Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.6% → 76.5%

Time: 24.4s

Alternatives: 29

Speedup: 3.7×

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: 76.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{M \cdot D}{d \cdot 2}\\ \mathbf{if}\;d \leq -1.85 \cdot 10^{-305}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({t\_0}^{2} \cdot \frac{-1}{2}\right)\right)\\ \mathbf{elif}\;d \leq 3.05 \cdot 10^{-214}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\left(M \cdot D\right) \cdot -0.25\right), \left(M \cdot D\right) \cdot \frac{h}{\ell \cdot \left(d \cdot 2\right)}, \frac{d}{\sqrt{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(t\_0, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ (* M D) (* d 2.0))))
   (if (<= d -1.85e-305)
     (*
      (* (pow (/ d h) (/ 1.0 2.0)) (* (sqrt (- d)) (sqrt (/ -1.0 l))))
      (+ 1.0 (* (/ h l) (* (pow t_0 2.0) (/ -1.0 2.0)))))
     (if (<= d 3.05e-214)
       (fma
        (* (sqrt (/ 1.0 (* h l))) (* (* M D) -0.25))
        (* (* M D) (/ h (* l (* d 2.0))))
        (/ d (sqrt (* h l))))
       (*
        (/ (/ d (sqrt h)) (sqrt l))
        (fma t_0 (* (/ h l) (* -0.25 (/ (* M D) d))) 1.0))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (M * D) / (d * 2.0);
	double tmp;
	if (d <= -1.85e-305) {
		tmp = (pow((d / h), (1.0 / 2.0)) * (sqrt(-d) * sqrt((-1.0 / l)))) * (1.0 + ((h / l) * (pow(t_0, 2.0) * (-1.0 / 2.0))));
	} else if (d <= 3.05e-214) {
		tmp = fma((sqrt((1.0 / (h * l))) * ((M * D) * -0.25)), ((M * D) * (h / (l * (d * 2.0)))), (d / sqrt((h * l))));
	} else {
		tmp = ((d / sqrt(h)) / sqrt(l)) * fma(t_0, ((h / l) * (-0.25 * ((M * D) / d))), 1.0);
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(M * D) / Float64(d * 2.0))
	tmp = 0.0
	if (d <= -1.85e-305)
		tmp = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * Float64(sqrt(Float64(-d)) * sqrt(Float64(-1.0 / l)))) * Float64(1.0 + Float64(Float64(h / l) * Float64((t_0 ^ 2.0) * Float64(-1.0 / 2.0)))));
	elseif (d <= 3.05e-214)
		tmp = fma(Float64(sqrt(Float64(1.0 / Float64(h * l))) * Float64(Float64(M * D) * -0.25)), Float64(Float64(M * D) * Float64(h / Float64(l * Float64(d * 2.0)))), Float64(d / sqrt(Float64(h * l))));
	else
		tmp = Float64(Float64(Float64(d / sqrt(h)) / sqrt(l)) * fma(t_0, Float64(Float64(h / l) * Float64(-0.25 * Float64(Float64(M * D) / d))), 1.0));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.85e-305], N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] * N[Sqrt[N[(-1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.05e-214], N[(N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * N[(h / N[(l * N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(N[(h / l), $MachinePrecision] * N[(-0.25 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.84999999999999989e-305

   1. Initial program 70.7%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. div-inv N/A

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

      7. sqrt-prod N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. neg-mul-1 N/A

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

      13. associate-/r* N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 79.0

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

   4. Applied egg-rr 79.0%

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

   if -1.84999999999999989e-305 < d < 3.05e-214

   1. Initial program 33.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

   4. Applied egg-rr 33.4%

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

      1. lift-/.f64 N/A

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

      2. rem-exp-log N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. rem-exp-log N/A

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

      6. lift-/.f64 N/A

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

      7. pow-prod-down N/A

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

      8. rem-exp-log N/A

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

      9. rem-exp-log N/A

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

      10. *-commutative N/A

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

      11. pow-prod-down N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. unpow1/2 N/A

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

      16. lift-/.f64 N/A

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

      17. clear-num N/A

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

      18. lift-/.f64 N/A

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

      19. sqrt-div N/A

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

      20. metadata-eval N/A

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

      21. lift-sqrt.f64 N/A

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

   6. Applied egg-rr 29.3%

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

   8. Applied egg-rr 51.9%

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

   9. Taylor expanded in d around 0

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 70.8

          \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-0.25 \cdot \color{blue}{\left(D \cdot M\right)}\right), \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

   11. Simplified 70.8%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-0.25 \cdot \left(D \cdot M\right)\right)}, \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

   if 3.05e-214 < d

   1. Initial program 84.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

   4. Applied egg-rr 85.3%

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

      1. lift-/.f64 N/A

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

      2. rem-exp-log N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. rem-exp-log N/A

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

      6. lift-/.f64 N/A

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

      7. pow-prod-down N/A

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

      8. rem-exp-log N/A

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

      9. rem-exp-log N/A

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

      10. *-commutative N/A

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

      11. pow-prod-down N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. unpow1/2 N/A

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

      16. lift-/.f64 N/A

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

      17. clear-num N/A

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

      18. lift-/.f64 N/A

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

      19. sqrt-div N/A

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

      20. metadata-eval N/A

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

      21. lift-sqrt.f64 N/A

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

   6. Applied egg-rr 87.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

   8. Applied egg-rr 87.6%

      \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right)} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-undiv N/A

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

      4. div-inv N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. times-frac N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/r* N/A

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

      14. sqrt-div N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. sqrt-div N/A

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

      18. lift-*.f64 N/A

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

      19. sqrt-prod N/A

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

      20. rem-square-sqrt N/A

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

      21. pow1/2 N/A

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

      22. lower-/.f64 N/A

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

      23. pow1/2 N/A

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

      24. lower-sqrt.f64 92.1

          \[\leadsto \frac{\frac{d}{\color{blue}{\sqrt{h}}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right) \]

   10. Applied egg-rr 92.1%

       \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.85 \cdot 10^{-305}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right)\\ \mathbf{elif}\;d \leq 3.05 \cdot 10^{-214}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\left(M \cdot D\right) \cdot -0.25\right), \left(M \cdot D\right) \cdot \frac{h}{\ell \cdot \left(d \cdot 2\right)}, \frac{d}{\sqrt{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 59.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -2 \cdot 10^{-169}:\\ \;\;\;\;\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{h \cdot \left(M \cdot D\right)}{2 \cdot \left(d \cdot \ell\right)}, \frac{\left(M \cdot D\right) \cdot -0.25}{d}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<=
      (*
       (+ 1.0 (* (/ h l) (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ -1.0 2.0))))
       (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))))
      -2e-169)
   (*
    (sqrt (/ (* d d) (* h l)))
    (fma (/ (* h (* M D)) (* 2.0 (* d l))) (/ (* (* M D) -0.25) d) 1.0))
   (/ (sqrt (/ d l)) (sqrt (/ h d)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((1.0 + ((h / l) * (pow(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0)))) * (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0)))) <= -2e-169) {
		tmp = sqrt(((d * d) / (h * l))) * fma(((h * (M * D)) / (2.0 * (d * l))), (((M * D) * -0.25) / d), 1.0);
	} else {
		tmp = sqrt((d / l)) / sqrt((h / d));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (Float64(Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))) * Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0)))) <= -2e-169)
		tmp = Float64(sqrt(Float64(Float64(d * d) / Float64(h * l))) * fma(Float64(Float64(h * Float64(M * D)) / Float64(2.0 * Float64(d * l))), Float64(Float64(Float64(M * D) * -0.25) / d), 1.0));
	else
		tmp = Float64(sqrt(Float64(d / l)) / sqrt(Float64(h / d)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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]), $MachinePrecision], -2e-169], N[(N[Sqrt[N[(N[(d * d), $MachinePrecision] / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(h * N[(M * D), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M * D), $MachinePrecision] * -0.25), $MachinePrecision] / d), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -2.00000000000000004e-169

   1. Initial program 90.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

   4. Applied egg-rr 91.8%

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

      1. lift-/.f64 N/A

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

      2. rem-exp-log N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. rem-exp-log N/A

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

      6. lift-/.f64 N/A

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

      7. pow-prod-down N/A

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

      8. rem-exp-log N/A

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

      9. rem-exp-log N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. frac-times N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 58.9

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

   6. Applied egg-rr 58.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

   8. Applied egg-rr 41.2%

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

   10. Applied egg-rr 58.9%

       \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{\left(M \cdot D\right) \cdot h}{2 \cdot \left(d \cdot \ell\right)}, \frac{\left(M \cdot D\right) \cdot -0.25}{d}, 1\right)} \]

   if -2.00000000000000004e-169 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 58.3%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 62.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. pow-to-exp N/A

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

      7. pow-to-exp N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-to-exp N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-lft-identity N/A

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

      13. *-lft-identity N/A

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

      14. *-rgt-identity 62.3

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-pow.f64 N/A

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

   7. Applied egg-rr 63.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -2 \cdot 10^{-169}:\\ \;\;\;\;\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{h \cdot \left(M \cdot D\right)}{2 \cdot \left(d \cdot \ell\right)}, \frac{\left(M \cdot D\right) \cdot -0.25}{d}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 58.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -0.002:\\ \;\;\;\;\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \mathsf{fma}\left(\left(M \cdot D\right) \cdot \frac{h}{\ell \cdot \left(d \cdot 2\right)}, -0.25 \cdot \frac{M \cdot D}{d}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<=
      (*
       (+ 1.0 (* (/ h l) (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ -1.0 2.0))))
       (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))))
      -0.002)
   (*
    (sqrt (/ (* d d) (* h l)))
    (fma (* (* M D) (/ h (* l (* d 2.0)))) (* -0.25 (/ (* M D) d)) 1.0))
   (/ (sqrt (/ d l)) (sqrt (/ h d)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((1.0 + ((h / l) * (pow(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0)))) * (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0)))) <= -0.002) {
		tmp = sqrt(((d * d) / (h * l))) * fma(((M * D) * (h / (l * (d * 2.0)))), (-0.25 * ((M * D) / d)), 1.0);
	} else {
		tmp = sqrt((d / l)) / sqrt((h / d));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (Float64(Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))) * Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0)))) <= -0.002)
		tmp = Float64(sqrt(Float64(Float64(d * d) / Float64(h * l))) * fma(Float64(Float64(M * D) * Float64(h / Float64(l * Float64(d * 2.0)))), Float64(-0.25 * Float64(Float64(M * D) / d)), 1.0));
	else
		tmp = Float64(sqrt(Float64(d / l)) / sqrt(Float64(h / d)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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]), $MachinePrecision], -0.002], N[(N[Sqrt[N[(N[(d * d), $MachinePrecision] / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(M * D), $MachinePrecision] * N[(h / N[(l * N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.25 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -2e-3

   1. Initial program 91.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

   4. Applied egg-rr 92.0%

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

      1. lift-/.f64 N/A

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

      2. rem-exp-log N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. rem-exp-log N/A

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

      6. lift-/.f64 N/A

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

      7. pow-prod-down N/A

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

      8. rem-exp-log N/A

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

      9. rem-exp-log N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. frac-times N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 61.5

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

   6. Applied egg-rr 61.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 61.5

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

   8. Applied egg-rr 60.6%

      \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \color{blue}{\mathsf{fma}\left(\left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, -0.25 \cdot \frac{M \cdot D}{d}, 1\right)} \]

   if -2e-3 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 59.3%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 60.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. pow-to-exp N/A

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

      7. pow-to-exp N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-to-exp N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-lft-identity N/A

