Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.6% → 79.5%

Time: 19.2s

Alternatives: 18

Speedup: 3.6×

Specification

Math

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

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)))))

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)));
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
end function

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)));
}

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)))

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

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

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]

Initial Program: 66.6% accurate, 1.0× speedup

Math

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

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)))))

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)));
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
end function

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)));
}

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)))

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

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

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]

Alternative 1: 79.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{\left(\frac{0.5}{d} \cdot D\right) \cdot M}{{h}^{-1}} \cdot \frac{\frac{M}{d} \cdot \left(\left(D \cdot 0.5\right) \cdot 0.5\right)}{\ell}\\ t_1 := {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\\ \mathbf{if}\;d \leq -1.9 \cdot 10^{-169}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right)\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-296}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\frac{h}{\ell}\right)}^{1.5}, \frac{{\left(M \cdot D\right)}^{2}}{d} \cdot -0.125, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{-68}:\\ \;\;\;\;\left(1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot t\_1\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (-
          1.0
          (*
           (/ (* (* (/ 0.5 d) D) M) (pow h -1.0))
           (/ (* (/ M d) (* (* D 0.5) 0.5)) l))))
        (t_1 (pow (/ d l) (/ 1.0 2.0))))
   (if (<= d -1.9e-169)
     (* t_0 (* t_1 (/ (sqrt (- d)) (sqrt (- h)))))
     (if (<= d 3.8e-296)
       (/
        (fma
         (pow (/ h l) 1.5)
         (* (/ (pow (* M D) 2.0) d) -0.125)
         (* (sqrt (/ h l)) d))
        h)
       (if (<= d 1.7e-68)
         (*
          (- 1.0 (* (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ 1.0 2.0)) (/ h l)))
          (* (sqrt (/ 1.0 (* l h))) d))
         (* (* (/ (sqrt d) (sqrt h)) t_1) t_0))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 - (((((0.5 / d) * D) * M) / pow(h, -1.0)) * (((M / d) * ((D * 0.5) * 0.5)) / l));
	double t_1 = pow((d / l), (1.0 / 2.0));
	double tmp;
	if (d <= -1.9e-169) {
		tmp = t_0 * (t_1 * (sqrt(-d) / sqrt(-h)));
	} else if (d <= 3.8e-296) {
		tmp = fma(pow((h / l), 1.5), ((pow((M * D), 2.0) / d) * -0.125), (sqrt((h / l)) * d)) / h;
	} else if (d <= 1.7e-68) {
		tmp = (1.0 - ((pow(((M * D) / (2.0 * d)), 2.0) * (1.0 / 2.0)) * (h / l))) * (sqrt((1.0 / (l * h))) * d);
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * t_1) * t_0;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(1.0 - Float64(Float64(Float64(Float64(Float64(0.5 / d) * D) * M) / (h ^ -1.0)) * Float64(Float64(Float64(M / d) * Float64(Float64(D * 0.5) * 0.5)) / l)))
	t_1 = Float64(d / l) ^ Float64(1.0 / 2.0)
	tmp = 0.0
	if (d <= -1.9e-169)
		tmp = Float64(t_0 * Float64(t_1 * Float64(sqrt(Float64(-d)) / sqrt(Float64(-h)))));
	elseif (d <= 3.8e-296)
		tmp = Float64(fma((Float64(h / l) ^ 1.5), Float64(Float64((Float64(M * D) ^ 2.0) / d) * -0.125), Float64(sqrt(Float64(h / l)) * d)) / h);
	elseif (d <= 1.7e-68)
		tmp = Float64(Float64(1.0 - Float64(Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)) * Float64(h / l))) * Float64(sqrt(Float64(1.0 / Float64(l * h))) * d));
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * t_1) * t_0);
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(1.0 - N[(N[(N[(N[(N[(0.5 / d), $MachinePrecision] * D), $MachinePrecision] * M), $MachinePrecision] / N[Power[h, -1.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M / d), $MachinePrecision] * N[(N[(D * 0.5), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -1.9e-169], N[(t$95$0 * N[(t$95$1 * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.8e-296], N[(N[(N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision] * N[(N[(N[Power[N[(M * D), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision] + N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[d, 1.7e-68], N[(N[(1.0 - N[(N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.9e-169

   1. Initial program 72.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 77.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\ell} \cdot \frac{\left(\frac{0.5}{d} \cdot D\right) \cdot M}{{h}^{-1}}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. metadata-eval 77.1

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 88.7

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

   6. Applied rewrites 88.7%

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

   if -1.9e-169 < d < 3.8000000000000002e-296

   1. Initial program 36.3%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   5. Applied rewrites 39.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \frac{D \cdot D}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]
   6. Step-by-step derivation

   7. Applied rewrites 68.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\left(\frac{h}{\ell}\right)}^{1.5}, -0.125 \cdot \frac{{\left(M \cdot D\right)}^{2}}{d}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   if 3.8000000000000002e-296 < d < 1.70000000000000009e-68

   1. Initial program 50.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 50.8

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

   4. Applied rewrites 50.8%

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

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 73.8

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

   7. Applied rewrites 73.8%

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

   if 1.70000000000000009e-68 < d

   1. Initial program 72.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 77.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\ell} \cdot \frac{\left(\frac{0.5}{d} \cdot D\right) \cdot M}{{h}^{-1}}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. metadata-eval 77.9

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 92.3

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

   6. Applied rewrites 92.3%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.9 \cdot 10^{-169}:\\ \;\;\;\;\left(1 - \frac{\left(\frac{0.5}{d} \cdot D\right) \cdot M}{{h}^{-1}} \cdot \frac{\frac{M}{d} \cdot \left(\left(D \cdot 0.5\right) \cdot 0.5\right)}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right)\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-296}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\frac{h}{\ell}\right)}^{1.5}, \frac{{\left(M \cdot D\right)}^{2}}{d} \cdot -0.125, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{-68}:\\ \;\;\;\;\left(1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot D\right) \cdot M}{{h}^{-1}} \cdot \frac{\frac{M}{d} \cdot \left(\left(D \cdot 0.5\right) \cdot 0.5\right)}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\ell \cdot h}}\\ t_1 := \left(\frac{0.5}{d} \cdot D\right) \cdot M\\ \mathbf{if}\;d \leq -2.95 \cdot 10^{-67}:\\ \;\;\;\;\left(1 - \frac{t\_1 \cdot 0.5}{{h}^{-1}} \cdot \frac{t\_1}{\ell}\right) \cdot \left(t\_0 \cdot \left(-d\right)\right)\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-296}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\frac{h}{\ell}\right)}^{1.5}, \frac{{\left(M \cdot D\right)}^{2}}{d} \cdot -0.125, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{-68}:\\ \;\;\;\;\left(1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left(t\_0 \cdot d\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{t\_1}{{h}^{-1}} \cdot \frac{\frac{M}{d} \cdot \left(\left(D \cdot 0.5\right) \cdot 0.5\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (* l h)))) (t_1 (* (* (/ 0.5 d) D) M)))
   (if (<= d -2.95e-67)
     (* (- 1.0 (* (/ (* t_1 0.5) (pow h -1.0)) (/ t_1 l))) (* t_0 (- d)))
     (if (<= d 3.8e-296)
       (/
        (fma
         (pow (/ h l) 1.5)
         (* (/ (pow (* M D) 2.0) d) -0.125)
         (* (sqrt (/ h l)) d))
        h)
       (if (<= d 1.7e-68)
         (*
          (- 1.0 (* (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ 1.0 2.0)) (/ h l)))
          (* t_0 d))
         (*
          (* (/ (sqrt d) (sqrt h)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (/ t_1 (pow h -1.0)) (/ (* (/ M d) (* (* D 0.5) 0.5)) l)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((1.0 / (l * h)));
	double t_1 = ((0.5 / d) * D) * M;
	double tmp;
	if (d <= -2.95e-67) {
		tmp = (1.0 - (((t_1 * 0.5) / pow(h, -1.0)) * (t_1 / l))) * (t_0 * -d);
	} else if (d <= 3.8e-296) {
		tmp = fma(pow((h / l), 1.5), ((pow((M * D), 2.0) / d) * -0.125), (sqrt((h / l)) * d)) / h;
	} else if (d <= 1.7e-68) {
		tmp = (1.0 - ((pow(((M * D) / (2.0 * d)), 2.0) * (1.0 / 2.0)) * (h / l))) * (t_0 * d);
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * pow((d / l), (1.0 / 2.0))) * (1.0 - ((t_1 / pow(h, -1.0)) * (((M / d) * ((D * 0.5) * 0.5)) / l)));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(1.0 / Float64(l * h)))
	t_1 = Float64(Float64(Float64(0.5 / d) * D) * M)
	tmp = 0.0
	if (d <= -2.95e-67)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(t_1 * 0.5) / (h ^ -1.0)) * Float64(t_1 / l))) * Float64(t_0 * Float64(-d)));
	elseif (d <= 3.8e-296)
		tmp = Float64(fma((Float64(h / l) ^ 1.5), Float64(Float64((Float64(M * D) ^ 2.0) / d) * -0.125), Float64(sqrt(Float64(h / l)) * d)) / h);
	elseif (d <= 1.7e-68)
		tmp = Float64(Float64(1.0 - Float64(Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)) * Float64(h / l))) * Float64(t_0 * d));
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(t_1 / (h ^ -1.0)) * Float64(Float64(Float64(M / d) * Float64(Float64(D * 0.5) * 0.5)) / l))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(0.5 / d), $MachinePrecision] * D), $MachinePrecision] * M), $MachinePrecision]}, If[LessEqual[d, -2.95e-67], N[(N[(1.0 - N[(N[(N[(t$95$1 * 0.5), $MachinePrecision] / N[Power[h, -1.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * (-d)), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.8e-296], N[(N[(N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision] * N[(N[(N[Power[N[(M * D), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision] + N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[d, 1.7e-68], N[(N[(1.0 - N[(N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * d), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(t$95$1 / N[Power[h, -1.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M / d), $MachinePrecision] * N[(N[(D * 0.5), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.95e-67

   1. Initial program 75.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 74.4

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

   4. Applied rewrites 74.4%

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

   5. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 74.1

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

   7. Applied rewrites 74.1%

      \[\leadsto \color{blue}{\left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. div-inv N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

   9. Applied rewrites 85.7%

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

   if -2.95e-67 < d < 3.8000000000000002e-296

   1. Initial program 46.8%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   5. Applied rewrites 48.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \frac{D \cdot D}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]
   6. Step-by-step derivation

