Henrywood and Agarwal, Equation (12)

Average Error: 26.5 → 14.5

Time: 35.4s

Precision: binary64

Cost: 104464

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

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right)\\ t_3 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \frac{{\left(D \cdot \frac{0.5 \cdot M}{d}\right)}^{2}}{\ell} \cdot \left(h \cdot -0.5\right)\right)\\ \mathbf{elif}\;t_2 \leq -1 \cdot 10^{-164}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq 10^{+198}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

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 l)))
        (t_1 (sqrt (/ d h)))
        (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 (- INFINITY))
     (*
      (/ d (* (sqrt h) (sqrt l)))
      (+ 1.0 (* (/ (pow (* D (/ (* 0.5 M) d)) 2.0) l) (* h -0.5))))
     (if (<= t_2 -1e-164)
       (*
        t_0
        (* t_1 (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D d) (/ M 2.0)) 2.0))))))
       (if (<= t_2 0.0)
         t_3
         (if (<= t_2 1e+198)
           (*
            t_0
            (*
             t_1
             (- 1.0 (* 0.5 (* h (/ (pow (* D (/ (/ M d) 2.0)) 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 / l));
	double t_1 = sqrt((d / h));
	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 <= -((double) INFINITY)) {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 + ((pow((D * ((0.5 * M) / d)), 2.0) / l) * (h * -0.5)));
	} else if (t_2 <= -1e-164) {
		tmp = t_0 * (t_1 * (1.0 - (0.5 * ((h / l) * pow(((D / d) * (M / 2.0)), 2.0)))));
	} else if (t_2 <= 0.0) {
		tmp = t_3;
	} else if (t_2 <= 1e+198) {
		tmp = t_0 * (t_1 * (1.0 - (0.5 * (h * (pow((D * ((M / d) / 2.0)), 2.0) / l)))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Java

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

↓

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((d / l));
	double t_1 = Math.sqrt((d / h));
	double t_2 = (Math.pow((d / h), 0.5) * Math.pow((d / l), 0.5)) * (1.0 + ((h / l) * (Math.pow(((M * D) / (d * 2.0)), 2.0) * -0.5)));
	double t_3 = Math.abs((d / Math.sqrt((h * l))));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * (1.0 + ((Math.pow((D * ((0.5 * M) / d)), 2.0) / l) * (h * -0.5)));
	} else if (t_2 <= -1e-164) {
		tmp = t_0 * (t_1 * (1.0 - (0.5 * ((h / l) * Math.pow(((D / d) * (M / 2.0)), 2.0)))));
	} else if (t_2 <= 0.0) {
		tmp = t_3;
	} else if (t_2 <= 1e+198) {
		tmp = t_0 * (t_1 * (1.0 - (0.5 * (h * (Math.pow((D * ((M / d) / 2.0)), 2.0) / l)))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	return (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))

↓

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / l))
	t_1 = math.sqrt((d / h))
	t_2 = (math.pow((d / h), 0.5) * math.pow((d / l), 0.5)) * (1.0 + ((h / l) * (math.pow(((M * D) / (d * 2.0)), 2.0) * -0.5)))
	t_3 = math.fabs((d / math.sqrt((h * l))))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = (d / (math.sqrt(h) * math.sqrt(l))) * (1.0 + ((math.pow((D * ((0.5 * M) / d)), 2.0) / l) * (h * -0.5)))
	elif t_2 <= -1e-164:
		tmp = t_0 * (t_1 * (1.0 - (0.5 * ((h / l) * math.pow(((D / d) * (M / 2.0)), 2.0)))))
	elif t_2 <= 0.0:
		tmp = t_3
	elif t_2 <= 1e+198:
		tmp = t_0 * (t_1 * (1.0 - (0.5 * (h * (math.pow((D * ((M / d) / 2.0)), 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 / l))
	t_1 = sqrt(Float64(d / h))
	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 <= Float64(-Inf))
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 + Float64(Float64((Float64(D * Float64(Float64(0.5 * M) / d)) ^ 2.0) / l) * Float64(h * -0.5))));
	elseif (t_2 <= -1e-164)
		tmp = Float64(t_0 * Float64(t_1 * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D / d) * Float64(M / 2.0)) ^ 2.0))))));
	elseif (t_2 <= 0.0)
		tmp = t_3;
	elseif (t_2 <= 1e+198)
		tmp = Float64(t_0 * Float64(t_1 * Float64(1.0 - Float64(0.5 * Float64(h * Float64((Float64(D * Float64(Float64(M / d) / 2.0)) ^ 2.0) / l))))));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
end