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

      13. *-lft-identity N/A

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

      14. *-rgt-identity 60.3

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-pow.f64 N/A

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

   7. Applied egg-rr 61.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -0.002:\\ \;\;\;\;\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \mathsf{fma}\left(\left(M \cdot D\right) \cdot \frac{h}{\ell \cdot \left(d \cdot 2\right)}, -0.25 \cdot \frac{M \cdot D}{d}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -0.002:\\ \;\;\;\;\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \mathsf{fma}\left(M \cdot \left(D \cdot \left(D \cdot \left(M \cdot -0.5\right)\right)\right), \frac{h}{\ell \cdot \left(\left(d \cdot d\right) \cdot 4\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<=
      (*
       (+ 1.0 (* (/ h l) (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ -1.0 2.0))))
       (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))))
      -0.002)
   (*
    (sqrt (/ (* d d) (* h l)))
    (fma (* M (* D (* D (* M -0.5)))) (/ h (* l (* (* d d) 4.0))) 1.0))
   (/ (sqrt (/ d l)) (sqrt (/ h d)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((1.0 + ((h / l) * (pow(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0)))) * (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0)))) <= -0.002) {
		tmp = sqrt(((d * d) / (h * l))) * fma((M * (D * (D * (M * -0.5)))), (h / (l * ((d * d) * 4.0))), 1.0);
	} else {
		tmp = sqrt((d / l)) / sqrt((h / d));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (Float64(Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))) * Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0)))) <= -0.002)
		tmp = Float64(sqrt(Float64(Float64(d * d) / Float64(h * l))) * fma(Float64(M * Float64(D * Float64(D * Float64(M * -0.5)))), Float64(h / Float64(l * Float64(Float64(d * d) * 4.0))), 1.0));
	else
		tmp = Float64(sqrt(Float64(d / l)) / sqrt(Float64(h / d)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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]), $MachinePrecision], -0.002], N[(N[Sqrt[N[(N[(d * d), $MachinePrecision] / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(M * N[(D * N[(D * N[(M * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / N[(l * N[(N[(d * d), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -2e-3

   1. Initial program 91.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

   4. Applied egg-rr 92.0%

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

      1. lift-/.f64 N/A

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

      2. rem-exp-log N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. rem-exp-log N/A

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

      6. lift-/.f64 N/A

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

      7. pow-prod-down N/A

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

      8. rem-exp-log N/A

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

      9. rem-exp-log N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. frac-times N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 61.5

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

   6. Applied egg-rr 61.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

   8. Applied egg-rr 43.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. frac-times N/A

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

      13. associate-/l* N/A

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

   10. Applied egg-rr 39.2%

       \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \color{blue}{\mathsf{fma}\left(M \cdot \left(D \cdot \left(D \cdot \left(M \cdot -0.5\right)\right)\right), \frac{h}{\ell \cdot \left(\left(d \cdot d\right) \cdot 4\right)}, 1\right)} \]

   if -2e-3 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 59.3%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 60.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. pow-to-exp N/A

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

      7. pow-to-exp N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-to-exp N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-lft-identity N/A

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

      13. *-lft-identity N/A

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

      14. *-rgt-identity 60.3

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-pow.f64 N/A

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

   7. Applied egg-rr 61.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -0.002:\\ \;\;\;\;\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \mathsf{fma}\left(M \cdot \left(D \cdot \left(D \cdot \left(M \cdot -0.5\right)\right)\right), \frac{h}{\ell \cdot \left(\left(d \cdot d\right) \cdot 4\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 45.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -2 \cdot 10^{-169}:\\ \;\;\;\;d \cdot \sqrt{\sqrt{\frac{1}{\ell} \cdot \frac{1}{\ell}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<=
      (*
       (+ 1.0 (* (/ h l) (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ -1.0 2.0))))
       (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))))
      -2e-169)
   (* d (sqrt (sqrt (* (/ 1.0 l) (/ 1.0 l)))))
   (/ (sqrt (/ d l)) (sqrt (/ h d)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((1.0 + ((h / l) * (pow(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0)))) * (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0)))) <= -2e-169) {
		tmp = d * sqrt(sqrt(((1.0 / l) * (1.0 / l))));
	} else {
		tmp = sqrt((d / l)) / sqrt((h / 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) :: tmp
    if (((1.0d0 + ((h / l) * ((((m * d_1) / (d * 2.0d0)) ** 2.0d0) * ((-1.0d0) / 2.0d0)))) * (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0)))) <= (-2d-169)) then
        tmp = d * sqrt(sqrt(((1.0d0 / l) * (1.0d0 / l))))
    else
        tmp = sqrt((d / l)) / sqrt((h / d))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((1.0 + ((h / l) * (Math.pow(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0)))) * (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0)))) <= -2e-169) {
		tmp = d * Math.sqrt(Math.sqrt(((1.0 / l) * (1.0 / l))));
	} else {
		tmp = Math.sqrt((d / l)) / Math.sqrt((h / d));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if ((1.0 + ((h / l) * (math.pow(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0)))) * (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0)))) <= -2e-169:
		tmp = d * math.sqrt(math.sqrt(((1.0 / l) * (1.0 / l))))
	else:
		tmp = math.sqrt((d / l)) / math.sqrt((h / d))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (Float64(Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))) * Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0)))) <= -2e-169)
		tmp = Float64(d * sqrt(sqrt(Float64(Float64(1.0 / l) * Float64(1.0 / l)))));
	else
		tmp = Float64(sqrt(Float64(d / l)) / sqrt(Float64(h / d)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (((1.0 + ((h / l) * ((((M * D) / (d * 2.0)) ^ 2.0) * (-1.0 / 2.0)))) * (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0)))) <= -2e-169)
		tmp = d * sqrt(sqrt(((1.0 / l) * (1.0 / l))));
	else
		tmp = sqrt((d / l)) / sqrt((h / d));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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]), $MachinePrecision], -2e-169], N[(d * N[Sqrt[N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -2.00000000000000004e-169

   1. Initial program 90.9%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 0.7%

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

   6. Taylor expanded in h around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 0.4

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

   8. Simplified 0.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell}}} \]
   9. Step-by-step derivation

      1. lift-/.f64 0.4

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

      2. rem-square-sqrt N/A

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

      3. sqrt-unprod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 16.9

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

   10. Applied egg-rr 16.9%

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

   if -2.00000000000000004e-169 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 58.3%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 62.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. pow-to-exp N/A

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

      7. pow-to-exp N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-to-exp N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-lft-identity N/A

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

      13. *-lft-identity N/A

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

      14. *-rgt-identity 62.3

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-pow.f64 N/A

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

   7. Applied egg-rr 63.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -2 \cdot 10^{-169}:\\ \;\;\;\;d \cdot \sqrt{\sqrt{\frac{1}{\ell} \cdot \frac{1}{\ell}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 44.1% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<=
      (*
       (+ 1.0 (* (/ h l) (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ -1.0 2.0))))
       (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))))
      0.0)
   (* (- d) (sqrt (/ 1.0 (* h l))))
   (* (sqrt (/ d l)) (sqrt (/ d h)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((1.0 + ((h / l) * (pow(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0)))) * (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0)))) <= 0.0) {
		tmp = -d * sqrt((1.0 / (h * l)));
	} else {
		tmp = sqrt((d / l)) * sqrt((d / 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 (((1.0d0 + ((h / l) * ((((m * d_1) / (d * 2.0d0)) ** 2.0d0) * ((-1.0d0) / 2.0d0)))) * (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0)))) <= 0.0d0) then
        tmp = -d * sqrt((1.0d0 / (h * l)))
    else
        tmp = sqrt((d / l)) * sqrt((d / 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 (((1.0 + ((h / l) * (Math.pow(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0)))) * (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0)))) <= 0.0) {
		tmp = -d * Math.sqrt((1.0 / (h * l)));
	} else {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if ((1.0 + ((h / l) * (math.pow(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0)))) * (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0)))) <= 0.0:
		tmp = -d * math.sqrt((1.0 / (h * l)))
	else:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (Float64(Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))) * Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0)))) <= 0.0)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l))));
	else
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (((1.0 + ((h / l) * ((((M * D) / (d * 2.0)) ^ 2.0) * (-1.0 / 2.0)))) * (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0)))) <= 0.0)
		tmp = -d * sqrt((1.0 / (h * l)));
	else
		tmp = sqrt((d / l)) * sqrt((d / h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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]), $MachinePrecision], 0.0], N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   1. Initial program 84.1%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 3.8%

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

   6. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. 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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 15.9

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

   8. Simplified 15.9%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 61.6%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 66.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. pow-to-exp N/A

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

      7. pow-to-exp N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-to-exp N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-lft-identity N/A

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

      13. *-lft-identity N/A

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

      14. *-rgt-identity 66.0

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 66.0

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

   7. Applied egg-rr 66.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 33.3% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<=
      (*
       (+ 1.0 (* (/ h l) (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ -1.0 2.0))))
       (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))))
      0.0)
   (* d (sqrt (/ 1.0 (* h l))))
   (sqrt (* d (/ d (* h l))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((1.0 + ((h / l) * (pow(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0)))) * (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0)))) <= 0.0) {
		tmp = d * sqrt((1.0 / (h * l)));
	} else {
		tmp = sqrt((d * (d / (h * l))));
	}
	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 (((1.0d0 + ((h / l) * ((((m * d_1) / (d * 2.0d0)) ** 2.0d0) * ((-1.0d0) / 2.0d0)))) * (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0)))) <= 0.0d0) then
        tmp = d * sqrt((1.0d0 / (h * l)))
    else
        tmp = sqrt((d * (d / (h * l))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((1.0 + ((h / l) * (Math.pow(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0)))) * (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0)))) <= 0.0) {
		tmp = d * Math.sqrt((1.0 / (h * l)));
	} else {
		tmp = Math.sqrt((d * (d / (h * l))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if ((1.0 + ((h / l) * (math.pow(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0)))) * (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0)))) <= 0.0:
		tmp = d * math.sqrt((1.0 / (h * l)))
	else:
		tmp = math.sqrt((d * (d / (h * l))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (Float64(Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))) * Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0)))) <= 0.0)
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(h * l))));
	else
		tmp = sqrt(Float64(d * Float64(d / Float64(h * l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (((1.0 + ((h / l) * ((((M * D) / (d * 2.0)) ^ 2.0) * (-1.0 / 2.0)))) * (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0)))) <= 0.0)
		tmp = d * sqrt((1.0 / (h * l)));
	else
		tmp = sqrt((d * (d / (h * l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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]), $MachinePrecision], 0.0], N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(d * N[(d / N[(h * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   1. Initial program 84.1%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 3.8%

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

   6. Taylor expanded in d around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 15.1

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

   8. Simplified 15.1%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 61.6%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 66.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. pow-to-exp N/A

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

      7. pow-to-exp N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-to-exp N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-lft-identity N/A

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

      13. *-lft-identity N/A

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

      14. *-rgt-identity 66.0

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

      15. lift-*.f64 N/A

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

      16. lift-pow.f64 N/A

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

      17. lift-pow.f64 N/A

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

   7. Applied egg-rr 25.9%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. sqrt-undiv N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lift-/.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. times-frac N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-sqrt.f64 43.8