   7. Applied rewrites 73.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\left(\frac{h}{\ell}\right)}^{1.5}, -0.125 \cdot \frac{{\left(M \cdot D\right)}^{2}}{d}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   if 3.8000000000000002e-296 < d < 1.70000000000000009e-68

   1. Initial program 50.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 50.8

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

   4. Applied rewrites 50.8%

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

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 73.8

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

   7. Applied rewrites 73.8%

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

   if 1.70000000000000009e-68 < d

   1. Initial program 72.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 77.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{\ell} \cdot \frac{\left(\frac{0.5}{d} \cdot D\right) \cdot M}{{h}^{-1}}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. metadata-eval 77.9

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 92.3

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

   6. Applied rewrites 92.3%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.95 \cdot 10^{-67}:\\ \;\;\;\;\left(1 - \frac{\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right) \cdot 0.5}{{h}^{-1}} \cdot \frac{\left(\frac{0.5}{d} \cdot D\right) \cdot M}{\ell}\right) \cdot \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right)\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-296}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\frac{h}{\ell}\right)}^{1.5}, \frac{{\left(M \cdot D\right)}^{2}}{d} \cdot -0.125, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{-68}:\\ \;\;\;\;\left(1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot D\right) \cdot M}{{h}^{-1}} \cdot \frac{\frac{M}{d} \cdot \left(\left(D \cdot 0.5\right) \cdot 0.5\right)}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{0.5}{d} \cdot D\right) \cdot M\\ \mathbf{if}\;d \leq -2.95 \cdot 10^{-67}:\\ \;\;\;\;\left(1 - \frac{t\_0 \cdot 0.5}{{h}^{-1}} \cdot \frac{t\_0}{\ell}\right) \cdot \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right)\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{-297}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\frac{h}{\ell}\right)}^{1.5}, \frac{{\left(M \cdot D\right)}^{2}}{d} \cdot -0.125, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{elif}\;d \leq 5 \cdot 10^{+264}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(-0.5 \cdot h\right) \cdot t\_0}{\ell}, t\_0, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.5 \cdot \frac{h}{\ell}\right) \cdot t\_0, t\_0, 1\right) \cdot \left(\frac{1}{\sqrt{\frac{\ell}{d}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (* (/ 0.5 d) D) M)))
   (if (<= d -2.95e-67)
     (*
      (- 1.0 (* (/ (* t_0 0.5) (pow h -1.0)) (/ t_0 l)))
      (* (sqrt (/ 1.0 (* l h))) (- d)))
     (if (<= d 1.35e-297)
       (/
        (fma
         (pow (/ h l) 1.5)
         (* (/ (pow (* M D) 2.0) d) -0.125)
         (* (sqrt (/ h l)) d))
        h)
       (if (<= d 5e+264)
         (* (fma (/ (* (* -0.5 h) t_0) l) t_0 1.0) (/ d (sqrt (* l h))))
         (*
          (fma (* (* -0.5 (/ h l)) t_0) t_0 1.0)
          (* (/ 1.0 (sqrt (/ l d))) (pow (/ d h) (/ 1.0 2.0)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = ((0.5 / d) * D) * M;
	double tmp;
	if (d <= -2.95e-67) {
		tmp = (1.0 - (((t_0 * 0.5) / pow(h, -1.0)) * (t_0 / l))) * (sqrt((1.0 / (l * h))) * -d);
	} else if (d <= 1.35e-297) {
		tmp = fma(pow((h / l), 1.5), ((pow((M * D), 2.0) / d) * -0.125), (sqrt((h / l)) * d)) / h;
	} else if (d <= 5e+264) {
		tmp = fma((((-0.5 * h) * t_0) / l), t_0, 1.0) * (d / sqrt((l * h)));
	} else {
		tmp = fma(((-0.5 * (h / l)) * t_0), t_0, 1.0) * ((1.0 / sqrt((l / d))) * pow((d / h), (1.0 / 2.0)));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(Float64(0.5 / d) * D) * M)
	tmp = 0.0
	if (d <= -2.95e-67)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(t_0 * 0.5) / (h ^ -1.0)) * Float64(t_0 / l))) * Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d)));
	elseif (d <= 1.35e-297)
		tmp = Float64(fma((Float64(h / l) ^ 1.5), Float64(Float64((Float64(M * D) ^ 2.0) / d) * -0.125), Float64(sqrt(Float64(h / l)) * d)) / h);
	elseif (d <= 5e+264)
		tmp = Float64(fma(Float64(Float64(Float64(-0.5 * h) * t_0) / l), t_0, 1.0) * Float64(d / sqrt(Float64(l * h))));
	else
		tmp = Float64(fma(Float64(Float64(-0.5 * Float64(h / l)) * t_0), t_0, 1.0) * Float64(Float64(1.0 / sqrt(Float64(l / d))) * (Float64(d / h) ^ Float64(1.0 / 2.0))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[(0.5 / d), $MachinePrecision] * D), $MachinePrecision] * M), $MachinePrecision]}, If[LessEqual[d, -2.95e-67], N[(N[(1.0 - N[(N[(N[(t$95$0 * 0.5), $MachinePrecision] / N[Power[h, -1.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.35e-297], N[(N[(N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision] * N[(N[(N[Power[N[(M * D), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision] + N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[d, 5e+264], N[(N[(N[(N[(N[(-0.5 * h), $MachinePrecision] * t$95$0), $MachinePrecision] / l), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] * N[(N[(1.0 / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.95e-67

   1. Initial program 75.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 74.4

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

   4. Applied rewrites 74.4%

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

   5. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 74.1

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

   7. Applied rewrites 74.1%

      \[\leadsto \color{blue}{\left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. div-inv N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

   9. Applied rewrites 85.7%

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

   if -2.95e-67 < d < 1.3500000000000001e-297

   1. Initial program 46.8%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   5. Applied rewrites 48.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \frac{D \cdot D}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]
   6. Step-by-step derivation

   7. Applied rewrites 73.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\left(\frac{h}{\ell}\right)}^{1.5}, -0.125 \cdot \frac{{\left(M \cdot D\right)}^{2}}{d}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   if 1.3500000000000001e-297 < d < 5.00000000000000033e264

   1. Initial program 64.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 65.1

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

   4. Applied rewrites 65.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

   6. Applied rewrites 65.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. div-inv N/A

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

      10. associate-/l* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. associate-*r/ N/A

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

      17. *-commutative N/A

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

      18. lower-/.f64 N/A

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

   8. Applied rewrites 67.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. pow1/2 N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. sqrt-div N/A

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

      10. lift-/.f64 N/A

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

      11. clear-num N/A

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

      12. lift-/.f64 N/A

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

      13. sqrt-unprod N/A

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

      14. lift-/.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. frac-times N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. sqrt-div N/A

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

      20. sqrt-unprod N/A

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

      21. rem-square-sqrt N/A

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

      22. lift-sqrt.f64 N/A

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

      23. lower-/.f64 82.1

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

   10. Applied rewrites 82.1%

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

   if 5.00000000000000033e264 < d

   1. Initial program 88.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

   6. Applied rewrites 99.6%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.95 \cdot 10^{-67}:\\ \;\;\;\;\left(1 - \frac{\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right) \cdot 0.5}{{h}^{-1}} \cdot \frac{\left(\frac{0.5}{d} \cdot D\right) \cdot M}{\ell}\right) \cdot \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right)\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{-297}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\frac{h}{\ell}\right)}^{1.5}, \frac{{\left(M \cdot D\right)}^{2}}{d} \cdot -0.125, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{elif}\;d \leq 5 \cdot 10^{+264}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(-0.5 \cdot h\right) \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right)}{\ell}, \left(\frac{0.5}{d} \cdot D\right) \cdot M, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.5 \cdot \frac{h}{\ell}\right) \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{0.5}{d} \cdot D\right) \cdot M, 1\right) \cdot \left(\frac{1}{\sqrt{\frac{\ell}{d}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{0.5}{d} \cdot D\right) \cdot M\\ t_1 := \mathsf{fma}\left(\frac{\left(-0.5 \cdot h\right) \cdot t\_0}{\ell}, t\_0, 1\right)\\ \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(1 - \frac{t\_0 \cdot 0.5}{{h}^{-1}} \cdot \frac{t\_0}{\ell}\right) \cdot \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right)\\ \mathbf{elif}\;h \leq 7.7 \cdot 10^{+136}:\\ \;\;\;\;t\_1 \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (* (/ 0.5 d) D) M))
        (t_1 (fma (/ (* (* -0.5 h) t_0) l) t_0 1.0)))
   (if (<= h -5e-310)
     (*
      (- 1.0 (* (/ (* t_0 0.5) (pow h -1.0)) (/ t_0 l)))
      (* (sqrt (/ 1.0 (* l h))) (- d)))
     (if (<= h 7.7e+136)
       (* t_1 (/ d (sqrt (* l h))))
       (* (* (sqrt (/ d h)) (sqrt (/ d l))) t_1)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = ((0.5 / d) * D) * M;
	double t_1 = fma((((-0.5 * h) * t_0) / l), t_0, 1.0);
	double tmp;
	if (h <= -5e-310) {
		tmp = (1.0 - (((t_0 * 0.5) / pow(h, -1.0)) * (t_0 / l))) * (sqrt((1.0 / (l * h))) * -d);
	} else if (h <= 7.7e+136) {
		tmp = t_1 * (d / sqrt((l * h)));
	} else {
		tmp = (sqrt((d / h)) * sqrt((d / l))) * t_1;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(Float64(0.5 / d) * D) * M)
	t_1 = fma(Float64(Float64(Float64(-0.5 * h) * t_0) / l), t_0, 1.0)
	tmp = 0.0
	if (h <= -5e-310)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(t_0 * 0.5) / (h ^ -1.0)) * Float64(t_0 / l))) * Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d)));
	elseif (h <= 7.7e+136)
		tmp = Float64(t_1 * Float64(d / sqrt(Float64(l * h))));
	else
		tmp = Float64(Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))) * t_1);
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[(0.5 / d), $MachinePrecision] * D), $MachinePrecision] * M), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(-0.5 * h), $MachinePrecision] * t$95$0), $MachinePrecision] / l), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]}, If[LessEqual[h, -5e-310], N[(N[(1.0 - N[(N[(N[(t$95$0 * 0.5), $MachinePrecision] / N[Power[h, -1.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 7.7e+136], N[(t$95$1 * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -4.999999999999985e-310