↓

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / l));
	t_1 = sqrt((d / h));
	t_2 = (((d / h) ^ 0.5) * ((d / l) ^ 0.5)) * (1.0 + ((h / l) * ((((M * D) / (d * 2.0)) ^ 2.0) * -0.5)));
	t_3 = abs((d / sqrt((h * l))));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 + ((((D * ((0.5 * M) / d)) ^ 2.0) / l) * (h * -0.5)));
	elseif (t_2 <= -1e-164)
		tmp = t_0 * (t_1 * (1.0 - (0.5 * ((h / l) * (((D / d) * (M / 2.0)) ^ 2.0)))));
	elseif (t_2 <= 0.0)
		tmp = t_3;
	elseif (t_2 <= 1e+198)
		tmp = t_0 * (t_1 * (1.0 - (0.5 * (h * (((D * ((M / d) / 2.0)) ^ 2.0) / l)))));
	else
		tmp = t_3;
	end
	tmp_2 = 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 / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $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, (-Infinity)], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(N[Power[N[(D * N[(N[(0.5 * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(h * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1e-164], N[(t$95$0 * N[(t$95$1 * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], t$95$3, If[LessEqual[t$95$2, 1e+198], N[(t$95$0 * N[(t$95$1 * N[(1.0 - N[(0.5 * N[(h * N[(N[Power[N[(D * N[(N[(M / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $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)))) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 60.9

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
      Proof
      (*.f64 (*.f64 (sqrt.f64 (/.f64 d h)) (sqrt.f64 (/.f64 d l))) (-.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 (*.f64 (Rewrite<= unpow1/2_binary64 (pow.f64 (/.f64 d h) 1/2)) (sqrt.f64 (/.f64 d l))) (-.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 (*.f64 (pow.f64 (/.f64 d h) (Rewrite<= metadata-eval (/.f64 1 2))) (sqrt.f64 (/.f64 d l))) (-.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 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (Rewrite<= unpow1/2_binary64 (pow.f64 (/.f64 d l) 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 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (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 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.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 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.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))))): 4 points increase in error, 3 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.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

   3. Applied egg-rr 53.4

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

   4. Applied egg-rr 56.9

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

   5. Simplified 56.9

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \frac{{\left(\frac{M \cdot 0.5}{d} \cdot D\right)}^{2}}{\ell} \cdot \left(h \cdot -0.5\right)\right)} \]
      Proof
      (*.f64 (/.f64 d (*.f64 (sqrt.f64 h) (sqrt.f64 l))) (+.f64 1 (*.f64 (/.f64 (pow.f64 (*.f64 (/.f64 (*.f64 M 1/2) d) D) 2) l) (*.f64 h -1/2)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 d (*.f64 (sqrt.f64 h) (sqrt.f64 l))) (+.f64 1 (*.f64 (/.f64 (pow.f64 (Rewrite<= associate-/r/_binary64 (/.f64 (*.f64 M 1/2) (/.f64 d D))) 2) l) (*.f64 h -1/2)))): 3 points increase in error, 6 points decrease in error
      (*.f64 (/.f64 d (*.f64 (sqrt.f64 h) (sqrt.f64 l))) (+.f64 1 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (pow.f64 (/.f64 (*.f64 M 1/2) (/.f64 d D)) 2) l) h) -1/2)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 d (*.f64 (sqrt.f64 h) (sqrt.f64 l))) (Rewrite<= rem-log-exp_binary64 (log.f64 (exp.f64 (+.f64 1 (*.f64 (*.f64 (/.f64 (pow.f64 (/.f64 (*.f64 M 1/2) (/.f64 d D)) 2) l) h) -1/2)))))): 23 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 d (*.f64 (sqrt.f64 h) (sqrt.f64 l))) (Rewrite=> rem-log-exp_binary64 (+.f64 1 (*.f64 (*.f64 (/.f64 (pow.f64 (/.f64 (*.f64 M 1/2) (/.f64 d D)) 2) l) h) -1/2)))): 0 points increase in error, 23 points decrease in error
      (Rewrite=> distribute-rgt-in_binary64 (+.f64 (*.f64 1 (/.f64 d (*.f64 (sqrt.f64 h) (sqrt.f64 l)))) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 (/.f64 (*.f64 M 1/2) (/.f64 d D)) 2) l) h) -1/2) (/.f64 d (*.f64 (sqrt.f64 h) (sqrt.f64 l)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite=> *-lft-identity_binary64 (/.f64 d (*.f64 (sqrt.f64 h) (sqrt.f64 l)))) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 (/.f64 (*.f64 M 1/2) (/.f64 d D)) 2) l) h) -1/2) (/.f64 d (*.f64 (sqrt.f64 h) (sqrt.f64 l))))): 0 points increase in error, 0 points decrease in error