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

      17. lift-/.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-/l* N/A

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

      20. lower-*.f64 N/A

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

      21. lower-/.f64 52.3

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

   9. Applied egg-rr 52.3%

      \[\leadsto \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 33.2% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<=
      (*
       (+ 1.0 (* (/ h l) (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ -1.0 2.0))))
       (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))))
      0.0)
   (/ d (sqrt (* h l)))
   (sqrt (* d (/ d (* h l))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((1.0 + ((h / l) * (pow(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0)))) * (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0)))) <= 0.0) {
		tmp = d / sqrt((h * l));
	} else {
		tmp = sqrt((d * (d / (h * l))));
	}
	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 (((1.0d0 + ((h / l) * ((((m * d_1) / (d * 2.0d0)) ** 2.0d0) * ((-1.0d0) / 2.0d0)))) * (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0)))) <= 0.0d0) then
        tmp = d / sqrt((h * l))
    else
        tmp = sqrt((d * (d / (h * l))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((1.0 + ((h / l) * (Math.pow(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0)))) * (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0)))) <= 0.0) {
		tmp = d / Math.sqrt((h * l));
	} else {
		tmp = Math.sqrt((d * (d / (h * l))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if ((1.0 + ((h / l) * (math.pow(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0)))) * (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0)))) <= 0.0:
		tmp = d / math.sqrt((h * l))
	else:
		tmp = math.sqrt((d * (d / (h * l))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (Float64(Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))) * Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0)))) <= 0.0)
		tmp = Float64(d / sqrt(Float64(h * l)));
	else
		tmp = sqrt(Float64(d * Float64(d / Float64(h * l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (((1.0 + ((h / l) * ((((M * D) / (d * 2.0)) ^ 2.0) * (-1.0 / 2.0)))) * (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0)))) <= 0.0)
		tmp = d / sqrt((h * l));
	else
		tmp = sqrt((d * (d / (h * l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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]), $MachinePrecision], 0.0], N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(d * N[(d / N[(h * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   1. Initial program 84.1%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 3.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. pow-to-exp N/A

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

      7. pow-to-exp N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-to-exp N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-lft-identity N/A

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

      13. *-lft-identity N/A

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

      14. *-rgt-identity 3.8

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

      15. lift-*.f64 N/A

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

      16. lift-pow.f64 N/A

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

      17. lift-pow.f64 N/A

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

   7. Applied egg-rr 4.4%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. sqrt-undiv N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lift-/.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. times-frac N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. sqrt-div N/A

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

      16. lift-*.f64 N/A

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

      17. sqrt-prod N/A

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

      18. rem-square-sqrt N/A

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

      19. lower-/.f64 N/A

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

      20. lower-sqrt.f64 15.1

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

   9. Applied egg-rr 15.1%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 61.6%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 66.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. pow-to-exp N/A

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

      7. pow-to-exp N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-to-exp N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-lft-identity N/A

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

      13. *-lft-identity N/A

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

      14. *-rgt-identity 66.0

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

      15. lift-*.f64 N/A

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

      16. lift-pow.f64 N/A

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

      17. lift-pow.f64 N/A

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

   7. Applied egg-rr 25.9%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. sqrt-undiv N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lift-/.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. times-frac N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-sqrt.f64 43.8

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

      17. lift-/.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-/l* N/A

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

      20. lower-*.f64 N/A

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

      21. lower-/.f64 52.3

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

   9. Applied egg-rr 52.3%

      \[\leadsto \color{blue}{\sqrt{d \cdot \frac{d}{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{M \cdot D}{d \cdot 2}\\ \mathbf{if}\;d \leq -1.85 \cdot 10^{-305}:\\ \;\;\;\;\left(1 + \frac{h}{\ell} \cdot \left({t\_0}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right)\\ \mathbf{elif}\;d \leq 3.05 \cdot 10^{-214}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\left(M \cdot D\right) \cdot -0.25\right), \left(M \cdot D\right) \cdot \frac{h}{\ell \cdot \left(d \cdot 2\right)}, \frac{d}{\sqrt{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(t\_0, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ (* M D) (* d 2.0))))
   (if (<= d -1.85e-305)
     (*
      (+ 1.0 (* (/ h l) (* (pow t_0 2.0) (/ -1.0 2.0))))
      (* (pow (/ d h) (/ 1.0 2.0)) (/ (sqrt (- d)) (sqrt (- l)))))
     (if (<= d 3.05e-214)
       (fma
        (* (sqrt (/ 1.0 (* h l))) (* (* M D) -0.25))
        (* (* M D) (/ h (* l (* d 2.0))))
        (/ d (sqrt (* h l))))
       (*
        (/ (/ d (sqrt h)) (sqrt l))
        (fma t_0 (* (/ h l) (* -0.25 (/ (* M D) d))) 1.0))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (M * D) / (d * 2.0);
	double tmp;
	if (d <= -1.85e-305) {
		tmp = (1.0 + ((h / l) * (pow(t_0, 2.0) * (-1.0 / 2.0)))) * (pow((d / h), (1.0 / 2.0)) * (sqrt(-d) / sqrt(-l)));
	} else if (d <= 3.05e-214) {
		tmp = fma((sqrt((1.0 / (h * l))) * ((M * D) * -0.25)), ((M * D) * (h / (l * (d * 2.0)))), (d / sqrt((h * l))));
	} else {
		tmp = ((d / sqrt(h)) / sqrt(l)) * fma(t_0, ((h / l) * (-0.25 * ((M * D) / d))), 1.0);
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(M * D) / Float64(d * 2.0))
	tmp = 0.0
	if (d <= -1.85e-305)
		tmp = Float64(Float64(1.0 + Float64(Float64(h / l) * Float64((t_0 ^ 2.0) * Float64(-1.0 / 2.0)))) * Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-l)))));
	elseif (d <= 3.05e-214)
		tmp = fma(Float64(sqrt(Float64(1.0 / Float64(h * l))) * Float64(Float64(M * D) * -0.25)), Float64(Float64(M * D) * Float64(h / Float64(l * Float64(d * 2.0)))), Float64(d / sqrt(Float64(h * l))));
	else
		tmp = Float64(Float64(Float64(d / sqrt(h)) / sqrt(l)) * fma(t_0, Float64(Float64(h / l) * Float64(-0.25 * Float64(Float64(M * D) / d))), 1.0));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.85e-305], N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.05e-214], N[(N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * N[(h / N[(l * N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(N[(h / l), $MachinePrecision] * N[(-0.25 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.84999999999999989e-305

   1. Initial program 70.7%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-neg.f64 79.0

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

   4. Applied egg-rr 79.0%

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

   if -1.84999999999999989e-305 < d < 3.05e-214

   1. Initial program 33.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

   4. Applied egg-rr 33.4%

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

      1. lift-/.f64 N/A

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

      2. rem-exp-log N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. rem-exp-log N/A

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

      6. lift-/.f64 N/A

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

      7. pow-prod-down N/A

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

      8. rem-exp-log N/A

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

      9. rem-exp-log N/A

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

      10. *-commutative N/A

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

      11. pow-prod-down N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. unpow1/2 N/A

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

      16. lift-/.f64 N/A

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

      17. clear-num N/A

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

      18. lift-/.f64 N/A

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

      19. sqrt-div N/A

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

      20. metadata-eval N/A

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

      21. lift-sqrt.f64 N/A

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

   6. Applied egg-rr 29.3%

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

   8. Applied egg-rr 51.9%

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

   9. Taylor expanded in d around 0

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 70.8

          \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-0.25 \cdot \color{blue}{\left(D \cdot M\right)}\right), \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

   11. Simplified 70.8%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-0.25 \cdot \left(D \cdot M\right)\right)}, \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

   if 3.05e-214 < d

   1. Initial program 84.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

   4. Applied egg-rr 85.3%

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

      1. lift-/.f64 N/A

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

      2. rem-exp-log N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. rem-exp-log N/A

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

      6. lift-/.f64 N/A

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

      7. pow-prod-down N/A

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

      8. rem-exp-log N/A

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

      9. rem-exp-log N/A

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

      10. *-commutative N/A

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

      11. pow-prod-down N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. unpow1/2 N/A

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

      16. lift-/.f64 N/A

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

      17. clear-num N/A

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

      18. lift-/.f64 N/A

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

      19. sqrt-div N/A

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

      20. metadata-eval N/A

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

      21. lift-sqrt.f64 N/A

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

   6. Applied egg-rr 87.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

   8. Applied egg-rr 87.6%

      \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right)} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-undiv N/A

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

      4. div-inv N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. times-frac N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/r* N/A

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

      14. sqrt-div N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. sqrt-div N/A

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

      18. lift-*.f64 N/A

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

      19. sqrt-prod N/A

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

      20. rem-square-sqrt N/A

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

      21. pow1/2 N/A

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

      22. lower-/.f64 N/A

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

      23. pow1/2 N/A

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

      24. lower-sqrt.f64 92.1

          \[\leadsto \frac{\frac{d}{\color{blue}{\sqrt{h}}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right) \]

   10. Applied egg-rr 92.1%

       \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.85 \cdot 10^{-305}:\\ \;\;\;\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right)\\ \mathbf{elif}\;d \leq 3.05 \cdot 10^{-214}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\left(M \cdot D\right) \cdot -0.25\right), \left(M \cdot D\right) \cdot \frac{h}{\ell \cdot \left(d \cdot 2\right)}, \frac{d}{\sqrt{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right)\\ \mathbf{if}\;d \leq -1.85 \cdot 10^{-305}:\\ \;\;\;\;t\_0 \cdot \frac{\frac{\sqrt{-d}}{\sqrt{-\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;d \leq 3.05 \cdot 10^{-214}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\left(M \cdot D\right) \cdot -0.25\right), \left(M \cdot D\right) \cdot \frac{h}{\ell \cdot \left(d \cdot 2\right)}, \frac{d}{\sqrt{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (fma (/ (* M D) (* d 2.0)) (* (/ h l) (* -0.25 (/ (* M D) d))) 1.0)))
   (if (<= d -1.85e-305)
     (* t_0 (/ (/ (sqrt (- d)) (sqrt (- l))) (sqrt (/ h d))))
     (if (<= d 3.05e-214)
       (fma
        (* (sqrt (/ 1.0 (* h l))) (* (* M D) -0.25))
        (* (* M D) (/ h (* l (* d 2.0))))
        (/ d (sqrt (* h l))))
       (* (/ (/ d (sqrt h)) (sqrt l)) t_0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fma(((M * D) / (d * 2.0)), ((h / l) * (-0.25 * ((M * D) / d))), 1.0);
	double tmp;
	if (d <= -1.85e-305) {
		tmp = t_0 * ((sqrt(-d) / sqrt(-l)) / sqrt((h / d)));
	} else if (d <= 3.05e-214) {
		tmp = fma((sqrt((1.0 / (h * l))) * ((M * D) * -0.25)), ((M * D) * (h / (l * (d * 2.0)))), (d / sqrt((h * l))));
	} else {
		tmp = ((d / sqrt(h)) / sqrt(l)) * t_0;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = fma(Float64(Float64(M * D) / Float64(d * 2.0)), Float64(Float64(h / l) * Float64(-0.25 * Float64(Float64(M * D) / d))), 1.0)
	tmp = 0.0
	if (d <= -1.85e-305)
		tmp = Float64(t_0 * Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))) / sqrt(Float64(h / d))));
	elseif (d <= 3.05e-214)
		tmp = fma(Float64(sqrt(Float64(1.0 / Float64(h * l))) * Float64(Float64(M * D) * -0.25)), Float64(Float64(M * D) * Float64(h / Float64(l * Float64(d * 2.0)))), Float64(d / sqrt(Float64(h * l))));
	else
		tmp = Float64(Float64(Float64(d / sqrt(h)) / sqrt(l)) * t_0);
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(-0.25 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[d, -1.85e-305], N[(t$95$0 * N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.05e-214], N[(N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * N[(h / N[(l * N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.84999999999999989e-305

   1. Initial program 70.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

   4. Applied egg-rr 70.8%

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

      1. lift-/.f64 N/A

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

      2. rem-exp-log N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. rem-exp-log N/A

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

      6. lift-/.f64 N/A

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

      7. pow-prod-down N/A

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

      8. rem-exp-log N/A

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

      9. rem-exp-log N/A

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

      10. *-commutative N/A

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

      11. pow-prod-down N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. unpow1/2 N/A