   1. Initial program 64.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 63.8

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

   4. Applied rewrites 63.8%

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

   5. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 66.2

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

   7. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. div-inv N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

   9. Applied rewrites 78.0%

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

   if -4.999999999999985e-310 < h < 7.7000000000000003e136

   1. Initial program 65.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 67.1

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

   4. Applied rewrites 67.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

   6. Applied rewrites 67.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. div-inv N/A

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

      10. associate-/l* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. associate-*r/ N/A

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

      17. *-commutative N/A

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

      18. lower-/.f64 N/A

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

   8. Applied rewrites 68.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. pow1/2 N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. sqrt-div N/A

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

      10. lift-/.f64 N/A

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

      11. clear-num N/A

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

      12. lift-/.f64 N/A

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

      13. sqrt-unprod N/A

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

      14. lift-/.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. frac-times N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. sqrt-div N/A

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

      20. sqrt-unprod N/A

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

      21. rem-square-sqrt N/A

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

      22. lift-sqrt.f64 N/A

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

      23. lower-/.f64 84.8

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

   10. Applied rewrites 84.8%

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

   if 7.7000000000000003e136 < h

   1. Initial program 63.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 63.8

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

   4. Applied rewrites 63.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

   6. Applied rewrites 63.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. div-inv N/A

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

      10. associate-/l* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. associate-*r/ N/A

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

      17. *-commutative N/A

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

      18. lower-/.f64 N/A

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

   8. Applied rewrites 72.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. lift-/.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lower-*.f64 72.9

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

      12. lift-/.f64 N/A

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

      13. metadata-eval 72.9

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

      14. lift-pow.f64 N/A

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

      15. pow1/2 N/A

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

      16. lift-sqrt.f64 72.9

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

   10. Applied rewrites 72.9%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(1 - \frac{\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right) \cdot 0.5}{{h}^{-1}} \cdot \frac{\left(\frac{0.5}{d} \cdot D\right) \cdot M}{\ell}\right) \cdot \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right)\\ \mathbf{elif}\;h \leq 7.7 \cdot 10^{+136}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(-0.5 \cdot h\right) \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right)}{\ell}, \left(\frac{0.5}{d} \cdot D\right) \cdot M, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{\left(-0.5 \cdot h\right) \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right)}{\ell}, \left(\frac{0.5}{d} \cdot D\right) \cdot M, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{0.5}{d} \cdot D\right) \cdot M\\ t_1 := \mathsf{fma}\left(\frac{\left(-0.5 \cdot h\right) \cdot t\_0}{\ell}, t\_0, 1\right)\\ \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t\_1 \cdot \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right)\\ \mathbf{elif}\;h \leq 7.7 \cdot 10^{+136}:\\ \;\;\;\;t\_1 \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (* (/ 0.5 d) D) M))
        (t_1 (fma (/ (* (* -0.5 h) t_0) l) t_0 1.0)))
   (if (<= h -5e-310)
     (* t_1 (* (sqrt (/ 1.0 (* l h))) (- d)))
     (if (<= h 7.7e+136)
       (* t_1 (/ d (sqrt (* l h))))
       (* (* (sqrt (/ d h)) (sqrt (/ d l))) t_1)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = ((0.5 / d) * D) * M;
	double t_1 = fma((((-0.5 * h) * t_0) / l), t_0, 1.0);
	double tmp;
	if (h <= -5e-310) {
		tmp = t_1 * (sqrt((1.0 / (l * h))) * -d);
	} else if (h <= 7.7e+136) {
		tmp = t_1 * (d / sqrt((l * h)));
	} else {
		tmp = (sqrt((d / h)) * sqrt((d / l))) * t_1;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(Float64(0.5 / d) * D) * M)
	t_1 = fma(Float64(Float64(Float64(-0.5 * h) * t_0) / l), t_0, 1.0)
	tmp = 0.0
	if (h <= -5e-310)
		tmp = Float64(t_1 * Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d)));
	elseif (h <= 7.7e+136)
		tmp = Float64(t_1 * Float64(d / sqrt(Float64(l * h))));
	else
		tmp = Float64(Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))) * t_1);
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[(0.5 / d), $MachinePrecision] * D), $MachinePrecision] * M), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(-0.5 * h), $MachinePrecision] * t$95$0), $MachinePrecision] / l), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]}, If[LessEqual[h, -5e-310], N[(t$95$1 * N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 7.7e+136], N[(t$95$1 * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -4.999999999999985e-310

   1. Initial program 64.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 63.8

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

   4. Applied rewrites 63.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

   6. Applied rewrites 65.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. div-inv N/A

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

      10. associate-/l* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. associate-*r/ N/A

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

      17. *-commutative N/A

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

      18. lower-/.f64 N/A

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

   8. Applied rewrites 68.7%

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

   9. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. rem-square-sqrt N/A

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

       4. lower-*.f64 N/A

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

       5. mul-1-neg N/A

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

       6. lower-neg.f64 N/A

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

       7. lower-sqrt.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 77.3

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

   11. Applied rewrites 77.3%

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

   if -4.999999999999985e-310 < h < 7.7000000000000003e136

   1. Initial program 65.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 67.1

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

   4. Applied rewrites 67.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

   6. Applied rewrites 67.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. div-inv N/A

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

      10. associate-/l* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. associate-*r/ N/A

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

      17. *-commutative N/A

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

      18. lower-/.f64 N/A

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

   8. Applied rewrites 68.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. pow1/2 N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. sqrt-div N/A

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

      10. lift-/.f64 N/A

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

      11. clear-num N/A

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

      12. lift-/.f64 N/A

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

      13. sqrt-unprod N/A

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

      14. lift-/.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. frac-times N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. sqrt-div N/A

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

      20. sqrt-unprod N/A

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

      21. rem-square-sqrt N/A

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

      22. lift-sqrt.f64 N/A

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

      23. lower-/.f64 84.8

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

   10. Applied rewrites 84.8%

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

   if 7.7000000000000003e136 < h

   1. Initial program 63.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 63.8

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

   4. Applied rewrites 63.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

   6. Applied rewrites 63.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. div-inv N/A

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

      10. associate-/l* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. associate-*r/ N/A

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

      17. *-commutative N/A

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

      18. lower-/.f64 N/A

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

   8. Applied rewrites 72.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. lift-/.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lower-*.f64 72.9

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

      12. lift-/.f64 N/A

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

      13. metadata-eval 72.9

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

      14. lift-pow.f64 N/A

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

      15. pow1/2 N/A

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

      16. lift-sqrt.f64 72.9

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

   10. Applied rewrites 72.9%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(-0.5 \cdot h\right) \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right)}{\ell}, \left(\frac{0.5}{d} \cdot D\right) \cdot M, 1\right) \cdot \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right)\\ \mathbf{elif}\;h \leq 7.7 \cdot 10^{+136}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(-0.5 \cdot h\right) \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right)}{\ell}, \left(\frac{0.5}{d} \cdot D\right) \cdot M, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{\left(-0.5 \cdot h\right) \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right)}{\ell}, \left(\frac{0.5}{d} \cdot D\right) \cdot M, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{0.5}{d} \cdot D\right) \cdot M\\ t_1 := \mathsf{fma}\left(\frac{\left(-0.5 \cdot h\right) \cdot t\_0}{\ell}, t\_0, 1\right)\\ \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t\_1 \cdot \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right)\\ \mathbf{elif}\;h \leq 3.6 \cdot 10^{+238}:\\ \;\;\;\;t\_1 \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{h}{\ell} \cdot t\_0, -0.25 \cdot \frac{M \cdot D}{d}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (* (/ 0.5 d) D) M))
        (t_1 (fma (/ (* (* -0.5 h) t_0) l) t_0 1.0)))
   (if (<= h -5e-310)
     (* t_1 (* (sqrt (/ 1.0 (* l h))) (- d)))
     (if (<= h 3.6e+238)
       (* t_1 (/ d (sqrt (* l h))))
       (*
        (* (fma (* (/ h l) t_0) (* -0.25 (/ (* M D) d)) 1.0) (sqrt (/ d l)))
        (sqrt (/ d h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = ((0.5 / d) * D) * M;
	double t_1 = fma((((-0.5 * h) * t_0) / l), t_0, 1.0);
	double tmp;
	if (h <= -5e-310) {
		tmp = t_1 * (sqrt((1.0 / (l * h))) * -d);
	} else if (h <= 3.6e+238) {
		tmp = t_1 * (d / sqrt((l * h)));
	} else {
		tmp = (fma(((h / l) * t_0), (-0.25 * ((M * D) / d)), 1.0) * sqrt((d / l))) * sqrt((d / h));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(Float64(0.5 / d) * D) * M)
	t_1 = fma(Float64(Float64(Float64(-0.5 * h) * t_0) / l), t_0, 1.0)
	tmp = 0.0
	if (h <= -5e-310)
		tmp = Float64(t_1 * Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d)));
	elseif (h <= 3.6e+238)
		tmp = Float64(t_1 * Float64(d / sqrt(Float64(l * h))));
	else
		tmp = Float64(Float64(fma(Float64(Float64(h / l) * t_0), Float64(-0.25 * Float64(Float64(M * D) / d)), 1.0) * sqrt(Float64(d / l))) * sqrt(Float64(d / h)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[(0.5 / d), $MachinePrecision] * D), $MachinePrecision] * M), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(-0.5 * h), $MachinePrecision] * t$95$0), $MachinePrecision] / l), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]}, If[LessEqual[h, -5e-310], N[(t$95$1 * N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 3.6e+238], N[(t$95$1 * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(h / l), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(-0.25 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -4.999999999999985e-310

   1. Initial program 64.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 63.8

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

   4. Applied rewrites 63.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

   6. Applied rewrites 65.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. div-inv N/A

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

      10. associate-/l* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. associate-*r/ N/A

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

      17. *-commutative N/A

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

      18. lower-/.f64 N/A

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

   8. Applied rewrites 68.7%

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

   9. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. rem-square-sqrt N/A

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

       4. lower-*.f64 N/A

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

       5. mul-1-neg N/A

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

       6. lower-neg.f64 N/A

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

       7. lower-sqrt.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 77.3

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

   11. Applied rewrites 77.3%

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

   if -4.999999999999985e-310 < h < 3.59999999999999971e238

   1. Initial program 64.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 66.0

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

   4. Applied rewrites 66.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

   6. Applied rewrites 65.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. div-inv N/A

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

      10. associate-/l* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. associate-*r/ N/A