   if -inf.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)))) < -9.99999999999999962e-165

   1. Initial program 1.5

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

   2. Simplified 5.4

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} \]
      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)))))): 6 points increase in error, 3 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))))): 5 points increase in error, 8 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

   if -9.99999999999999962e-165 < (*.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 1.00000000000000002e198 < (*.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 52.7

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

   2. Simplified 52.9

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
      Proof
      (*.f64 (*.f64 (sqrt.f64 (/.f64 d h)) (sqrt.f64 (/.f64 d l))) (-.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 (*.f64 (Rewrite<= unpow1/2_binary64 (pow.f64 (/.f64 d h) 1/2)) (sqrt.f64 (/.f64 d l))) (-.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 (*.f64 (pow.f64 (/.f64 d h) (Rewrite<= metadata-eval (/.f64 1 2))) (sqrt.f64 (/.f64 d l))) (-.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 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (Rewrite<= unpow1/2_binary64 (pow.f64 (/.f64 d l) 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 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (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 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.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 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.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))))): 4 points increase in error, 3 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.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

   3. Taylor expanded in d around inf 40.4

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

   4. Simplified 40.4

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
      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))))): 26 points increase in error, 14 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 51.7

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

   6. Simplified 23.0

      \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{\ell \cdot h}}\right|} \]
      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, 1 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))))): 153 points increase in error, 26 points decrease in error
      (fabs.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (exp.f64 (*.f64 (log.f64 (*.f64 l h)) 1/2))) d))): 22 points increase in error, 29 points decrease in error
      (fabs.f64 (*.f64 (Rewrite<= exp-neg_binary64 (exp.f64 (neg.f64 (*.f64 (log.f64 (*.f64 l h)) 1/2)))) d)): 23 points increase in error, 23 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)): 14 points increase in error, 8 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)): 8 points increase in error, 11 points decrease in error
      (fabs.f64 (*.f64 (pow.f64 (/.f64 1 (Rewrite=> rem-exp-log_binary64 (*.f64 l h))) 1/2) d)): 27 points increase in error, 153 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)): 55 points increase in error, 13 points decrease in error
      (fabs.f64 (*.f64 (Rewrite=> pow-sqr_binary64 (pow.f64 (/.f64 1 (*.f64 l h)) (*.f64 2 1/4))) d)): 13 points increase in error, 55 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)))): 70 points increase in error, 18 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)))): 37 points increase in error, 9 points decrease in error
      (sqrt.f64 (*.f64 (Rewrite=> rem-square-sqrt_binary64 (/.f64 1 (*.f64 l h))) (*.f64 d d))): 4 points increase in error, 15 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)))) < 1.00000000000000002e198

   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)))))): 6 points increase in error, 3 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))))): 5 points increase in error, 8 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

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)} - 1\right)}\right)\right) \]