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

      16. lift-/.f64 N/A

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

      17. clear-num N/A

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

      18. lift-/.f64 N/A

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

      19. sqrt-div N/A

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

      20. metadata-eval N/A

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

      21. lift-sqrt.f64 N/A

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

   6. Applied egg-rr 70.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

   8. Applied egg-rr 70.7%

      \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right)} \]
   9. Step-by-step derivation

      1. frac-2neg N/A

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

      2. sqrt-div N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 79.0

         \[\leadsto \frac{\frac{\sqrt{-d}}{\sqrt{\color{blue}{-\ell}}}}{\sqrt{\frac{h}{d}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right) \]

   10. Applied egg-rr 79.0%

       \[\leadsto \frac{\color{blue}{\frac{\sqrt{-d}}{\sqrt{-\ell}}}}{\sqrt{\frac{h}{d}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right) \]

   if -1.84999999999999989e-305 < d < 3.05e-214

   1. Initial program 33.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

   4. Applied egg-rr 33.4%

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

      1. lift-/.f64 N/A

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

      2. rem-exp-log N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. rem-exp-log N/A

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

      6. lift-/.f64 N/A

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

      7. pow-prod-down N/A

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

      8. rem-exp-log N/A

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

      9. rem-exp-log N/A

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

      10. *-commutative N/A

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

      11. pow-prod-down N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. unpow1/2 N/A

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

      16. lift-/.f64 N/A

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

      17. clear-num N/A

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

      18. lift-/.f64 N/A

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

      19. sqrt-div N/A

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

      20. metadata-eval N/A

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

      21. lift-sqrt.f64 N/A

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

   6. Applied egg-rr 29.3%

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

   8. Applied egg-rr 51.9%

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

   9. Taylor expanded in d around 0

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 70.8

          \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-0.25 \cdot \color{blue}{\left(D \cdot M\right)}\right), \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

   11. Simplified 70.8%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-0.25 \cdot \left(D \cdot M\right)\right)}, \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

   if 3.05e-214 < d

   1. Initial program 84.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

   4. Applied egg-rr 85.3%

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

      1. lift-/.f64 N/A

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

      2. rem-exp-log N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. rem-exp-log N/A

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

      6. lift-/.f64 N/A

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

      7. pow-prod-down N/A

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

      8. rem-exp-log N/A

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

      9. rem-exp-log N/A

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

      10. *-commutative N/A

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

      11. pow-prod-down N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. unpow1/2 N/A

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

      16. lift-/.f64 N/A

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

      17. clear-num N/A

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

      18. lift-/.f64 N/A

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

      19. sqrt-div N/A

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

      20. metadata-eval N/A

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

      21. lift-sqrt.f64 N/A

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

   6. Applied egg-rr 87.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right)} + 1\right) \]

      10. *-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)} + 1\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right)} \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) + 1\right) \]

      12. associate-*l* N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\frac{M \cdot D}{d \cdot 2} \cdot \left(\frac{h}{\ell} \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)\right)} + 1\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right), 1\right)} \]

   8. Applied egg-rr 87.6%

      \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right)} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{\ell}}}}{\sqrt{\frac{h}{d}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\color{blue}{\frac{h}{d}}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      3. sqrt-undiv N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{d}{\ell}}{\frac{h}{d}}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      4. div-inv N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{1}{\frac{h}{d}}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell}} \cdot \frac{1}{\frac{h}{d}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{1}{\color{blue}{\frac{h}{d}}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      7. clear-num N/A

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \color{blue}{\frac{d}{h}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      8. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      13. associate-/r* N/A

          \[\leadsto \sqrt{\color{blue}{\frac{\frac{d \cdot d}{h}}{\ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      14. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{\frac{d \cdot d}{h}}}{\sqrt{\ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      15. lift-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{d \cdot d}{h}}}{\color{blue}{\sqrt{\ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{\frac{d \cdot d}{h}}}{\sqrt{\ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      17. sqrt-div N/A

          \[\leadsto \frac{\color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h}}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      19. sqrt-prod N/A

          \[\leadsto \frac{\frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      20. rem-square-sqrt N/A

          \[\leadsto \frac{\frac{\color{blue}{d}}{\sqrt{h}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      21. pow1/2 N/A

          \[\leadsto \frac{\frac{d}{\color{blue}{{h}^{\frac{1}{2}}}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      22. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{d}{{h}^{\frac{1}{2}}}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      23. pow1/2 N/A

          \[\leadsto \frac{\frac{d}{\color{blue}{\sqrt{h}}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      24. lower-sqrt.f64 92.1

          \[\leadsto \frac{\frac{d}{\color{blue}{\sqrt{h}}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right) \]

   10. Applied egg-rr 92.1%

       \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.85 \cdot 10^{-305}:\\ \;\;\;\;\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right) \cdot \frac{\frac{\sqrt{-d}}{\sqrt{-\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;d \leq 3.05 \cdot 10^{-214}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\left(M \cdot D\right) \cdot -0.25\right), \left(M \cdot D\right) \cdot \frac{h}{\ell \cdot \left(d \cdot 2\right)}, \frac{d}{\sqrt{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{h \cdot \ell}\\ t_1 := \frac{M \cdot D}{d \cdot 2}\\ t_2 := \mathsf{fma}\left(t\_1, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right)\\ \mathbf{if}\;d \leq -4.2 \cdot 10^{-80}:\\ \;\;\;\;t\_2 \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;d \leq -1.85 \cdot 10^{-305}:\\ \;\;\;\;\frac{-1}{\frac{t\_0}{d}} \cdot \mathsf{fma}\left(t\_1 \cdot -0.5, \frac{h}{\ell} \cdot t\_1, 1\right)\\ \mathbf{elif}\;d \leq 3.05 \cdot 10^{-214}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\left(M \cdot D\right) \cdot -0.25\right), \left(M \cdot D\right) \cdot \frac{h}{\ell \cdot \left(d \cdot 2\right)}, \frac{d}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (* h l)))
        (t_1 (/ (* M D) (* d 2.0)))
        (t_2 (fma t_1 (* (/ h l) (* -0.25 (/ (* M D) d))) 1.0)))
   (if (<= d -4.2e-80)
     (* t_2 (* (sqrt (/ d l)) (sqrt (/ d h))))
     (if (<= d -1.85e-305)
       (* (/ -1.0 (/ t_0 d)) (fma (* t_1 -0.5) (* (/ h l) t_1) 1.0))
       (if (<= d 3.05e-214)
         (fma
          (* (sqrt (/ 1.0 (* h l))) (* (* M D) -0.25))
          (* (* M D) (/ h (* l (* d 2.0))))
          (/ d t_0))
         (* (/ (/ d (sqrt h)) (sqrt l)) t_2))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h * l));
	double t_1 = (M * D) / (d * 2.0);
	double t_2 = fma(t_1, ((h / l) * (-0.25 * ((M * D) / d))), 1.0);
	double tmp;
	if (d <= -4.2e-80) {
		tmp = t_2 * (sqrt((d / l)) * sqrt((d / h)));
	} else if (d <= -1.85e-305) {
		tmp = (-1.0 / (t_0 / d)) * fma((t_1 * -0.5), ((h / l) * t_1), 1.0);
	} else if (d <= 3.05e-214) {
		tmp = fma((sqrt((1.0 / (h * l))) * ((M * D) * -0.25)), ((M * D) * (h / (l * (d * 2.0)))), (d / t_0));
	} else {
		tmp = ((d / sqrt(h)) / sqrt(l)) * t_2;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h * l))
	t_1 = Float64(Float64(M * D) / Float64(d * 2.0))
	t_2 = fma(t_1, Float64(Float64(h / l) * Float64(-0.25 * Float64(Float64(M * D) / d))), 1.0)
	tmp = 0.0
	if (d <= -4.2e-80)
		tmp = Float64(t_2 * Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))));
	elseif (d <= -1.85e-305)
		tmp = Float64(Float64(-1.0 / Float64(t_0 / d)) * fma(Float64(t_1 * -0.5), Float64(Float64(h / l) * t_1), 1.0));
	elseif (d <= 3.05e-214)
		tmp = fma(Float64(sqrt(Float64(1.0 / Float64(h * l))) * Float64(Float64(M * D) * -0.25)), Float64(Float64(M * D) * Float64(h / Float64(l * Float64(d * 2.0)))), Float64(d / t_0));
	else
		tmp = Float64(Float64(Float64(d / sqrt(h)) / sqrt(l)) * t_2);
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(h / l), $MachinePrecision] * N[(-0.25 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[d, -4.2e-80], N[(t$95$2 * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.85e-305], N[(N[(-1.0 / N[(t$95$0 / d), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 * -0.5), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.05e-214], N[(N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * N[(h / N[(l * N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(d / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -4.20000000000000003e-80

   1. Initial program 82.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      9. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      10. +-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

   4. Applied egg-rr 82.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      2. rem-exp-log N/A

         \[\leadsto \left({\color{blue}{\left(e^{\log \left(\frac{d}{h}\right)}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      5. rem-exp-log N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      7. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(e^{\log \left(\frac{d}{h}\right)} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      8. rem-exp-log N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      9. rem-exp-log N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      10. *-commutative N/A

          \[\leadsto {\color{blue}{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      11. pow-prod-down N/A

          \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      14. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      15. unpow1/2 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      17. clear-num N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{1}{\color{blue}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      19. sqrt-div N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      20. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      21. lift-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

   6. Applied egg-rr 82.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}} \cdot \mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{\color{blue}{M \cdot D}}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{\color{blue}{d \cdot 2}}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \color{blue}{\frac{M \cdot D}{d \cdot 2}}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)} \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{\color{blue}{M \cdot D}}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{\color{blue}{d \cdot 2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\color{blue}{\frac{M \cdot D}{d \cdot 2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \color{blue}{\frac{h}{\ell}}\right) + 1\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right)} + 1\right) \]

      10. *-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)} + 1\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right)} \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) + 1\right) \]

      12. associate-*l* N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\frac{M \cdot D}{d \cdot 2} \cdot \left(\frac{h}{\ell} \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)\right)} + 1\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right), 1\right)} \]

   8. Applied egg-rr 82.2%

      \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right)} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{\ell}}}}{\sqrt{\frac{h}{d}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell}}}}{\sqrt{\frac{h}{d}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\color{blue}{\frac{h}{d}}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\color{blue}{\sqrt{\frac{h}{d}}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      5. div-inv N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \frac{1}{\sqrt{\frac{h}{d}}}\right)} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \frac{1}{\sqrt{\frac{h}{d}}}\right)} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\color{blue}{\sqrt{1}}}{\sqrt{\frac{h}{d}}}\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      9. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{1}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{1}{\color{blue}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      11. clear-num N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      13. lower-/.f64 82.3

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right) \]

   10. Applied egg-rr 82.3%

       \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right) \]

   if -4.20000000000000003e-80 < d < -1.84999999999999989e-305

   1. Initial program 48.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      9. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      10. +-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

   4. Applied egg-rr 48.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      2. rem-exp-log N/A

         \[\leadsto \left({\color{blue}{\left(e^{\log \left(\frac{d}{h}\right)}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      5. rem-exp-log N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      7. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(e^{\log \left(\frac{d}{h}\right)} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      8. rem-exp-log N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      9. rem-exp-log N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      10. lift-/.f64 N/A

          \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      11. metadata-eval N/A

          \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      12. unpow1/2 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      14. clear-num N/A

          \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{h}{d}}} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\frac{h}{d}}} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{1}{\frac{h}{d}} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      17. clear-num N/A

          \[\leadsto \sqrt{\frac{1}{\frac{h}{d}} \cdot \color{blue}{\frac{1}{\frac{\ell}{d}}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      18. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{1 \cdot 1}{\frac{h}{d} \cdot \frac{\ell}{d}}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      19. metadata-eval N/A

          \[\leadsto \sqrt{\frac{\color{blue}{1}}{\frac{h}{d} \cdot \frac{\ell}{d}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      20. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d} \cdot \frac{\ell}{d}}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      21. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{\frac{h}{d} \cdot \frac{\ell}{d}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      22. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\sqrt{\frac{h}{d} \cdot \frac{\ell}{d}}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