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

      17. *-commutative N/A

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

      18. lower-/.f64 N/A

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

   8. Applied rewrites 68.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. pow1/2 N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. sqrt-div N/A

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

      10. lift-/.f64 N/A

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

      11. clear-num N/A

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

      12. lift-/.f64 N/A

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

      13. sqrt-unprod N/A

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

      14. lift-/.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. frac-times N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. sqrt-div N/A

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

      20. sqrt-unprod N/A

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

      21. rem-square-sqrt N/A

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

      22. lift-sqrt.f64 N/A

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

      23. lower-/.f64 82.5

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

   10. Applied rewrites 82.5%

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

   if 3.59999999999999971e238 < h

   1. Initial program 75.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 75.0%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-pow N/A

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

      6. inv-pow N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. clear-num N/A

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

      14. lift-/.f64 N/A

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

      15. lift-pow.f64 N/A

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

   6. Applied rewrites 50.0%

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

   8. Applied rewrites 75.6%

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

   9. Taylor expanded in M around 0

      \[\leadsto \left(\mathsf{fma}\left(\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \frac{h}{\ell}, \color{blue}{\frac{-1}{4} \cdot \frac{D \cdot M}{d}}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \frac{h}{\ell}, \color{blue}{\frac{D \cdot M}{d} \cdot \frac{-1}{4}}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \frac{h}{\ell}, \color{blue}{\frac{D \cdot M}{d} \cdot \frac{-1}{4}}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

       3. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \frac{h}{\ell}, \color{blue}{\frac{D \cdot M}{d}} \cdot \frac{-1}{4}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

       4. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \frac{h}{\ell}, \frac{\color{blue}{M \cdot D}}{d} \cdot \frac{-1}{4}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

       5. lower-*.f64 75.6

          \[\leadsto \left(\mathsf{fma}\left(\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right) \cdot \frac{h}{\ell}, \frac{\color{blue}{M \cdot D}}{d} \cdot -0.25, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

   11. Applied rewrites 75.6%

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right) \cdot \frac{h}{\ell}, \color{blue}{\frac{M \cdot D}{d} \cdot -0.25}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(-0.5 \cdot h\right) \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right)}{\ell}, \left(\frac{0.5}{d} \cdot D\right) \cdot M, 1\right) \cdot \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right)\\ \mathbf{elif}\;h \leq 3.6 \cdot 10^{+238}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(-0.5 \cdot h\right) \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right)}{\ell}, \left(\frac{0.5}{d} \cdot D\right) \cdot M, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{h}{\ell} \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), -0.25 \cdot \frac{M \cdot D}{d}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ t_1 := \left(\frac{0.5}{d} \cdot D\right) \cdot M\\ \mathbf{if}\;d \leq -1.5 \cdot 10^{+95}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -2.7 \cdot 10^{-151}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -2.05 \cdot 10^{-303}:\\ \;\;\;\;\left(\left(\frac{\frac{M \cdot M}{d}}{d} \cdot h\right) \cdot \frac{\left(D \cdot D\right) \cdot -0.125}{\ell}\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(-0.5 \cdot h\right) \cdot t\_1}{\ell}, t\_1, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (sqrt (/ 1.0 (* l h))) (- d))) (t_1 (* (* (/ 0.5 d) D) M)))
   (if (<= d -1.5e+95)
     t_0
     (if (<= d -2.7e-151)
       (* (sqrt (/ d l)) (sqrt (/ d h)))
       (if (<= d -2.05e-303)
         (* (* (* (/ (/ (* M M) d) d) h) (/ (* (* D D) -0.125) l)) t_0)
         (* (fma (/ (* (* -0.5 h) t_1) l) t_1 1.0) (/ d (sqrt (* l h)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((1.0 / (l * h))) * -d;
	double t_1 = ((0.5 / d) * D) * M;
	double tmp;
	if (d <= -1.5e+95) {
		tmp = t_0;
	} else if (d <= -2.7e-151) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (d <= -2.05e-303) {
		tmp = (((((M * M) / d) / d) * h) * (((D * D) * -0.125) / l)) * t_0;
	} else {
		tmp = fma((((-0.5 * h) * t_1) / l), t_1, 1.0) * (d / sqrt((l * h)));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d))
	t_1 = Float64(Float64(Float64(0.5 / d) * D) * M)
	tmp = 0.0
	if (d <= -1.5e+95)
		tmp = t_0;
	elseif (d <= -2.7e-151)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (d <= -2.05e-303)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(M * M) / d) / d) * h) * Float64(Float64(Float64(D * D) * -0.125) / l)) * t_0);
	else
		tmp = Float64(fma(Float64(Float64(Float64(-0.5 * h) * t_1) / l), t_1, 1.0) * Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(0.5 / d), $MachinePrecision] * D), $MachinePrecision] * M), $MachinePrecision]}, If[LessEqual[d, -1.5e+95], t$95$0, If[LessEqual[d, -2.7e-151], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -2.05e-303], N[(N[(N[(N[(N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision] * h), $MachinePrecision] * N[(N[(N[(D * D), $MachinePrecision] * -0.125), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(-0.5 * h), $MachinePrecision] * t$95$1), $MachinePrecision] / l), $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision] * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.49999999999999996e95

   1. Initial program 70.8%

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

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 65.1

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

   5. Applied rewrites 65.1%

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

   if -1.49999999999999996e95 < d < -2.70000000000000007e-151

   1. Initial program 76.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 76.1%

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

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}} \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 61.4

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

   7. Applied rewrites 61.4%

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

   if -2.70000000000000007e-151 < d < -2.05000000000000009e-303

   1. Initial program 32.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 32.9

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

   4. Applied rewrites 32.9%

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

   5. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 54.5

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

   7. Applied rewrites 54.5%

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

   8. Taylor expanded in h around inf

      \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \color{blue}{\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \frac{\color{blue}{\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \left({M}^{2} \cdot h\right)}}{{d}^{2} \cdot \ell} \]

      3. *-commutative N/A

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \frac{\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{\ell \cdot {d}^{2}}} \]

      4. times-frac N/A

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \color{blue}{\left(\frac{\frac{-1}{8} \cdot {D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \color{blue}{\left(\frac{\frac{-1}{8} \cdot {D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\color{blue}{\frac{\frac{-1}{8} \cdot {D}^{2}}{\ell}} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{\color{blue}{\frac{-1}{8} \cdot {D}^{2}}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{\frac{-1}{8} \cdot \color{blue}{\left(D \cdot D\right)}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{\frac{-1}{8} \cdot \color{blue}{\left(D \cdot D\right)}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right) \]

      10. associate-*l/ N/A

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{\frac{-1}{8} \cdot \left(D \cdot D\right)}{\ell} \cdot \color{blue}{\left(\frac{{M}^{2}}{{d}^{2}} \cdot h\right)}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{\frac{-1}{8} \cdot \left(D \cdot D\right)}{\ell} \cdot \color{blue}{\left(\frac{{M}^{2}}{{d}^{2}} \cdot h\right)}\right) \]

      12. unpow2 N/A

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{\frac{-1}{8} \cdot \left(D \cdot D\right)}{\ell} \cdot \left(\frac{{M}^{2}}{\color{blue}{d \cdot d}} \cdot h\right)\right) \]

      13. associate-/r* N/A

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{\frac{-1}{8} \cdot \left(D \cdot D\right)}{\ell} \cdot \left(\color{blue}{\frac{\frac{{M}^{2}}{d}}{d}} \cdot h\right)\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{\frac{-1}{8} \cdot \left(D \cdot D\right)}{\ell} \cdot \left(\color{blue}{\frac{\frac{{M}^{2}}{d}}{d}} \cdot h\right)\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{\frac{-1}{8} \cdot \left(D \cdot D\right)}{\ell} \cdot \left(\frac{\color{blue}{\frac{{M}^{2}}{d}}}{d} \cdot h\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{\frac{-1}{8} \cdot \left(D \cdot D\right)}{\ell} \cdot \left(\frac{\frac{\color{blue}{M \cdot M}}{d}}{d} \cdot h\right)\right) \]

      17. lower-*.f64 53.9

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{-0.125 \cdot \left(D \cdot D\right)}{\ell} \cdot \left(\frac{\frac{\color{blue}{M \cdot M}}{d}}{d} \cdot h\right)\right) \]

   10. Applied rewrites 53.9%

       \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \color{blue}{\left(\frac{-0.125 \cdot \left(D \cdot D\right)}{\ell} \cdot \left(\frac{\frac{M \cdot M}{d}}{d} \cdot h\right)\right)} \]

   if -2.05000000000000009e-303 < d

   1. Initial program 65.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 66.5

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

   4. Applied rewrites 66.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

   6. Applied rewrites 66.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. div-inv N/A

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

      10. associate-/l* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. associate-*r/ N/A

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

      17. *-commutative N/A

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

      18. lower-/.f64 N/A

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

   8. Applied rewrites 68.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. pow1/2 N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. sqrt-div N/A

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

      10. lift-/.f64 N/A

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

      11. clear-num N/A

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

      12. lift-/.f64 N/A

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

      13. sqrt-unprod N/A

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

      14. lift-/.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. frac-times N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. sqrt-div N/A