   4. Simplified 1.1

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}}{\ell}\right)}\right)\right) \]
      Proof
      (*.f64 h (/.f64 (pow.f64 (*.f64 D (/.f64 (/.f64 M d) 2)) 2) l)): 0 points increase in error, 0 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (*.f64 D (Rewrite<= associate-/r*_binary64 (/.f64 M (*.f64 d 2)))) 2) l)): 1 points increase in error, 0 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (*.f64 D (/.f64 M (Rewrite<= *-commutative_binary64 (*.f64 2 d)))) 2) l)): 0 points increase in error, 0 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 M (*.f64 2 d)) D)) 2) l)): 0 points increase in error, 0 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 M D) (*.f64 2 d))) 2) l)): 15 points increase in error, 17 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 (*.f64 M D) 2) d)) 2) l)): 1 points increase in error, 0 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (/.f64 (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 D M)) 2) d) 2) l)): 0 points increase in error, 0 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (/.f64 (Rewrite=> associate-/l*_binary64 (/.f64 D (/.f64 2 M))) d) 2) l)): 12 points increase in error, 8 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (/.f64 (/.f64 D (/.f64 (Rewrite<= metadata-eval (/.f64 -2 -1)) M)) d) 2) l)): 0 points increase in error, 0 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (/.f64 (/.f64 D (Rewrite<= associate-/r*_binary64 (/.f64 -2 (*.f64 -1 M)))) d) 2) l)): 0 points increase in error, 0 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (/.f64 (/.f64 D (/.f64 -2 (Rewrite<= neg-mul-1_binary64 (neg.f64 M)))) d) 2) l)): 0 points increase in error, 0 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (/.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 D (neg.f64 M)) -2)) d) 2) l)): 8 points increase in error, 12 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (/.f64 (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 (neg.f64 M) D)) -2) d) 2) l)): 0 points increase in error, 0 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (Rewrite<= associate-/r*_binary64 (/.f64 (*.f64 (neg.f64 M) D) (*.f64 -2 d))) 2) l)): 0 points increase in error, 1 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (/.f64 (Rewrite=> distribute-lft-neg-out_binary64 (neg.f64 (*.f64 M D))) (*.f64 -2 d)) 2) l)): 0 points increase in error, 0 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (/.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (*.f64 M D))) (*.f64 -2 d)) 2) l)): 0 points increase in error, 0 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (Rewrite=> times-frac_binary64 (*.f64 (/.f64 -1 -2) (/.f64 (*.f64 M D) d))) 2) l)): 0 points increase in error, 0 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (*.f64 (Rewrite=> metadata-eval 1/2) (/.f64 (*.f64 M D) d)) 2) l)): 0 points increase in error, 0 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (*.f64 1/2 (Rewrite<= associate-*r/_binary64 (*.f64 M (/.f64 D d)))) 2) l)): 18 points increase in error, 15 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (*.f64 1/2 (Rewrite=> *-commutative_binary64 (*.f64 (/.f64 D d) M))) 2) l)): 0 points increase in error, 0 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 1/2 (/.f64 D d)) M)) 2) l)): 0 points increase in error, 0 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (Rewrite<= *-commutative_binary64 (*.f64 M (*.f64 1/2 (/.f64 D d)))) 2) l)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (pow.f64 (*.f64 M (*.f64 1/2 (/.f64 D d))) 2) l) h)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (pow.f64 (*.f64 M (*.f64 1/2 (/.f64 D d))) 2) h) l)): 27 points increase in error, 20 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 (pow.f64 (*.f64 M (*.f64 1/2 (/.f64 D d))) 2) (/.f64 h l))): 33 points increase in error, 29 points decrease in error
      (Rewrite<= expm1-log1p_binary64 (expm1.f64 (log1p.f64 (*.f64 (pow.f64 (*.f64 M (*.f64 1/2 (/.f64 D d))) 2) (/.f64 h l))))): 20 points increase in error, 18 points decrease in error
      (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (log1p.f64 (*.f64 (pow.f64 (*.f64 M (*.f64 1/2 (/.f64 D d))) 2) (/.f64 h l)))) 1)): 36 points increase in error, 10 points decrease in error
3. Recombined 4 regimes into one program.