   6. Applied egg-rr 33.0%

      \[\leadsto \color{blue}{\frac{1}{\sqrt{\frac{h}{d} \cdot \frac{\ell}{d}}}} \cdot \mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

   7. Taylor expanded in d around -inf

      \[\leadsto \frac{1}{\color{blue}{-1 \cdot \left(\frac{1}{d} \cdot \sqrt{h \cdot \ell}\right)}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{neg}\left(\frac{1}{d} \cdot \sqrt{h \cdot \ell}\right)}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      2. associate-*l/ N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(\color{blue}{\frac{1 \cdot \sqrt{h \cdot \ell}}{d}}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      3. *-lft-identity N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(\frac{\color{blue}{\sqrt{h \cdot \ell}}}{d}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{h \cdot \ell}}{\mathsf{neg}\left(d\right)}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      5. mul-1-neg N/A

         \[\leadsto \frac{1}{\frac{\sqrt{h \cdot \ell}}{\color{blue}{-1 \cdot d}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{h \cdot \ell}}{-1 \cdot d}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{h \cdot \ell}}}{-1 \cdot d}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{h \cdot \ell}}}{-1 \cdot d}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      9. mul-1-neg N/A

         \[\leadsto \frac{1}{\frac{\sqrt{h \cdot \ell}}{\color{blue}{\mathsf{neg}\left(d\right)}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      10. lower-neg.f64 68.9

          \[\leadsto \frac{1}{\frac{\sqrt{h \cdot \ell}}{\color{blue}{-d}}} \cdot \mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

   9. Simplified 68.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{h \cdot \ell}}{-d}}} \cdot \mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

   if -1.84999999999999989e-305 < d < 3.05e-214

   1. Initial program 33.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      9. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      10. +-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

   4. Applied egg-rr 33.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      2. rem-exp-log N/A

         \[\leadsto \left({\color{blue}{\left(e^{\log \left(\frac{d}{h}\right)}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      5. rem-exp-log N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      7. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(e^{\log \left(\frac{d}{h}\right)} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      8. rem-exp-log N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      9. rem-exp-log N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      10. *-commutative N/A

          \[\leadsto {\color{blue}{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      11. pow-prod-down N/A

          \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      14. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      15. unpow1/2 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      17. clear-num N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{1}{\color{blue}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      19. sqrt-div N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      20. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      21. lift-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

   6. Applied egg-rr 29.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}} \cdot \mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

   8. Applied egg-rr 51.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right)} \]

   9. Taylor expanded in d around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{4} \cdot \left(\left(D \cdot M\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}, \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}}, \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right)}, \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right)}, \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right), \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

       5. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right), \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right), \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right)}, \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

       8. lower-*.f64 70.8

          \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-0.25 \cdot \color{blue}{\left(D \cdot M\right)}\right), \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

   11. Simplified 70.8%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-0.25 \cdot \left(D \cdot M\right)\right)}, \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

   if 3.05e-214 < d

   1. Initial program 84.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      9. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      10. +-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

   4. Applied egg-rr 85.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      2. rem-exp-log N/A

         \[\leadsto \left({\color{blue}{\left(e^{\log \left(\frac{d}{h}\right)}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      5. rem-exp-log N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      7. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(e^{\log \left(\frac{d}{h}\right)} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      8. rem-exp-log N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      9. rem-exp-log N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      10. *-commutative N/A

          \[\leadsto {\color{blue}{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      11. pow-prod-down N/A

          \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      14. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      15. unpow1/2 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      17. clear-num N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{1}{\color{blue}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      19. sqrt-div N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      20. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      21. lift-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

   6. Applied egg-rr 87.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}} \cdot \mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{\color{blue}{M \cdot D}}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{\color{blue}{d \cdot 2}}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \color{blue}{\frac{M \cdot D}{d \cdot 2}}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)} \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{\color{blue}{M \cdot D}}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{\color{blue}{d \cdot 2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\color{blue}{\frac{M \cdot D}{d \cdot 2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \color{blue}{\frac{h}{\ell}}\right) + 1\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right)} + 1\right) \]

      10. *-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)} + 1\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right)} \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) + 1\right) \]

      12. associate-*l* N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\frac{M \cdot D}{d \cdot 2} \cdot \left(\frac{h}{\ell} \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)\right)} + 1\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right), 1\right)} \]

   8. Applied egg-rr 87.6%

      \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right)} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{\ell}}}}{\sqrt{\frac{h}{d}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\color{blue}{\frac{h}{d}}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      3. sqrt-undiv N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{d}{\ell}}{\frac{h}{d}}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      4. div-inv N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{1}{\frac{h}{d}}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell}} \cdot \frac{1}{\frac{h}{d}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{1}{\color{blue}{\frac{h}{d}}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      7. clear-num N/A

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \color{blue}{\frac{d}{h}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      8. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      13. associate-/r* N/A

          \[\leadsto \sqrt{\color{blue}{\frac{\frac{d \cdot d}{h}}{\ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      14. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{\frac{d \cdot d}{h}}}{\sqrt{\ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      15. lift-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{d \cdot d}{h}}}{\color{blue}{\sqrt{\ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{\frac{d \cdot d}{h}}}{\sqrt{\ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      17. sqrt-div N/A

          \[\leadsto \frac{\color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h}}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      19. sqrt-prod N/A

          \[\leadsto \frac{\frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      20. rem-square-sqrt N/A

          \[\leadsto \frac{\frac{\color{blue}{d}}{\sqrt{h}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      21. pow1/2 N/A

          \[\leadsto \frac{\frac{d}{\color{blue}{{h}^{\frac{1}{2}}}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      22. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{d}{{h}^{\frac{1}{2}}}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      23. pow1/2 N/A

          \[\leadsto \frac{\frac{d}{\color{blue}{\sqrt{h}}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      24. lower-sqrt.f64 92.1

          \[\leadsto \frac{\frac{d}{\color{blue}{\sqrt{h}}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right) \]

   10. Applied egg-rr 92.1%

       \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.2 \cdot 10^{-80}:\\ \;\;\;\;\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;d \leq -1.85 \cdot 10^{-305}:\\ \;\;\;\;\frac{-1}{\frac{\sqrt{h \cdot \ell}}{d}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2} \cdot -0.5, \frac{h}{\ell} \cdot \frac{M \cdot D}{d \cdot 2}, 1\right)\\ \mathbf{elif}\;d \leq 3.05 \cdot 10^{-214}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\left(M \cdot D\right) \cdot -0.25\right), \left(M \cdot D\right) \cdot \frac{h}{\ell \cdot \left(d \cdot 2\right)}, \frac{d}{\sqrt{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{M \cdot D}{d \cdot 2}\\ t_1 := \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;d \leq -1.85 \cdot 10^{-305}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right) \cdot \left(\left(-d\right) \cdot t\_1\right)\\ \mathbf{elif}\;d \leq 7.8 \cdot 10^{-212}:\\ \;\;\;\;\mathsf{fma}\left(t\_1 \cdot \left(\left(M \cdot D\right) \cdot -0.25\right), \left(M \cdot D\right) \cdot \frac{h}{\ell \cdot \left(d \cdot 2\right)}, \frac{d}{\sqrt{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq 10^{+152}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{\left(M \cdot D\right) \cdot \left(\left(M \cdot D\right) \cdot -0.5\right)}{\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot -0.5, \frac{h}{\ell} \cdot t\_0, 1\right) \cdot \left(d \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ (* M D) (* d 2.0))) (t_1 (sqrt (/ 1.0 (* h l)))))
   (if (<= d -1.85e-305)
     (* (fma t_0 (* (/ h l) (* -0.25 (/ (* M D) d))) 1.0) (* (- d) t_1))
     (if (<= d 7.8e-212)
       (fma
        (* t_1 (* (* M D) -0.25))
        (* (* M D) (/ h (* l (* d 2.0))))
        (/ d (sqrt (* h l))))
       (if (<= d 1e+152)
         (*
          (/ (/ d (sqrt h)) (sqrt l))
          (fma
           (/ (* (* M D) (* (* M D) -0.5)) (* (* d 2.0) (* d 2.0)))
           (/ h l)
           1.0))
         (* (fma (* t_0 -0.5) (* (/ h l) t_0) 1.0) (* d t_1)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (M * D) / (d * 2.0);
	double t_1 = sqrt((1.0 / (h * l)));
	double tmp;
	if (d <= -1.85e-305) {
		tmp = fma(t_0, ((h / l) * (-0.25 * ((M * D) / d))), 1.0) * (-d * t_1);
	} else if (d <= 7.8e-212) {
		tmp = fma((t_1 * ((M * D) * -0.25)), ((M * D) * (h / (l * (d * 2.0)))), (d / sqrt((h * l))));
	} else if (d <= 1e+152) {
		tmp = ((d / sqrt(h)) / sqrt(l)) * fma((((M * D) * ((M * D) * -0.5)) / ((d * 2.0) * (d * 2.0))), (h / l), 1.0);
	} else {
		tmp = fma((t_0 * -0.5), ((h / l) * t_0), 1.0) * (d * t_1);
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(M * D) / Float64(d * 2.0))
	t_1 = sqrt(Float64(1.0 / Float64(h * l)))
	tmp = 0.0
	if (d <= -1.85e-305)
		tmp = Float64(fma(t_0, Float64(Float64(h / l) * Float64(-0.25 * Float64(Float64(M * D) / d))), 1.0) * Float64(Float64(-d) * t_1));
	elseif (d <= 7.8e-212)
		tmp = fma(Float64(t_1 * Float64(Float64(M * D) * -0.25)), Float64(Float64(M * D) * Float64(h / Float64(l * Float64(d * 2.0)))), Float64(d / sqrt(Float64(h * l))));
	elseif (d <= 1e+152)
		tmp = Float64(Float64(Float64(d / sqrt(h)) / sqrt(l)) * fma(Float64(Float64(Float64(M * D) * Float64(Float64(M * D) * -0.5)) / Float64(Float64(d * 2.0) * Float64(d * 2.0))), Float64(h / l), 1.0));
	else
		tmp = Float64(fma(Float64(t_0 * -0.5), Float64(Float64(h / l) * t_0), 1.0) * Float64(d * t_1));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -1.85e-305], N[(N[(t$95$0 * N[(N[(h / l), $MachinePrecision] * N[(-0.25 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[((-d) * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7.8e-212], N[(N[(t$95$1 * N[(N[(M * D), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * N[(h / N[(l * N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1e+152], N[(N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(M * D), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / N[(N[(d * 2.0), $MachinePrecision] * N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 * -0.5), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.84999999999999989e-305

   1. Initial program 70.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      9. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      10. +-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

   4. Applied egg-rr 70.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      2. rem-exp-log N/A

         \[\leadsto \left({\color{blue}{\left(e^{\log \left(\frac{d}{h}\right)}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      5. rem-exp-log N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      7. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(e^{\log \left(\frac{d}{h}\right)} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      8. rem-exp-log N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      9. rem-exp-log N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      10. *-commutative N/A

          \[\leadsto {\color{blue}{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      11. pow-prod-down N/A

          \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      14. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      15. unpow1/2 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      17. clear-num N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{1}{\color{blue}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      19. sqrt-div N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      20. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      21. lift-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

   6. Applied egg-rr 70.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}} \cdot \mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{\color{blue}{M \cdot D}}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{\color{blue}{d \cdot 2}}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \color{blue}{\frac{M \cdot D}{d \cdot 2}}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)} \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{\color{blue}{M \cdot D}}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{\color{blue}{d \cdot 2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\color{blue}{\frac{M \cdot D}{d \cdot 2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \color{blue}{\frac{h}{\ell}}\right) + 1\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right)} + 1\right) \]

