+86}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -5.8 \cdot 10^{-211}:\\ \;\;\;\;\left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\left(0.5 \cdot D\right) \cdot \frac{M}{d}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 1.55 \cdot 10^{-221}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 12
Error	23.6
Cost	13516

\[\begin{array}{l} t_0 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{if}\;d \leq -4.6 \cdot 10^{+112}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -3.1 \cdot 10^{-182}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 1.48 \cdot 10^{-221}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 13
Error	28.6
Cost	13384

\[\begin{array}{l} t_0 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{if}\;d \leq 3.8 \cdot 10^{-138}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+48}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 14
Error	24.2
Cost	13252

\[\begin{array}{l} \mathbf{if}\;h \leq 2.9 \cdot 10^{-288}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 15
Error	35.4
Cost	6980

\[\begin{array}{l} \mathbf{if}\;\ell \leq -6.5 \cdot 10^{-196}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 16
Error	33.4
Cost	6980

\[\begin{array}{l} \mathbf{if}\;\ell \leq 9.5 \cdot 10^{-283}:\\ \;\;\;\;\sqrt{d \cdot \frac{\frac{d}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 17
Error	34.2
Cost	6980

\[\begin{array}{l} \mathbf{if}\;h \leq 4.7 \cdot 10^{-247}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 18
Error	62.2
Cost	6720

\[d \cdot \sqrt{h \cdot \ell} \]

Alternative 19
Error	44.0
Cost	6720

\[\frac{d}{\sqrt{h \cdot \ell}} \]

Reproduce

herbie shell --seed 2022332 
(FPCore (d h l M D)
  :name "Henrywood and Agarwal, Equation (12)"
  :precision binary64
  (* (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))) (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))