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

      20. sqrt-unprod N/A

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

      21. rem-square-sqrt N/A

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

      22. lift-sqrt.f64 N/A

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

      23. lower-/.f64 78.1

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

   10. Applied rewrites 78.1%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.5 \cdot 10^{+95}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -2.7 \cdot 10^{-151}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -2.05 \cdot 10^{-303}:\\ \;\;\;\;\left(\left(\frac{\frac{M \cdot M}{d}}{d} \cdot h\right) \cdot \frac{\left(D \cdot D\right) \cdot -0.125}{\ell}\right) \cdot \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(-0.5 \cdot h\right) \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right)}{\ell}, \left(\frac{0.5}{d} \cdot D\right) \cdot M, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ t_1 := \left(\frac{0.5}{d} \cdot D\right) \cdot M\\ \mathbf{if}\;d \leq -1.5 \cdot 10^{+95}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -2.7 \cdot 10^{-151}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -2.05 \cdot 10^{-303}:\\ \;\;\;\;\left(\left(\frac{\frac{M \cdot M}{d}}{d} \cdot h\right) \cdot \frac{\left(D \cdot D\right) \cdot -0.125}{\ell}\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.5 \cdot \frac{h}{\ell}\right) \cdot t\_1, t\_1, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (sqrt (/ 1.0 (* l h))) (- d))) (t_1 (* (* (/ 0.5 d) D) M)))
   (if (<= d -1.5e+95)
     t_0
     (if (<= d -2.7e-151)
       (* (sqrt (/ d l)) (sqrt (/ d h)))
       (if (<= d -2.05e-303)
         (* (* (* (/ (/ (* M M) d) d) h) (/ (* (* D D) -0.125) l)) t_0)
         (* (fma (* (* -0.5 (/ h l)) t_1) t_1 1.0) (/ d (sqrt (* l h)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((1.0 / (l * h))) * -d;
	double t_1 = ((0.5 / d) * D) * M;
	double tmp;
	if (d <= -1.5e+95) {
		tmp = t_0;
	} else if (d <= -2.7e-151) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (d <= -2.05e-303) {
		tmp = (((((M * M) / d) / d) * h) * (((D * D) * -0.125) / l)) * t_0;
	} else {
		tmp = fma(((-0.5 * (h / l)) * t_1), t_1, 1.0) * (d / sqrt((l * h)));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d))
	t_1 = Float64(Float64(Float64(0.5 / d) * D) * M)
	tmp = 0.0
	if (d <= -1.5e+95)
		tmp = t_0;
	elseif (d <= -2.7e-151)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (d <= -2.05e-303)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(M * M) / d) / d) * h) * Float64(Float64(Float64(D * D) * -0.125) / l)) * t_0);
	else
		tmp = Float64(fma(Float64(Float64(-0.5 * Float64(h / l)) * t_1), t_1, 1.0) * Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(0.5 / d), $MachinePrecision] * D), $MachinePrecision] * M), $MachinePrecision]}, If[LessEqual[d, -1.5e+95], t$95$0, If[LessEqual[d, -2.7e-151], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -2.05e-303], N[(N[(N[(N[(N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision] * h), $MachinePrecision] * N[(N[(N[(D * D), $MachinePrecision] * -0.125), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[(N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision] * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.49999999999999996e95

   1. Initial program 70.8%

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

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 65.1

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

   5. Applied rewrites 65.1%

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

   if -1.49999999999999996e95 < d < -2.70000000000000007e-151

   1. Initial program 76.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \]

   4. Applied rewrites 76.1%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}} \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}} \]

      2. lower-/.f64 61.4

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}} \]

   7. Applied rewrites 61.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}} \]

   if -2.70000000000000007e-151 < d < -2.05000000000000009e-303

   1. Initial program 32.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-/.f64 32.9

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 32.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow2 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lower-neg.f64 N/A

         \[\leadsto \left(\color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. *-commutative N/A

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-*.f64 54.5

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   7. Applied rewrites 54.5%

      \[\leadsto \color{blue}{\left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   8. Taylor expanded in h around inf

      \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \color{blue}{\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \frac{\color{blue}{\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \left({M}^{2} \cdot h\right)}}{{d}^{2} \cdot \ell} \]

      3. *-commutative N/A

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \frac{\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{\ell \cdot {d}^{2}}} \]

      4. times-frac N/A

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \color{blue}{\left(\frac{\frac{-1}{8} \cdot {D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \color{blue}{\left(\frac{\frac{-1}{8} \cdot {D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\color{blue}{\frac{\frac{-1}{8} \cdot {D}^{2}}{\ell}} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{\color{blue}{\frac{-1}{8} \cdot {D}^{2}}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{\frac{-1}{8} \cdot \color{blue}{\left(D \cdot D\right)}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{\frac{-1}{8} \cdot \color{blue}{\left(D \cdot D\right)}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right) \]

      10. associate-*l/ N/A

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{\frac{-1}{8} \cdot \left(D \cdot D\right)}{\ell} \cdot \color{blue}{\left(\frac{{M}^{2}}{{d}^{2}} \cdot h\right)}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{\frac{-1}{8} \cdot \left(D \cdot D\right)}{\ell} \cdot \color{blue}{\left(\frac{{M}^{2}}{{d}^{2}} \cdot h\right)}\right) \]

      12. unpow2 N/A

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{\frac{-1}{8} \cdot \left(D \cdot D\right)}{\ell} \cdot \left(\frac{{M}^{2}}{\color{blue}{d \cdot d}} \cdot h\right)\right) \]

      13. associate-/r* N/A

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{\frac{-1}{8} \cdot \left(D \cdot D\right)}{\ell} \cdot \left(\color{blue}{\frac{\frac{{M}^{2}}{d}}{d}} \cdot h\right)\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{\frac{-1}{8} \cdot \left(D \cdot D\right)}{\ell} \cdot \left(\color{blue}{\frac{\frac{{M}^{2}}{d}}{d}} \cdot h\right)\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{\frac{-1}{8} \cdot \left(D \cdot D\right)}{\ell} \cdot \left(\frac{\color{blue}{\frac{{M}^{2}}{d}}}{d} \cdot h\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{\frac{-1}{8} \cdot \left(D \cdot D\right)}{\ell} \cdot \left(\frac{\frac{\color{blue}{M \cdot M}}{d}}{d} \cdot h\right)\right) \]

      17. lower-*.f64 53.9

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{-0.125 \cdot \left(D \cdot D\right)}{\ell} \cdot \left(\frac{\frac{\color{blue}{M \cdot M}}{d}}{d} \cdot h\right)\right) \]

   10. Applied rewrites 53.9%

       \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \color{blue}{\left(\frac{-0.125 \cdot \left(D \cdot D\right)}{\ell} \cdot \left(\frac{\frac{M \cdot M}{d}}{d} \cdot h\right)\right)} \]

   if -2.05000000000000009e-303 < d

   1. Initial program 65.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-/.f64 66.5

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 66.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

   6. Applied rewrites 66.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.5\right) \cdot \left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right), M \cdot \left(D \cdot \frac{0.5}{d}\right), 1\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right)} \cdot \mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      5. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{\sqrt{1}}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      9. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{1}{\color{blue}{\frac{\ell}{d}}}}\right) \cdot \mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      11. clear-num N/A

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      14. sqrt-div N/A

          \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      16. sqrt-div N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      17. frac-times N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h} \cdot \sqrt{\ell}}} \cdot \mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      18. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{d}}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      19. sqrt-prod N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      20. *-commutative N/A

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      21. lift-*.f64 N/A

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      22. lift-sqrt.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      23. lift-/.f64 75.1

          \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.5\right) \cdot \left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right), M \cdot \left(D \cdot \frac{0.5}{d}\right), 1\right) \]

   8. Applied rewrites 75.1%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.5\right) \cdot \left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right), M \cdot \left(D \cdot \frac{0.5}{d}\right), 1\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.5 \cdot 10^{+95}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -2.7 \cdot 10^{-151}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -2.05 \cdot 10^{-303}:\\ \;\;\;\;\left(\left(\frac{\frac{M \cdot M}{d}}{d} \cdot h\right) \cdot \frac{\left(D \cdot D\right) \cdot -0.125}{\ell}\right) \cdot \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.5 \cdot \frac{h}{\ell}\right) \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{0.5}{d} \cdot D\right) \cdot M, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 47.8% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{if}\;d \leq -1.5 \cdot 10^{+95}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -2.7 \cdot 10^{-151}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-312}:\\ \;\;\;\;\left(\left(\frac{\frac{M \cdot M}{d}}{d} \cdot h\right) \cdot \frac{\left(D \cdot D\right) \cdot -0.125}{\ell}\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (sqrt (/ 1.0 (* l h))) (- d))))
   (if (<= d -1.5e+95)
     t_0
     (if (<= d -2.7e-151)
       (* (sqrt (/ d l)) (sqrt (/ d h)))
       (if (<= d -5e-312)
         (* (* (* (/ (/ (* M M) d) d) h) (/ (* (* D D) -0.125) l)) t_0)
         (/ d (* (sqrt l) (sqrt h))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((1.0 / (l * h))) * -d;
	double tmp;
	if (d <= -1.5e+95) {
		tmp = t_0;
	} else if (d <= -2.7e-151) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (d <= -5e-312) {
		tmp = (((((M * M) / d) / d) * h) * (((D * D) * -0.125) / l)) * t_0;
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 / (l * h))) * -d
    if (d <= (-1.5d+95)) then
        tmp = t_0
    else if (d <= (-2.7d-151)) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else if (d <= (-5d-312)) then
        tmp = (((((m * m) / d) / d) * h) * (((d_1 * d_1) * (-0.125d0)) / l)) * t_0
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((1.0 / (l * h))) * -d;
	double tmp;
	if (d <= -1.5e+95) {
		tmp = t_0;
	} else if (d <= -2.7e-151) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else if (d <= -5e-312) {
		tmp = (((((M * M) / d) / d) * h) * (((D * D) * -0.125) / l)) * t_0;
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((1.0 / (l * h))) * -d
	tmp = 0
	if d <= -1.5e+95:
		tmp = t_0
	elif d <= -2.7e-151:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	elif d <= -5e-312:
		tmp = (((((M * M) / d) / d) * h) * (((D * D) * -0.125) / l)) * t_0
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d))
	tmp = 0.0
	if (d <= -1.5e+95)
		tmp = t_0;
	elseif (d <= -2.7e-151)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (d <= -5e-312)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(M * M) / d) / d) * h) * Float64(Float64(Float64(D * D) * -0.125) / l)) * t_0);
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((1.0 / (l * h))) * -d;
	tmp = 0.0;
	if (d <= -1.5e+95)
		tmp = t_0;
	elseif (d <= -2.7e-151)
		tmp = sqrt((d / l)) * sqrt((d / h));
	elseif (d <= -5e-312)
		tmp = (((((M * M) / d) / d) * h) * (((D * D) * -0.125) / l)) * t_0;
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision]}, If[LessEqual[d, -1.5e+95], t$95$0, If[LessEqual[d, -2.7e-151], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-312], N[(N[(N[(N[(N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision] * h), $MachinePrecision] * N[(N[(N[(D * D), $MachinePrecision] * -0.125), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.49999999999999996e95

   1. Initial program 70.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      10. lower-*.f64 65.1

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   5. Applied rewrites 65.1%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if -1.49999999999999996e95 < d < -2.70000000000000007e-151