      10. *-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)} + 1\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right)} \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) + 1\right) \]

      12. associate-*l* N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\frac{M \cdot D}{d \cdot 2} \cdot \left(\frac{h}{\ell} \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)\right)} + 1\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right), 1\right)} \]

   8. Applied egg-rr 70.7%

      \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right)} \]

   9. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)\right)} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)\right)} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right)\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

       5. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right)\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right)\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

       7. mul-1-neg N/A

          \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

       8. lower-neg.f64 72.4

          \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)}\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right) \]

   11. Simplified 72.4%

       \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\right)} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right) \]

   if -1.84999999999999989e-305 < d < 7.8e-212

   1. Initial program 33.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      9. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      10. +-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

   4. Applied egg-rr 33.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      2. rem-exp-log N/A

         \[\leadsto \left({\color{blue}{\left(e^{\log \left(\frac{d}{h}\right)}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      5. rem-exp-log N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      7. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(e^{\log \left(\frac{d}{h}\right)} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      8. rem-exp-log N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      9. rem-exp-log N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      10. *-commutative N/A

          \[\leadsto {\color{blue}{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      11. pow-prod-down N/A

          \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      14. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      15. unpow1/2 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      17. clear-num N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{1}{\color{blue}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      19. sqrt-div N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      20. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      21. lift-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

   6. Applied egg-rr 29.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}} \cdot \mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

   8. Applied egg-rr 51.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right)} \]

   9. Taylor expanded in d around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{4} \cdot \left(\left(D \cdot M\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}, \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}}, \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right)}, \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right)}, \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right), \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

       5. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right), \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right), \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right)}, \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

       8. lower-*.f64 70.8

          \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-0.25 \cdot \color{blue}{\left(D \cdot M\right)}\right), \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

   11. Simplified 70.8%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-0.25 \cdot \left(D \cdot M\right)\right)}, \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

   if 7.8e-212 < d < 1e152

   1. Initial program 84.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      9. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      10. +-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

   4. Applied egg-rr 85.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      2. rem-exp-log N/A

         \[\leadsto \left({\color{blue}{\left(e^{\log \left(\frac{d}{h}\right)}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      5. rem-exp-log N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      7. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(e^{\log \left(\frac{d}{h}\right)} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      8. rem-exp-log N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      9. rem-exp-log N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      10. lift-/.f64 N/A

          \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      11. metadata-eval N/A

          \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      12. unpow1/2 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      16. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      18. lower-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      19. lower-*.f64 64.1

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

   6. Applied egg-rr 64.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}}} \cdot \mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{\color{blue}{M \cdot D}}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{\color{blue}{d \cdot 2}}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \color{blue}{\frac{M \cdot D}{d \cdot 2}}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)} \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{\color{blue}{M \cdot D}}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{\color{blue}{d \cdot 2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\color{blue}{\frac{M \cdot D}{d \cdot 2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \color{blue}{\frac{h}{\ell}}\right) + 1\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right)} + 1\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right)} + 1\right) \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \frac{h}{\ell}} + 1\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \color{blue}{\mathsf{fma}\left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell}, 1\right)} \]

   8. Applied egg-rr 62.7%

      \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{\left(-0.5 \cdot \left(M \cdot D\right)\right) \cdot \left(M \cdot D\right)}{\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{\left(\frac{-1}{2} \cdot \left(M \cdot D\right)\right) \cdot \left(M \cdot D\right)}{\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \]

      2. associate-/r* N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{d \cdot d}{h}}{\ell}}} \cdot \mathsf{fma}\left(\frac{\left(\frac{-1}{2} \cdot \left(M \cdot D\right)\right) \cdot \left(M \cdot D\right)}{\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \]

      3. sqrt-div N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{d \cdot d}{h}}}{\sqrt{\ell}}} \cdot \mathsf{fma}\left(\frac{\left(\frac{-1}{2} \cdot \left(M \cdot D\right)\right) \cdot \left(M \cdot D\right)}{\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d \cdot d}{h}}}{\color{blue}{\sqrt{\ell}}} \cdot \mathsf{fma}\left(\frac{\left(\frac{-1}{2} \cdot \left(M \cdot D\right)\right) \cdot \left(M \cdot D\right)}{\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{d \cdot d}{h}}}{\sqrt{\ell}}} \cdot \mathsf{fma}\left(\frac{\left(\frac{-1}{2} \cdot \left(M \cdot D\right)\right) \cdot \left(M \cdot D\right)}{\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \]

      6. sqrt-div N/A

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h}}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{\left(\frac{-1}{2} \cdot \left(M \cdot D\right)\right) \cdot \left(M \cdot D\right)}{\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{\left(\frac{-1}{2} \cdot \left(M \cdot D\right)\right) \cdot \left(M \cdot D\right)}{\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \]

      8. sqrt-prod N/A

         \[\leadsto \frac{\frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{\left(\frac{-1}{2} \cdot \left(M \cdot D\right)\right) \cdot \left(M \cdot D\right)}{\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \frac{\frac{\color{blue}{d}}{\sqrt{h}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{\left(\frac{-1}{2} \cdot \left(M \cdot D\right)\right) \cdot \left(M \cdot D\right)}{\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \]

      10. pow1/2 N/A

          \[\leadsto \frac{\frac{d}{\color{blue}{{h}^{\frac{1}{2}}}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{\left(\frac{-1}{2} \cdot \left(M \cdot D\right)\right) \cdot \left(M \cdot D\right)}{\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{d}{{h}^{\frac{1}{2}}}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{\left(\frac{-1}{2} \cdot \left(M \cdot D\right)\right) \cdot \left(M \cdot D\right)}{\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \]

      12. pow1/2 N/A

          \[\leadsto \frac{\frac{d}{\color{blue}{\sqrt{h}}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{\left(\frac{-1}{2} \cdot \left(M \cdot D\right)\right) \cdot \left(M \cdot D\right)}{\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \]

      13. lower-sqrt.f64 85.8

          \[\leadsto \frac{\frac{d}{\color{blue}{\sqrt{h}}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{\left(-0.5 \cdot \left(M \cdot D\right)\right) \cdot \left(M \cdot D\right)}{\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \]

   10. Applied egg-rr 85.8%

       \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}} \cdot \mathsf{fma}\left(\frac{\left(-0.5 \cdot \left(M \cdot D\right)\right) \cdot \left(M \cdot D\right)}{\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right) \]

   if 1e152 < d

   1. Initial program 85.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      9. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      10. +-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

   4. Applied egg-rr 85.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right)} \]

   5. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      4. lower-*.f64 88.6

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

   7. Simplified 88.6%

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.85 \cdot 10^{-305}:\\ \;\;\;\;\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right) \cdot \left(\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq 7.8 \cdot 10^{-212}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\left(M \cdot D\right) \cdot -0.25\right), \left(M \cdot D\right) \cdot \frac{h}{\ell \cdot \left(d \cdot 2\right)}, \frac{d}{\sqrt{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq 10^{+152}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{\left(M \cdot D\right) \cdot \left(\left(M \cdot D\right) \cdot -0.5\right)}{\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2} \cdot -0.5, \frac{h}{\ell} \cdot \frac{M \cdot D}{d \cdot 2}, 1\right) \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 75.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right)\\ t_1 := \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;d \leq -1.85 \cdot 10^{-305}:\\ \;\;\;\;t\_0 \cdot \left(\left(-d\right) \cdot t\_1\right)\\ \mathbf{elif}\;d \leq 3.05 \cdot 10^{-214}:\\ \;\;\;\;\mathsf{fma}\left(t\_1 \cdot \left(\left(M \cdot D\right) \cdot -0.25\right), \left(M \cdot D\right) \cdot \frac{h}{\ell \cdot \left(d \cdot 2\right)}, \frac{d}{\sqrt{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (fma (/ (* M D) (* d 2.0)) (* (/ h l) (* -0.25 (/ (* M D) d))) 1.0))
        (t_1 (sqrt (/ 1.0 (* h l)))))
   (if (<= d -1.85e-305)
     (* t_0 (* (- d) t_1))
     (if (<= d 3.05e-214)
       (fma
        (* t_1 (* (* M D) -0.25))
        (* (* M D) (/ h (* l (* d 2.0))))
        (/ d (sqrt (* h l))))
       (* (/ (/ d (sqrt h)) (sqrt l)) t_0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fma(((M * D) / (d * 2.0)), ((h / l) * (-0.25 * ((M * D) / d))), 1.0);
	double t_1 = sqrt((1.0 / (h * l)));
	double tmp;
	if (d <= -1.85e-305) {
		tmp = t_0 * (-d * t_1);
	} else if (d <= 3.05e-214) {
		tmp = fma((t_1 * ((M * D) * -0.25)), ((M * D) * (h / (l * (d * 2.0)))), (d / sqrt((h * l))));
	} else {
		tmp = ((d / sqrt(h)) / sqrt(l)) * t_0;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = fma(Float64(Float64(M * D) / Float64(d * 2.0)), Float64(Float64(h / l) * Float64(-0.25 * Float64(Float64(M * D) / d))), 1.0)
	t_1 = sqrt(Float64(1.0 / Float64(h * l)))
	tmp = 0.0
	if (d <= -1.85e-305)
		tmp = Float64(t_0 * Float64(Float64(-d) * t_1));
	elseif (d <= 3.05e-214)
		tmp = fma(Float64(t_1 * Float64(Float64(M * D) * -0.25)), Float64(Float64(M * D) * Float64(h / Float64(l * Float64(d * 2.0)))), Float64(d / sqrt(Float64(h * l))));
	else
		tmp = Float64(Float64(Float64(d / sqrt(h)) / sqrt(l)) * t_0);
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(-0.25 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -1.85e-305], N[(t$95$0 * N[((-d) * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.05e-214], N[(N[(t$95$1 * N[(N[(M * D), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * N[(h / N[(l * N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.84999999999999989e-305

   1. Initial program 70.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      9. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      10. +-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

   4. Applied egg-rr 70.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      2. rem-exp-log N/A

         \[\leadsto \left({\color{blue}{\left(e^{\log \left(\frac{d}{h}\right)}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      5. rem-exp-log N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      7. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(e^{\log \left(\frac{d}{h}\right)} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      8. rem-exp-log N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      9. rem-exp-log N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      10. *-commutative N/A

          \[\leadsto {\color{blue}{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      11. pow-prod-down N/A

          \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      14. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      15. unpow1/2 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      17. clear-num N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{1}{\color{blue}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      19. sqrt-div N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      20. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      21. lift-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

   6. Applied egg-rr 70.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}} \cdot \mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{\color{blue}{M \cdot D}}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{\color{blue}{d \cdot 2}}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \color{blue}{\frac{M \cdot D}{d \cdot 2}}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)} \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{\color{blue}{M \cdot D}}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{\color{blue}{d \cdot 2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\color{blue}{\frac{M \cdot D}{d \cdot 2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \color{blue}{\frac{h}{\ell}}\right) + 1\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right)} + 1\right) \]

      10. *-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)} + 1\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right)} \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) + 1\right) \]

      12. associate-*l* N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\frac{M \cdot D}{d \cdot 2} \cdot \left(\frac{h}{\ell} \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)\right)} + 1\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right), 1\right)} \]

   8. Applied egg-rr 70.7%

      \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right)} \]

   9. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)\right)} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)\right)} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right)\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

       5. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right)\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right)\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

       7. mul-1-neg N/A

          \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

       8. lower-neg.f64 72.4

          \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)}\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right) \]

   11. Simplified 72.4%

       \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\right)} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right) \]

   if -1.84999999999999989e-305 < d < 3.05e-214

   1. Initial program 33.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      9. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      10. +-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