4. Final simplification 14.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \leq -\infty:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \frac{{\left(D \cdot \frac{0.5 \cdot M}{d}\right)}^{2}}{\ell} \cdot \left(h \cdot -0.5\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \leq -1 \cdot 10^{-164}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \leq 0:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \leq 10^{+198}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]

Alternatives

Alternative 1
Error	13.7
Cost	83532

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right)\\ t_3 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{-164}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 - 0.5 \cdot {\left(\frac{\left(0.5 \cdot M\right) \cdot \frac{D}{d}}{\sqrt{\frac{\ell}{h}}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq 10^{+198}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 2
Error	14.1
Cost	83532

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right)\\ t_3 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{-164}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 - 0.5 \cdot {\left(M \cdot \frac{\frac{0.5}{\frac{d}{D}}}{\sqrt{\frac{\ell}{h}}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq 10^{+198}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 3
Error	13.7
Cost	83532

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right)\\ t_3 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{-164}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 - 0.5 \cdot {\left(\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq 10^{+198}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 4
Error	22.1
Cost	21004

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.4 \cdot 10^{-165}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + 0.5 \cdot \left(\left(h \cdot \left(M \cdot \frac{M}{\ell}\right)\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot -0.25\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 2.9 \cdot 10^{-48}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(M \cdot \frac{0.5 \cdot D}{d}\right)}^{2}\right)\\ \end{array} \]

Alternative 5
Error	20.8
Cost	21004

\[\begin{array}{l} \mathbf{if}\;d \leq -0.145:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;d \leq -5.2 \cdot 10^{-163}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{D \cdot D}{\ell} \cdot \left(\left(\frac{h}{d} \cdot \frac{M \cdot M}{d}\right) \cdot -0.125\right)\right)\\ \mathbf{elif}\;d \leq 2.6 \cdot 10^{-273}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \frac{{\left(D \cdot \frac{0.5 \cdot M}{d}\right)}^{2}}{\ell} \cdot \left(h \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 6
Error	18.4
Cost	21004

\[\begin{array}{l} \mathbf{if}\;d \leq -1.2 \cdot 10^{+215}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;d \leq -7 \cdot 10^{-164}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{elif}\;d \leq 8 \cdot 10^{-270}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \frac{{\left(D \cdot \frac{0.5 \cdot M}{d}\right)}^{2}}{\ell} \cdot \left(h \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 7
Error	23.7
Cost	14920

\[\begin{array}{l} \mathbf{if}\;\ell \leq -9.5 \cdot 10^{-172}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + 0.5 \cdot \left(\left(h \cdot \left(M \cdot \frac{M}{\ell}\right)\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot -0.25\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot \sqrt{\frac{1}{h}}\right)\\ \end{array} \]

Alternative 8
Error	24.7
Cost	14600

\[\begin{array}{l} \mathbf{if}\;\ell \leq -6.5 \cdot 10^{-57}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq -3.2 \cdot 10^{-266}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{\frac{d \cdot d}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot \sqrt{\frac{1}{h}}\right)\\ \end{array} \]

Alternative 9
Error	23.7
Cost	13444

\[\begin{array}{l} \mathbf{if}\;h \leq -1 \cdot 10^{-307}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot \sqrt{\frac{1}{h}}\right)\\ \end{array} \]

Alternative 10
Error	23.7
Cost	13380

\[\begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{-309}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 11
Error	23.7
Cost	13252

\[\begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{-309}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 12
Error	27.5
Cost	13120

\[\left|\frac{d}{\sqrt{h \cdot \ell}}\right| \]

Alternative 13
Error	27.7
Cost	7108

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.8 \cdot 10^{-270}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(-\frac{\ell}{\frac{-1}{h}}\right)}^{-0.5}\\ \end{array} \]

Alternative 14
Error	27.9
Cost	7044

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.8 \cdot 10^{-270}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 15
Error	27.7
Cost	7044

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.8 \cdot 10^{-270}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 16
Error	34.5
Cost	6980

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.9 \cdot 10^{-270}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 17
Error	33.1
Cost	6980

\[\begin{array}{l} \mathbf{if}\;h \leq 3.55 \cdot 10^{-298}:\\ \;\;\;\;\sqrt{d \cdot \frac{\frac{d}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 18
Error	33.0
Cost	6980

\[\begin{array}{l} \mathbf{if}\;h \leq 2.25 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 19
Error	43.8
Cost	6784

\[d \cdot {\left(h \cdot \ell\right)}^{-0.5} \]

Alternative 20
Error	43.8
Cost	6720

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

Reproduce

herbie shell --seed 2022330 
(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)))))