   1. Initial program 76.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \]

   4. Applied rewrites 76.1%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}} \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}} \]

      2. lower-/.f64 61.4

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}} \]

   7. Applied rewrites 61.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}} \]

   if -2.70000000000000007e-151 < d < -5.0000000000022e-312

   1. Initial program 36.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-/.f64 36.2

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 36.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow2 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lower-neg.f64 N/A

         \[\leadsto \left(\color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. *-commutative N/A

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-*.f64 55.7

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   7. Applied rewrites 55.7%

      \[\leadsto \color{blue}{\left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   8. Taylor expanded in h around inf

      \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \color{blue}{\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \frac{\color{blue}{\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \left({M}^{2} \cdot h\right)}}{{d}^{2} \cdot \ell} \]

      3. *-commutative N/A

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \frac{\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{\ell \cdot {d}^{2}}} \]

      4. times-frac N/A

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \color{blue}{\left(\frac{\frac{-1}{8} \cdot {D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \color{blue}{\left(\frac{\frac{-1}{8} \cdot {D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\color{blue}{\frac{\frac{-1}{8} \cdot {D}^{2}}{\ell}} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{\color{blue}{\frac{-1}{8} \cdot {D}^{2}}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{\frac{-1}{8} \cdot \color{blue}{\left(D \cdot D\right)}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{\frac{-1}{8} \cdot \color{blue}{\left(D \cdot D\right)}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right) \]

      10. associate-*l/ N/A

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{\frac{-1}{8} \cdot \left(D \cdot D\right)}{\ell} \cdot \color{blue}{\left(\frac{{M}^{2}}{{d}^{2}} \cdot h\right)}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{\frac{-1}{8} \cdot \left(D \cdot D\right)}{\ell} \cdot \color{blue}{\left(\frac{{M}^{2}}{{d}^{2}} \cdot h\right)}\right) \]

      12. unpow2 N/A

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{\frac{-1}{8} \cdot \left(D \cdot D\right)}{\ell} \cdot \left(\frac{{M}^{2}}{\color{blue}{d \cdot d}} \cdot h\right)\right) \]

      13. associate-/r* N/A

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{\frac{-1}{8} \cdot \left(D \cdot D\right)}{\ell} \cdot \left(\color{blue}{\frac{\frac{{M}^{2}}{d}}{d}} \cdot h\right)\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{\frac{-1}{8} \cdot \left(D \cdot D\right)}{\ell} \cdot \left(\color{blue}{\frac{\frac{{M}^{2}}{d}}{d}} \cdot h\right)\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{\frac{-1}{8} \cdot \left(D \cdot D\right)}{\ell} \cdot \left(\frac{\color{blue}{\frac{{M}^{2}}{d}}}{d} \cdot h\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{\frac{-1}{8} \cdot \left(D \cdot D\right)}{\ell} \cdot \left(\frac{\frac{\color{blue}{M \cdot M}}{d}}{d} \cdot h\right)\right) \]

      17. lower-*.f64 48.7

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{-0.125 \cdot \left(D \cdot D\right)}{\ell} \cdot \left(\frac{\frac{\color{blue}{M \cdot M}}{d}}{d} \cdot h\right)\right) \]

   10. Applied rewrites 48.7%

       \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \color{blue}{\left(\frac{-0.125 \cdot \left(D \cdot D\right)}{\ell} \cdot \left(\frac{\frac{M \cdot M}{d}}{d} \cdot h\right)\right)} \]

   if -5.0000000000022e-312 < d

   1. Initial program 65.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 41.1

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 41.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 41.1%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 49.0%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.5 \cdot 10^{+95}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -2.7 \cdot 10^{-151}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-312}:\\ \;\;\;\;\left(\left(\frac{\frac{M \cdot M}{d}}{d} \cdot h\right) \cdot \frac{\left(D \cdot D\right) \cdot -0.125}{\ell}\right) \cdot \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.9% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{0.5}{d} \cdot D\right) \cdot M\\ t_1 := \mathsf{fma}\left(\frac{\left(-0.5 \cdot h\right) \cdot t\_0}{\ell}, t\_0, 1\right)\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t\_1 \cdot \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right)\\ \mathbf{elif}\;\ell \leq 9.6 \cdot 10^{+160}:\\ \;\;\;\;t\_1 \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (* (/ 0.5 d) D) M))
        (t_1 (fma (/ (* (* -0.5 h) t_0) l) t_0 1.0)))
   (if (<= l -5e-310)
     (* t_1 (* (sqrt (/ 1.0 (* l h))) (- d)))
     (if (<= l 9.6e+160)
       (* t_1 (/ d (sqrt (* l h))))
       (/ d (* (sqrt l) (sqrt h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = ((0.5 / d) * D) * M;
	double t_1 = fma((((-0.5 * h) * t_0) / l), t_0, 1.0);
	double tmp;
	if (l <= -5e-310) {
		tmp = t_1 * (sqrt((1.0 / (l * h))) * -d);
	} else if (l <= 9.6e+160) {
		tmp = t_1 * (d / sqrt((l * h)));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(Float64(0.5 / d) * D) * M)
	t_1 = fma(Float64(Float64(Float64(-0.5 * h) * t_0) / l), t_0, 1.0)
	tmp = 0.0
	if (l <= -5e-310)
		tmp = Float64(t_1 * Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d)));
	elseif (l <= 9.6e+160)
		tmp = Float64(t_1 * Float64(d / sqrt(Float64(l * h))));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[(0.5 / d), $MachinePrecision] * D), $MachinePrecision] * M), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(-0.5 * h), $MachinePrecision] * t$95$0), $MachinePrecision] / l), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]}, If[LessEqual[l, -5e-310], N[(t$95$1 * N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 9.6e+160], N[(t$95$1 * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 64.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-/.f64 63.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 63.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

   6. Applied rewrites 65.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.5\right) \cdot \left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right), M \cdot \left(D \cdot \frac{0.5}{d}\right), 1\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right)}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      2. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\color{blue}{\left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right) \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\color{blue}{\left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\left(M \cdot \color{blue}{\left(D \cdot \frac{\frac{1}{2}}{d}\right)}\right) \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\left(M \cdot \left(D \cdot \color{blue}{\frac{\frac{1}{2}}{d}}\right)\right) \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\left(M \cdot \left(D \cdot \frac{\color{blue}{\frac{1}{2}}}{d}\right)\right) \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      7. associate-/r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\left(M \cdot \left(D \cdot \color{blue}{\frac{1}{2 \cdot d}}\right)\right) \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\left(M \cdot \left(D \cdot \frac{1}{\color{blue}{2 \cdot d}}\right)\right) \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      9. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\left(M \cdot \color{blue}{\frac{D}{2 \cdot d}}\right) \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      10. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\color{blue}{\frac{h}{\ell}} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      15. associate-*l/ N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\frac{h \cdot \frac{-1}{2}}{\ell}}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      16. associate-*r/ N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{\frac{M \cdot D}{2 \cdot d} \cdot \left(h \cdot \frac{-1}{2}\right)}{\ell}}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      17. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\frac{\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\left(\frac{-1}{2} \cdot h\right)}}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      18. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

   8. Applied rewrites 68.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right) \cdot \left(-0.5 \cdot h\right)}{\ell}}, M \cdot \left(D \cdot \frac{0.5}{d}\right), 1\right) \]

   9. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

       2. unpow2 N/A

          \[\leadsto \left(\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

       3. rem-square-sqrt N/A

          \[\leadsto \left(\left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

       5. mul-1-neg N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

       6. lower-neg.f64 N/A

          \[\leadsto \left(\color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

       7. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

       8. lower-/.f64 N/A

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right) \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

       9. *-commutative N/A

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right) \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

       10. lower-*.f64 77.3

           \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right) \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right) \cdot \left(-0.5 \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{0.5}{d}\right), 1\right) \]

   11. Applied rewrites 77.3%

       \[\leadsto \color{blue}{\left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right) \cdot \left(-0.5 \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{0.5}{d}\right), 1\right) \]

   if -4.999999999999985e-310 < l < 9.6000000000000006e160

   1. Initial program 75.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-/.f64 76.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 76.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

   6. Applied rewrites 75.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.5\right) \cdot \left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right), M \cdot \left(D \cdot \frac{0.5}{d}\right), 1\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right)}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      2. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\color{blue}{\left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right) \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\color{blue}{\left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\left(M \cdot \color{blue}{\left(D \cdot \frac{\frac{1}{2}}{d}\right)}\right) \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\left(M \cdot \left(D \cdot \color{blue}{\frac{\frac{1}{2}}{d}}\right)\right) \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\left(M \cdot \left(D \cdot \frac{\color{blue}{\frac{1}{2}}}{d}\right)\right) \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      7. associate-/r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\left(M \cdot \left(D \cdot \color{blue}{\frac{1}{2 \cdot d}}\right)\right) \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\left(M \cdot \left(D \cdot \frac{1}{\color{blue}{2 \cdot d}}\right)\right) \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      9. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\left(M \cdot \color{blue}{\frac{D}{2 \cdot d}}\right) \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      10. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\color{blue}{\frac{h}{\ell}} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      15. associate-*l/ N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\frac{h \cdot \frac{-1}{2}}{\ell}}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      16. associate-*r/ N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{\frac{M \cdot D}{2 \cdot d} \cdot \left(h \cdot \frac{-1}{2}\right)}{\ell}}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      17. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\frac{\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\left(\frac{-1}{2} \cdot h\right)}}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      18. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

   8. Applied rewrites 78.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right) \cdot \left(-0.5 \cdot h\right)}{\ell}}, M \cdot \left(D \cdot \frac{0.5}{d}\right), 1\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right)} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      5. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{\sqrt{1}}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      9. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{1}{\color{blue}{\frac{\ell}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      11. clear-num N/A

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      13. sqrt-unprod N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      16. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      17. *-commutative N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      19. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      20. sqrt-unprod N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      21. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      22. lift-sqrt.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      23. lower-/.f64 85.1

          \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right) \cdot \left(-0.5 \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{0.5}{d}\right), 1\right) \]