   4. Applied egg-rr 33.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      2. rem-exp-log N/A

         \[\leadsto \left({\color{blue}{\left(e^{\log \left(\frac{d}{h}\right)}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      5. rem-exp-log N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      7. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(e^{\log \left(\frac{d}{h}\right)} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      8. rem-exp-log N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      9. rem-exp-log N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      10. *-commutative N/A

          \[\leadsto {\color{blue}{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      11. pow-prod-down N/A

          \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      14. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      15. unpow1/2 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      17. clear-num N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{1}{\color{blue}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      19. sqrt-div N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      20. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      21. lift-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

   6. Applied egg-rr 29.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}} \cdot \mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

   8. Applied egg-rr 51.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right)} \]

   9. Taylor expanded in d around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{4} \cdot \left(\left(D \cdot M\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}, \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}}, \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right)}, \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right)}, \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right), \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

       5. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right), \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right), \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right)}, \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

       8. lower-*.f64 70.8

          \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-0.25 \cdot \color{blue}{\left(D \cdot M\right)}\right), \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

   11. Simplified 70.8%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-0.25 \cdot \left(D \cdot M\right)\right)}, \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

   if 3.05e-214 < d

   1. Initial program 84.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      9. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      10. +-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

   4. Applied egg-rr 85.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      2. rem-exp-log N/A

         \[\leadsto \left({\color{blue}{\left(e^{\log \left(\frac{d}{h}\right)}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      5. rem-exp-log N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      7. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(e^{\log \left(\frac{d}{h}\right)} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      8. rem-exp-log N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      9. rem-exp-log N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      10. *-commutative N/A

          \[\leadsto {\color{blue}{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      11. pow-prod-down N/A

          \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      14. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      15. unpow1/2 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      17. clear-num N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{1}{\color{blue}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      19. sqrt-div N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      20. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      21. lift-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

   6. Applied egg-rr 87.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}} \cdot \mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{\color{blue}{M \cdot D}}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{\color{blue}{d \cdot 2}}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \color{blue}{\frac{M \cdot D}{d \cdot 2}}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)} \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{\color{blue}{M \cdot D}}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{\color{blue}{d \cdot 2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\color{blue}{\frac{M \cdot D}{d \cdot 2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \color{blue}{\frac{h}{\ell}}\right) + 1\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right)} + 1\right) \]

      10. *-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)} + 1\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right)} \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) + 1\right) \]

      12. associate-*l* N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\frac{M \cdot D}{d \cdot 2} \cdot \left(\frac{h}{\ell} \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)\right)} + 1\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right), 1\right)} \]

   8. Applied egg-rr 87.6%

      \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right)} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{\ell}}}}{\sqrt{\frac{h}{d}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\color{blue}{\frac{h}{d}}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      3. sqrt-undiv N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{d}{\ell}}{\frac{h}{d}}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      4. div-inv N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{1}{\frac{h}{d}}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell}} \cdot \frac{1}{\frac{h}{d}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{1}{\color{blue}{\frac{h}{d}}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      7. clear-num N/A

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \color{blue}{\frac{d}{h}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      8. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      13. associate-/r* N/A

          \[\leadsto \sqrt{\color{blue}{\frac{\frac{d \cdot d}{h}}{\ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      14. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{\frac{d \cdot d}{h}}}{\sqrt{\ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      15. lift-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{d \cdot d}{h}}}{\color{blue}{\sqrt{\ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{\frac{d \cdot d}{h}}}{\sqrt{\ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      17. sqrt-div N/A

          \[\leadsto \frac{\color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h}}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      19. sqrt-prod N/A

          \[\leadsto \frac{\frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      20. rem-square-sqrt N/A

          \[\leadsto \frac{\frac{\color{blue}{d}}{\sqrt{h}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      21. pow1/2 N/A

          \[\leadsto \frac{\frac{d}{\color{blue}{{h}^{\frac{1}{2}}}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      22. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{d}{{h}^{\frac{1}{2}}}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      23. pow1/2 N/A

          \[\leadsto \frac{\frac{d}{\color{blue}{\sqrt{h}}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      24. lower-sqrt.f64 92.1

          \[\leadsto \frac{\frac{d}{\color{blue}{\sqrt{h}}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right) \]

   10. Applied egg-rr 92.1%

       \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.85 \cdot 10^{-305}:\\ \;\;\;\;\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right) \cdot \left(\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq 3.05 \cdot 10^{-214}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\left(M \cdot D\right) \cdot -0.25\right), \left(M \cdot D\right) \cdot \frac{h}{\ell \cdot \left(d \cdot 2\right)}, \frac{d}{\sqrt{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 72.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d}{\sqrt{h \cdot \ell}}\\ t_1 := \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right)\\ t_2 := \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t\_1 \cdot \left(\left(-d\right) \cdot t\_2\right)\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{+35}:\\ \;\;\;\;t\_0 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_2 \cdot \left(\left(M \cdot D\right) \cdot -0.25\right), \left(M \cdot D\right) \cdot \frac{h}{\ell \cdot \left(d \cdot 2\right)}, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ d (sqrt (* h l))))
        (t_1
         (fma (/ (* M D) (* d 2.0)) (* (/ h l) (* -0.25 (/ (* M D) d))) 1.0))
        (t_2 (sqrt (/ 1.0 (* h l)))))
   (if (<= l -2e-310)
     (* t_1 (* (- d) t_2))
     (if (<= l 2.3e+35)
       (* t_0 t_1)
       (fma
        (* t_2 (* (* M D) -0.25))
        (* (* M D) (/ h (* l (* d 2.0))))
        t_0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = d / sqrt((h * l));
	double t_1 = fma(((M * D) / (d * 2.0)), ((h / l) * (-0.25 * ((M * D) / d))), 1.0);
	double t_2 = sqrt((1.0 / (h * l)));
	double tmp;
	if (l <= -2e-310) {
		tmp = t_1 * (-d * t_2);
	} else if (l <= 2.3e+35) {
		tmp = t_0 * t_1;
	} else {
		tmp = fma((t_2 * ((M * D) * -0.25)), ((M * D) * (h / (l * (d * 2.0)))), t_0);
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(d / sqrt(Float64(h * l)))
	t_1 = fma(Float64(Float64(M * D) / Float64(d * 2.0)), Float64(Float64(h / l) * Float64(-0.25 * Float64(Float64(M * D) / d))), 1.0)
	t_2 = sqrt(Float64(1.0 / Float64(h * l)))
	tmp = 0.0
	if (l <= -2e-310)
		tmp = Float64(t_1 * Float64(Float64(-d) * t_2));
	elseif (l <= 2.3e+35)
		tmp = Float64(t_0 * t_1);
	else
		tmp = fma(Float64(t_2 * Float64(Float64(M * D) * -0.25)), Float64(Float64(M * D) * Float64(h / Float64(l * Float64(d * 2.0)))), t_0);
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(-0.25 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -2e-310], N[(t$95$1 * N[((-d) * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.3e+35], N[(t$95$0 * t$95$1), $MachinePrecision], N[(N[(t$95$2 * N[(N[(M * D), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * N[(h / N[(l * N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.999999999999994e-310

   1. Initial program 69.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      9. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      10. +-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

   4. Applied egg-rr 69.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      2. rem-exp-log N/A

         \[\leadsto \left({\color{blue}{\left(e^{\log \left(\frac{d}{h}\right)}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      5. rem-exp-log N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      7. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(e^{\log \left(\frac{d}{h}\right)} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      8. rem-exp-log N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      9. rem-exp-log N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      10. *-commutative N/A

          \[\leadsto {\color{blue}{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      11. pow-prod-down N/A

          \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      14. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      15. unpow1/2 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      17. clear-num N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{1}{\color{blue}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      19. sqrt-div N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      20. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      21. lift-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

   6. Applied egg-rr 69.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}} \cdot \mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{\color{blue}{M \cdot D}}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{\color{blue}{d \cdot 2}}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \color{blue}{\frac{M \cdot D}{d \cdot 2}}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)} \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{\color{blue}{M \cdot D}}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{\color{blue}{d \cdot 2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\color{blue}{\frac{M \cdot D}{d \cdot 2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \color{blue}{\frac{h}{\ell}}\right) + 1\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right)} + 1\right) \]

      10. *-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)} + 1\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right)} \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) + 1\right) \]

      12. associate-*l* N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\frac{M \cdot D}{d \cdot 2} \cdot \left(\frac{h}{\ell} \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)\right)} + 1\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right), 1\right)} \]

   8. Applied egg-rr 69.6%

      \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right)} \]

   9. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)\right)} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)\right)} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right)\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

       5. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right)\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right)\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

       7. mul-1-neg N/A

          \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

       8. lower-neg.f64 71.3

          \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)}\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right) \]

   11. Simplified 71.3%

       \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\right)} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right) \]

   if -1.999999999999994e-310 < l < 2.2999999999999998e35

   1. Initial program 80.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      9. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      10. +-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

   4. Applied egg-rr 81.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      2. rem-exp-log N/A

         \[\leadsto \left({\color{blue}{\left(e^{\log \left(\frac{d}{h}\right)}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      5. rem-exp-log N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      7. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(e^{\log \left(\frac{d}{h}\right)} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      8. rem-exp-log N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      9. rem-exp-log N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      10. *-commutative N/A

          \[\leadsto {\color{blue}{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      11. pow-prod-down N/A

          \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      14. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      15. unpow1/2 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      17. clear-num N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{1}{\color{blue}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      19. sqrt-div N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      20. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      21. lift-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

   6. Applied egg-rr 81.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}} \cdot \mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{\color{blue}{M \cdot D}}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{\color{blue}{d \cdot 2}}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \color{blue}{\frac{M \cdot D}{d \cdot 2}}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)} \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{\color{blue}{M \cdot D}}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{\color{blue}{d \cdot 2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\color{blue}{\frac{M \cdot D}{d \cdot 2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \color{blue}{\frac{h}{\ell}}\right) + 1\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right)} + 1\right) \]

      10. *-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)} + 1\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right)} \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) + 1\right) \]

      12. associate-*l* N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\frac{M \cdot D}{d \cdot 2} \cdot \left(\frac{h}{\ell} \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)\right)} + 1\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right), 1\right)} \]

   8. Applied egg-rr 81.6%

      \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right)} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{\ell}}}}{\sqrt{\frac{h}{d}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\color{blue}{\frac{h}{d}}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      3. sqrt-undiv N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{d}{\ell}}{\frac{h}{d}}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      4. div-inv N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{1}{\frac{h}{d}}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell}} \cdot \frac{1}{\frac{h}{d}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{1}{\color{blue}{\frac{h}{d}}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      7. clear-num N/A

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \color{blue}{\frac{d}{h}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      8. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      12. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      14. sqrt-prod N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      15. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      17. lift-/.f64 85.2

          \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right) \]

   10. Applied egg-rr 85.2%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right) \]

   if 2.2999999999999998e35 < l

   1. Initial program 68.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      9. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      10. +-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

   4. Applied egg-rr 68.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      2. rem-exp-log N/A

         \[\leadsto \left({\color{blue}{\left(e^{\log \left(\frac{d}{h}\right)}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      5. rem-exp-log N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      7. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(e^{\log \left(\frac{d}{h}\right)} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      8. rem-exp-log N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      9. rem-exp-log N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      10. *-commutative N/A

          \[\leadsto {\color{blue}{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      11. pow-prod-down N/A

          \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      14. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      15. unpow1/2 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      17. clear-num N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{1}{\color{blue}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      19. sqrt-div N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      20. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      21. lift-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

   6. Applied egg-rr 72.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}} \cdot \mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

   8. Applied egg-rr 65.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right)} \]

   9. Taylor expanded in d around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{4} \cdot \left(\left(D \cdot M\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}, \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}}, \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right)}, \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right)}, \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right), \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