   10. Applied rewrites 85.1%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right) \cdot \left(-0.5 \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{0.5}{d}\right), 1\right) \]

   if 9.6000000000000006e160 < l

   1. Initial program 35.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 58.1

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 58.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 58.2%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 72.5%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(-0.5 \cdot h\right) \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right)}{\ell}, \left(\frac{0.5}{d} \cdot D\right) \cdot M, 1\right) \cdot \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right)\\ \mathbf{elif}\;\ell \leq 9.6 \cdot 10^{+160}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(-0.5 \cdot h\right) \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right)}{\ell}, \left(\frac{0.5}{d} \cdot D\right) \cdot M, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.9% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{0.5}{d} \cdot D\right) \cdot M\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.5 \cdot \frac{h}{\ell}\right) \cdot t\_0, t\_0, 1\right) \cdot \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right)\\ \mathbf{elif}\;\ell \leq 9.6 \cdot 10^{+160}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(-0.5 \cdot h\right) \cdot t\_0}{\ell}, t\_0, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (* (/ 0.5 d) D) M)))
   (if (<= l -5e-310)
     (*
      (fma (* (* -0.5 (/ h l)) t_0) t_0 1.0)
      (* (sqrt (/ 1.0 (* l h))) (- d)))
     (if (<= l 9.6e+160)
       (* (fma (/ (* (* -0.5 h) t_0) l) t_0 1.0) (/ d (sqrt (* l h))))
       (/ d (* (sqrt l) (sqrt h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = ((0.5 / d) * D) * M;
	double tmp;
	if (l <= -5e-310) {
		tmp = fma(((-0.5 * (h / l)) * t_0), t_0, 1.0) * (sqrt((1.0 / (l * h))) * -d);
	} else if (l <= 9.6e+160) {
		tmp = fma((((-0.5 * h) * t_0) / l), t_0, 1.0) * (d / sqrt((l * h)));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(Float64(0.5 / d) * D) * M)
	tmp = 0.0
	if (l <= -5e-310)
		tmp = Float64(fma(Float64(Float64(-0.5 * Float64(h / l)) * t_0), t_0, 1.0) * Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d)));
	elseif (l <= 9.6e+160)
		tmp = Float64(fma(Float64(Float64(Float64(-0.5 * h) * t_0) / l), t_0, 1.0) * Float64(d / sqrt(Float64(l * h))));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[(0.5 / d), $MachinePrecision] * D), $MachinePrecision] * M), $MachinePrecision]}, If[LessEqual[l, -5e-310], N[(N[(N[(N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 9.6e+160], N[(N[(N[(N[(N[(-0.5 * h), $MachinePrecision] * t$95$0), $MachinePrecision] / l), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 64.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-/.f64 63.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 63.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

   6. Applied rewrites 65.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.5\right) \cdot \left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right), M \cdot \left(D \cdot \frac{0.5}{d}\right), 1\right)} \]

   7. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      2. unpow2 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      5. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      6. lower-neg.f64 N/A

         \[\leadsto \left(\color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \cdot \mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right) \cdot \mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      9. *-commutative N/A

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right) \cdot \mathsf{fma}\left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      10. lower-*.f64 67.1

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right) \cdot \mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.5\right) \cdot \left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right), M \cdot \left(D \cdot \frac{0.5}{d}\right), 1\right) \]

   9. Applied rewrites 67.1%

      \[\leadsto \color{blue}{\left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.5\right) \cdot \left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right), M \cdot \left(D \cdot \frac{0.5}{d}\right), 1\right) \]

   if -4.999999999999985e-310 < l < 9.6000000000000006e160

   1. Initial program 75.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-/.f64 76.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 76.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

   6. Applied rewrites 75.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.5\right) \cdot \left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right), M \cdot \left(D \cdot \frac{0.5}{d}\right), 1\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right)}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      2. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\color{blue}{\left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right) \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\color{blue}{\left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\left(M \cdot \color{blue}{\left(D \cdot \frac{\frac{1}{2}}{d}\right)}\right) \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\left(M \cdot \left(D \cdot \color{blue}{\frac{\frac{1}{2}}{d}}\right)\right) \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\left(M \cdot \left(D \cdot \frac{\color{blue}{\frac{1}{2}}}{d}\right)\right) \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      7. associate-/r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\left(M \cdot \left(D \cdot \color{blue}{\frac{1}{2 \cdot d}}\right)\right) \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\left(M \cdot \left(D \cdot \frac{1}{\color{blue}{2 \cdot d}}\right)\right) \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      9. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\left(M \cdot \color{blue}{\frac{D}{2 \cdot d}}\right) \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      10. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\color{blue}{\frac{h}{\ell}} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      15. associate-*l/ N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\frac{h \cdot \frac{-1}{2}}{\ell}}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      16. associate-*r/ N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{\frac{M \cdot D}{2 \cdot d} \cdot \left(h \cdot \frac{-1}{2}\right)}{\ell}}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      17. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\frac{\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\left(\frac{-1}{2} \cdot h\right)}}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      18. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

   8. Applied rewrites 78.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right) \cdot \left(-0.5 \cdot h\right)}{\ell}}, M \cdot \left(D \cdot \frac{0.5}{d}\right), 1\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right)} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      5. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{\sqrt{1}}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      9. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{1}{\color{blue}{\frac{\ell}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      11. clear-num N/A

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      13. sqrt-unprod N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      16. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      17. *-commutative N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      19. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      20. sqrt-unprod N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      21. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      22. lift-sqrt.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      23. lower-/.f64 85.1

          \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right) \cdot \left(-0.5 \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{0.5}{d}\right), 1\right) \]

   10. Applied rewrites 85.1%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right) \cdot \left(-0.5 \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{0.5}{d}\right), 1\right) \]

   if 9.6000000000000006e160 < l

   1. Initial program 35.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 58.1

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 58.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 58.2%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 72.5%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.5 \cdot \frac{h}{\ell}\right) \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{0.5}{d} \cdot D\right) \cdot M, 1\right) \cdot \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right)\\ \mathbf{elif}\;\ell \leq 9.6 \cdot 10^{+160}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(-0.5 \cdot h\right) \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right)}{\ell}, \left(\frac{0.5}{d} \cdot D\right) \cdot M, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.6% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{0.5}{d} \cdot D\right) \cdot M\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(\left(-0.5 \cdot \frac{h}{\ell}\right) \cdot M\right) \cdot D\right) \cdot \frac{0.5}{d}\right) \cdot \left(M \cdot D\right), \frac{0.5}{d}, 1\right) \cdot \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right)\\ \mathbf{elif}\;\ell \leq 9.6 \cdot 10^{+160}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(-0.5 \cdot h\right) \cdot t\_0}{\ell}, t\_0, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (* (/ 0.5 d) D) M)))
   (if (<= l -5e-310)
     (*
      (fma
       (* (* (* (* (* -0.5 (/ h l)) M) D) (/ 0.5 d)) (* M D))
       (/ 0.5 d)
       1.0)
      (* (sqrt (/ 1.0 (* l h))) (- d)))
     (if (<= l 9.6e+160)
       (* (fma (/ (* (* -0.5 h) t_0) l) t_0 1.0) (/ d (sqrt (* l h))))
       (/ d (* (sqrt l) (sqrt h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = ((0.5 / d) * D) * M;
	double tmp;
	if (l <= -5e-310) {
		tmp = fma((((((-0.5 * (h / l)) * M) * D) * (0.5 / d)) * (M * D)), (0.5 / d), 1.0) * (sqrt((1.0 / (l * h))) * -d);
	} else if (l <= 9.6e+160) {
		tmp = fma((((-0.5 * h) * t_0) / l), t_0, 1.0) * (d / sqrt((l * h)));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(Float64(0.5 / d) * D) * M)
	tmp = 0.0
	if (l <= -5e-310)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(Float64(-0.5 * Float64(h / l)) * M) * D) * Float64(0.5 / d)) * Float64(M * D)), Float64(0.5 / d), 1.0) * Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d)));
	elseif (l <= 9.6e+160)
		tmp = Float64(fma(Float64(Float64(Float64(-0.5 * h) * t_0) / l), t_0, 1.0) * Float64(d / sqrt(Float64(l * h))));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[(0.5 / d), $MachinePrecision] * D), $MachinePrecision] * M), $MachinePrecision]}, If[LessEqual[l, -5e-310], N[(N[(N[(N[(N[(N[(N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision] * D), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * N[(0.5 / d), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 9.6e+160], N[(N[(N[(N[(N[(-0.5 * h), $MachinePrecision] * t$95$0), $MachinePrecision] / l), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 64.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-/.f64 63.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 63.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow2 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lower-neg.f64 N/A

         \[\leadsto \left(\color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. *-commutative N/A

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-*.f64 66.2

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   7. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   9. Applied rewrites 65.7%

      \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\left(\left(M \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right) \cdot D\right) \cdot \frac{0.5}{d}\right) \cdot \left(D \cdot M\right), \frac{0.5}{d}, 1\right)} \]

   if -4.999999999999985e-310 < l < 9.6000000000000006e160

   1. Initial program 75.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-/.f64 76.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 76.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

   6. Applied rewrites 75.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.5\right) \cdot \left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right), M \cdot \left(D \cdot \frac{0.5}{d}\right), 1\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right)}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      2. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\color{blue}{\left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right) \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\color{blue}{\left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\left(M \cdot \color{blue}{\left(D \cdot \frac{\frac{1}{2}}{d}\right)}\right) \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\left(M \cdot \left(D \cdot \color{blue}{\frac{\frac{1}{2}}{d}}\right)\right) \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\left(M \cdot \left(D \cdot \frac{\color{blue}{\frac{1}{2}}}{d}\right)\right) \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      7. associate-/r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\left(M \cdot \left(D \cdot \color{blue}{\frac{1}{2 \cdot d}}\right)\right) \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\left(M \cdot \left(D \cdot \frac{1}{\color{blue}{2 \cdot d}}\right)\right) \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      9. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\left(M \cdot \color{blue}{\frac{D}{2 \cdot d}}\right) \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      10. associate-/l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\color{blue}{\frac{h}{\ell}} \cdot \frac{-1}{2}\right), M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      15. associate-*l/ N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\frac{h \cdot \frac{-1}{2}}{\ell}}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      16. associate-*r/ N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{\frac{M \cdot D}{2 \cdot d} \cdot \left(h \cdot \frac{-1}{2}\right)}{\ell}}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      17. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\frac{\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\left(\frac{-1}{2} \cdot h\right)}}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      18. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

   8. Applied rewrites 78.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right) \cdot \left(-0.5 \cdot h\right)}{\ell}}, M \cdot \left(D \cdot \frac{0.5}{d}\right), 1\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right)} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      5. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{\sqrt{1}}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      9. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{1}{\color{blue}{\frac{\ell}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      11. clear-num N/A

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      13. sqrt-unprod N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      16. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      17. *-commutative N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      19. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      20. sqrt-unprod N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      21. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      22. lift-sqrt.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right), 1\right) \]

      23. lower-/.f64 85.1

          \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right) \cdot \left(-0.5 \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{0.5}{d}\right), 1\right) \]

   10. Applied rewrites 85.1%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right) \cdot \left(-0.5 \cdot h\right)}{\ell}, M \cdot \left(D \cdot \frac{0.5}{d}\right), 1\right) \]

   if 9.6000000000000006e160 < l

   1. Initial program 35.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 58.1

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 58.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 58.2%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 72.5%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(\left(-0.5 \cdot \frac{h}{\ell}\right) \cdot M\right) \cdot D\right) \cdot \frac{0.5}{d}\right) \cdot \left(M \cdot D\right), \frac{0.5}{d}, 1\right) \cdot \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right)\\ \mathbf{elif}\;\ell \leq 9.6 \cdot 10^{+160}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(-0.5 \cdot h\right) \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right)}{\ell}, \left(\frac{0.5}{d} \cdot D\right) \cdot M, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 45.4% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{if}\;d \leq -1.5 \cdot 10^{+95}:\\ \;\;\;\;t\_0 \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -2.8 \cdot 10^{-157}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-312}:\\ \;\;\;\;t\_0 \cdot d\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (* l h)))))
   (if (<= d -1.5e+95)
     (* t_0 (- d))
     (if (<= d -2.8e-157)
       (* (sqrt (/ d l)) (sqrt (/ d h)))
       (if (<= d -5e-312) (* t_0 d) (/ d (* (sqrt l) (sqrt h))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((1.0 / (l * h)));
	double tmp;
	if (d <= -1.5e+95) {
		tmp = t_0 * -d;
	} else if (d <= -2.8e-157) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (d <= -5e-312) {
		tmp = t_0 * d;
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 / (l * h)))
    if (d <= (-1.5d+95)) then
        tmp = t_0 * -d
    else if (d <= (-2.8d-157)) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else if (d <= (-5d-312)) then
        tmp = t_0 * d
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((1.0 / (l * h)));
	double tmp;
	if (d <= -1.5e+95) {
		tmp = t_0 * -d;
	} else if (d <= -2.8e-157) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else if (d <= -5e-312) {
		tmp = t_0 * d;
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((1.0 / (l * h)))
	tmp = 0
	if d <= -1.5e+95:
		tmp = t_0 * -d
	elif d <= -2.8e-157:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	elif d <= -5e-312:
		tmp = t_0 * d
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(1.0 / Float64(l * h)))
	tmp = 0.0
	if (d <= -1.5e+95)
		tmp = Float64(t_0 * Float64(-d));
	elseif (d <= -2.8e-157)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (d <= -5e-312)
		tmp = Float64(t_0 * d);
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((1.0 / (l * h)));
	tmp = 0.0;
	if (d <= -1.5e+95)
		tmp = t_0 * -d;
	elseif (d <= -2.8e-157)
		tmp = sqrt((d / l)) * sqrt((d / h));
	elseif (d <= -5e-312)
		tmp = t_0 * d;
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -1.5e+95], N[(t$95$0 * (-d)), $MachinePrecision], If[LessEqual[d, -2.8e-157], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-312], N[(t$95$0 * d), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.49999999999999996e95

   1. Initial program 70.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      10. lower-*.f64 65.1

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   5. Applied rewrites 65.1%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if -1.49999999999999996e95 < d < -2.8000000000000001e-157

   1. Initial program 76.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \]

   4. Applied rewrites 76.1%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}} \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}} \]

      2. lower-/.f64 61.4

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}} \]

   7. Applied rewrites 61.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}} \]

   if -2.8000000000000001e-157 < d < -5.0000000000022e-312

   1. Initial program 36.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 24.6

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 24.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

   if -5.0000000000022e-312 < d

   1. Initial program 65.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 41.1

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 41.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 41.1%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 49.0%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.5 \cdot 10^{+95}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -2.8 \cdot 10^{-157}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-312}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot d\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 45.8% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.3 \cdot 10^{-212}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{-223}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1.3e-212)
   (* (sqrt (/ 1.0 (* l h))) (- d))
   (if (<= l 1.3e-223)
     (/ (* (sqrt (/ h l)) (- d)) h)
     (/ d (* (sqrt l) (sqrt h))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.3e-212) {
		tmp = sqrt((1.0 / (l * h))) * -d;
	} else if (l <= 1.3e-223) {
		tmp = (sqrt((h / l)) * -d) / h;
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-1.3d-212)) then
        tmp = sqrt((1.0d0 / (l * h))) * -d
    else if (l <= 1.3d-223) then
        tmp = (sqrt((h / l)) * -d) / h
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.3e-212) {
		tmp = Math.sqrt((1.0 / (l * h))) * -d;
	} else if (l <= 1.3e-223) {
		tmp = (Math.sqrt((h / l)) * -d) / h;
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -1.3e-212:
		tmp = math.sqrt((1.0 / (l * h))) * -d
	elif l <= 1.3e-223:
		tmp = (math.sqrt((h / l)) * -d) / h
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1.3e-212)
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d));
	elseif (l <= 1.3e-223)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * Float64(-d)) / h);
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -1.3e-212)
		tmp = sqrt((1.0 / (l * h))) * -d;
	elseif (l <= 1.3e-223)
		tmp = (sqrt((h / l)) * -d) / h;
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1.3e-212], N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], If[LessEqual[l, 1.3e-223], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision] / h), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.3e-212

   1. Initial program 62.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      10. lower-*.f64 49.0

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   5. Applied rewrites 49.0%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if -1.3e-212 < l < 1.3e-223

   1. Initial program 78.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   5. Applied rewrites 46.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \frac{D \cdot D}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   6. Taylor expanded in l around -inf

      \[\leadsto \frac{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   7. Step-by-step derivation

   8. Applied rewrites 36.3%

      \[\leadsto \frac{\left(-d\right) \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   if 1.3e-223 < l

   1. Initial program 62.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 46.4

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 46.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 46.4%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 55.3%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.3 \cdot 10^{-212}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{-223}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 44.7% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 9.8 \cdot 10^{-224}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l 9.8e-224)
   (* (sqrt (/ 1.0 (* l h))) (- d))
   (/ d (* (sqrt l) (sqrt h)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 9.8e-224) {
		tmp = sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= 9.8d-224) then
        tmp = sqrt((1.0d0 / (l * h))) * -d
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 9.8e-224) {
		tmp = Math.sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= 9.8e-224:
		tmp = math.sqrt((1.0 / (l * h))) * -d
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 9.8e-224)
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= 9.8e-224)
		tmp = sqrt((1.0 / (l * h))) * -d;
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, 9.8e-224], N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 9.7999999999999992e-224

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      10. lower-*.f64 42.7

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   5. Applied rewrites 42.7%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if 9.7999999999999992e-224 < l

   1. Initial program 62.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 46.4

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 46.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 46.4%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 55.3%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 9.8 \cdot 10^{-224}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 40.8% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{if}\;d \leq -2.8 \cdot 10^{-157}:\\ \;\;\;\;t\_0 \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot d\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (* l h)))))
   (if (<= d -2.8e-157) (* t_0 (- d)) (* t_0 d))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((1.0 / (l * h)));
	double tmp;
	if (d <= -2.8e-157) {
		tmp = t_0 * -d;
	} else {
		tmp = t_0 * d;
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 / (l * h)))
    if (d <= (-2.8d-157)) then
        tmp = t_0 * -d
    else
        tmp = t_0 * d
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((1.0 / (l * h)));
	double tmp;
	if (d <= -2.8e-157) {
		tmp = t_0 * -d;
	} else {
		tmp = t_0 * d;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((1.0 / (l * h)))
	tmp = 0
	if d <= -2.8e-157:
		tmp = t_0 * -d
	else:
		tmp = t_0 * d
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(1.0 / Float64(l * h)))
	tmp = 0.0
	if (d <= -2.8e-157)
		tmp = Float64(t_0 * Float64(-d));
	else
		tmp = Float64(t_0 * d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((1.0 / (l * h)));
	tmp = 0.0;
	if (d <= -2.8e-157)
		tmp = t_0 * -d;
	else
		tmp = t_0 * d;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -2.8e-157], N[(t$95$0 * (-d)), $MachinePrecision], N[(t$95$0 * d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -2.8000000000000001e-157

   1. Initial program 73.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      10. lower-*.f64 56.9

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   5. Applied rewrites 56.9%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if -2.8000000000000001e-157 < d

   1. Initial program 59.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 37.9

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 37.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.8 \cdot 10^{-157}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot d\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 25.6% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{\ell \cdot h}} \cdot d \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (* (sqrt (/ 1.0 (* l h))) d))

C

double code(double d, double h, double l, double M, double D) {
	return sqrt((1.0 / (l * h))) * d;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = sqrt((1.0d0 / (l * h))) * d
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return Math.sqrt((1.0 / (l * h))) * d;
}

Python

def code(d, h, l, M, D):
	return math.sqrt((1.0 / (l * h))) * d

Julia

function code(d, h, l, M, D)
	return Float64(sqrt(Float64(1.0 / Float64(l * h))) * d)
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = sqrt((1.0 / (l * h))) * d;
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]

Derivation

1. Initial program 64.9%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Add Preprocessing

3. Taylor expanded in h around 0

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

   4. lower-/.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

   5. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   6. lower-*.f64 26.4

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

5. Applied rewrites 26.4%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
6. Add Preprocessing

Alternative 18: 25.6% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \frac{d}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (/ d (sqrt (* l h))))

C

double code(double d, double h, double l, double M, double D) {
	return d / sqrt((l * h));
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = d / sqrt((l * h))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return d / Math.sqrt((l * h));
}

Python

def code(d, h, l, M, D):
	return d / math.sqrt((l * h))

Julia

function code(d, h, l, M, D)
	return Float64(d / sqrt(Float64(l * h)))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = d / sqrt((l * h));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.9%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Add Preprocessing

3. Taylor expanded in h around 0

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

   4. lower-/.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

   5. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   6. lower-*.f64 26.4

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

5. Applied rewrites 26.4%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
6. Step-by-step derivation

7. Applied rewrites 26.4%

   \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024244 
(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)))))