       5. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right), \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right), \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right)}, \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

       8. lower-*.f64 74.3

          \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-0.25 \cdot \color{blue}{\left(D \cdot M\right)}\right), \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

   11. Simplified 74.3%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-0.25 \cdot \left(D \cdot M\right)\right)}, \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right) \cdot \left(\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{+35}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\left(M \cdot D\right) \cdot -0.25\right), \left(M \cdot D\right) \cdot \frac{h}{\ell \cdot \left(d \cdot 2\right)}, \frac{d}{\sqrt{h \cdot \ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 63.9% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -3.4 \cdot 10^{+146}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq -2.2 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{h \cdot \left(M \cdot D\right)}{2 \cdot \left(d \cdot \ell\right)}, \frac{\left(M \cdot D\right) \cdot -0.25}{d}, 1\right)\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\sqrt{\frac{1}{\ell} \cdot \frac{1}{\ell}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -3.4e+146)
   (* (- d) (sqrt (/ 1.0 (* h l))))
   (if (<= d -2.2e-162)
     (*
      (sqrt (/ (* d d) (* h l)))
      (fma (/ (* h (* M D)) (* 2.0 (* d l))) (/ (* (* M D) -0.25) d) 1.0))
     (if (<= d -2e-310)
       (* d (sqrt (sqrt (* (/ 1.0 l) (/ 1.0 l)))))
       (*
        (/ d (sqrt (* h l)))
        (fma
         (/ (* M D) (* d 2.0))
         (* (/ h l) (* -0.25 (/ (* M D) d)))
         1.0))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -3.4e+146) {
		tmp = -d * sqrt((1.0 / (h * l)));
	} else if (d <= -2.2e-162) {
		tmp = sqrt(((d * d) / (h * l))) * fma(((h * (M * D)) / (2.0 * (d * l))), (((M * D) * -0.25) / d), 1.0);
	} else if (d <= -2e-310) {
		tmp = d * sqrt(sqrt(((1.0 / l) * (1.0 / l))));
	} else {
		tmp = (d / sqrt((h * l))) * fma(((M * D) / (d * 2.0)), ((h / l) * (-0.25 * ((M * D) / d))), 1.0);
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -3.4e+146)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l))));
	elseif (d <= -2.2e-162)
		tmp = Float64(sqrt(Float64(Float64(d * d) / Float64(h * l))) * fma(Float64(Float64(h * Float64(M * D)) / Float64(2.0 * Float64(d * l))), Float64(Float64(Float64(M * D) * -0.25) / d), 1.0));
	elseif (d <= -2e-310)
		tmp = Float64(d * sqrt(sqrt(Float64(Float64(1.0 / l) * Float64(1.0 / l)))));
	else
		tmp = Float64(Float64(d / sqrt(Float64(h * l))) * fma(Float64(Float64(M * D) / Float64(d * 2.0)), Float64(Float64(h / l) * Float64(-0.25 * Float64(Float64(M * D) / d))), 1.0));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -3.4e+146], N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -2.2e-162], N[(N[Sqrt[N[(N[(d * d), $MachinePrecision] / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(h * N[(M * D), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M * D), $MachinePrecision] * -0.25), $MachinePrecision] / d), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -2e-310], N[(d * N[Sqrt[N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(-0.25 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -3.39999999999999991e146

   1. Initial program 79.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 51.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. 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. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      9. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      10. lower-neg.f64 64.8

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   8. Simplified 64.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if -3.39999999999999991e146 < d < -2.1999999999999999e-162

   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-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      9. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      10. +-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

   4. Applied egg-rr 75.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      2. rem-exp-log N/A

         \[\leadsto \left({\color{blue}{\left(e^{\log \left(\frac{d}{h}\right)}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      5. rem-exp-log N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      7. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(e^{\log \left(\frac{d}{h}\right)} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      8. rem-exp-log N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      9. rem-exp-log N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      10. lift-/.f64 N/A

          \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      11. metadata-eval N/A

          \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      12. unpow1/2 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      16. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      18. lower-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      19. lower-*.f64 61.9

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

   6. Applied egg-rr 61.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}}} \cdot \mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{\color{blue}{M \cdot D}}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{\color{blue}{d \cdot 2}}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \color{blue}{\frac{M \cdot D}{d \cdot 2}}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)} \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{\color{blue}{M \cdot D}}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{\color{blue}{d \cdot 2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\color{blue}{\frac{M \cdot D}{d \cdot 2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \color{blue}{\frac{h}{\ell}}\right) + 1\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right)} + 1\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right)} + 1\right) \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \frac{h}{\ell}} + 1\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \color{blue}{\mathsf{fma}\left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell}, 1\right)} \]

   8. Applied egg-rr 58.9%

      \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{\left(-0.5 \cdot \left(M \cdot D\right)\right) \cdot \left(M \cdot D\right)}{\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right)} \]

   10. Applied egg-rr 66.5%

       \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{\left(M \cdot D\right) \cdot h}{2 \cdot \left(d \cdot \ell\right)}, \frac{\left(M \cdot D\right) \cdot -0.25}{d}, 1\right)} \]

   if -2.1999999999999999e-162 < d < -1.999999999999994e-310

   1. Initial program 45.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 11.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in h around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{\ell}}} \]

      3. lower-/.f64 0.0

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{\ell}}} \]

   8. Simplified 0.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell}}} \]
   9. Step-by-step derivation

      1. lift-/.f64 0.0

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{\ell}}} \]

      2. rem-square-sqrt N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\sqrt{\frac{1}{\ell}} \cdot \sqrt{\frac{1}{\ell}}}} \]

      3. sqrt-unprod N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\sqrt{\frac{1}{\ell} \cdot \frac{1}{\ell}}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\sqrt{\frac{1}{\ell} \cdot \frac{1}{\ell}}}} \]

      5. lower-*.f64 25.0

         \[\leadsto d \cdot \sqrt{\sqrt{\color{blue}{\frac{1}{\ell} \cdot \frac{1}{\ell}}}} \]

   10. Applied egg-rr 25.0%

       \[\leadsto d \cdot \sqrt{\color{blue}{\sqrt{\frac{1}{\ell} \cdot \frac{1}{\ell}}}} \]

   if -1.999999999999994e-310 < d

   1. Initial program 76.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      9. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      10. +-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

   4. Applied egg-rr 76.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      2. rem-exp-log N/A

         \[\leadsto \left({\color{blue}{\left(e^{\log \left(\frac{d}{h}\right)}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      5. rem-exp-log N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      7. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(e^{\log \left(\frac{d}{h}\right)} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      8. rem-exp-log N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      9. rem-exp-log N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      10. *-commutative N/A

          \[\leadsto {\color{blue}{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      11. pow-prod-down N/A

          \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      14. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      15. unpow1/2 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      17. clear-num N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{1}{\color{blue}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      19. sqrt-div N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      20. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      21. lift-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

   6. Applied egg-rr 77.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}} \cdot \mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{\color{blue}{M \cdot D}}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{\color{blue}{d \cdot 2}}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \color{blue}{\frac{M \cdot D}{d \cdot 2}}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)} \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{\color{blue}{M \cdot D}}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{\color{blue}{d \cdot 2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\color{blue}{\frac{M \cdot D}{d \cdot 2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \color{blue}{\frac{h}{\ell}}\right) + 1\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right)} + 1\right) \]

      10. *-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)} + 1\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right)} \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) + 1\right) \]

      12. associate-*l* N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\frac{M \cdot D}{d \cdot 2} \cdot \left(\frac{h}{\ell} \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)\right)} + 1\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right), 1\right)} \]

   8. Applied egg-rr 77.9%

      \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right)} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{\ell}}}}{\sqrt{\frac{h}{d}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\color{blue}{\frac{h}{d}}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      3. sqrt-undiv N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{d}{\ell}}{\frac{h}{d}}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      4. div-inv N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{1}{\frac{h}{d}}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell}} \cdot \frac{1}{\frac{h}{d}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{1}{\color{blue}{\frac{h}{d}}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      7. clear-num N/A

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \color{blue}{\frac{d}{h}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      8. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      12. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      14. sqrt-prod N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      15. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      17. lift-/.f64 76.5

          \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right) \]

   10. Applied egg-rr 76.5%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.4 \cdot 10^{+146}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq -2.2 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{h \cdot \left(M \cdot D\right)}{2 \cdot \left(d \cdot \ell\right)}, \frac{\left(M \cdot D\right) \cdot -0.25}{d}, 1\right)\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\sqrt{\frac{1}{\ell} \cdot \frac{1}{\ell}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 71.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d}{\sqrt{h \cdot \ell}}\\ t_1 := \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right)\\ \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t\_1 \cdot \left(\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{+141}:\\ \;\;\;\;t\_0 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(M \cdot D\right) \cdot -0.25\right) \cdot \sqrt{\frac{1}{\ell}}, \left(M \cdot D\right) \cdot \frac{h}{\ell \cdot \left(d \cdot 2\right)}, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ d (sqrt (* h l))))
        (t_1
         (fma (/ (* M D) (* d 2.0)) (* (/ h l) (* -0.25 (/ (* M D) d))) 1.0)))
   (if (<= l -2e-310)
     (* t_1 (* (- d) (sqrt (/ 1.0 (* h l)))))
     (if (<= l 1.75e+141)
       (* t_0 t_1)
       (fma
        (* (* (* M D) -0.25) (sqrt (/ 1.0 l)))
        (* (* M D) (/ h (* l (* d 2.0))))
        t_0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = d / sqrt((h * l));
	double t_1 = fma(((M * D) / (d * 2.0)), ((h / l) * (-0.25 * ((M * D) / d))), 1.0);
	double tmp;
	if (l <= -2e-310) {
		tmp = t_1 * (-d * sqrt((1.0 / (h * l))));
	} else if (l <= 1.75e+141) {
		tmp = t_0 * t_1;
	} else {
		tmp = fma((((M * D) * -0.25) * sqrt((1.0 / l))), ((M * D) * (h / (l * (d * 2.0)))), t_0);
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(d / sqrt(Float64(h * l)))
	t_1 = fma(Float64(Float64(M * D) / Float64(d * 2.0)), Float64(Float64(h / l) * Float64(-0.25 * Float64(Float64(M * D) / d))), 1.0)
	tmp = 0.0
	if (l <= -2e-310)
		tmp = Float64(t_1 * Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l)))));
	elseif (l <= 1.75e+141)
		tmp = Float64(t_0 * t_1);
	else
		tmp = fma(Float64(Float64(Float64(M * D) * -0.25) * sqrt(Float64(1.0 / l))), Float64(Float64(M * D) * Float64(h / Float64(l * Float64(d * 2.0)))), t_0);
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(-0.25 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[l, -2e-310], N[(t$95$1 * N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.75e+141], N[(t$95$0 * t$95$1), $MachinePrecision], N[(N[(N[(N[(M * D), $MachinePrecision] * -0.25), $MachinePrecision] * N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * N[(h / N[(l * N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.999999999999994e-310

   1. Initial program 69.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      9. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      10. +-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

   4. Applied egg-rr 69.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      2. rem-exp-log N/A

         \[\leadsto \left({\color{blue}{\left(e^{\log \left(\frac{d}{h}\right)}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      5. rem-exp-log N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      7. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(e^{\log \left(\frac{d}{h}\right)} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      8. rem-exp-log N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      9. rem-exp-log N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      10. *-commutative N/A

          \[\leadsto {\color{blue}{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      11. pow-prod-down N/A

          \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      14. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      15. unpow1/2 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      17. clear-num N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{1}{\color{blue}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      19. sqrt-div N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      20. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      21. lift-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

   6. Applied egg-rr 69.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}} \cdot \mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{\color{blue}{M \cdot D}}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{\color{blue}{d \cdot 2}}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \color{blue}{\frac{M \cdot D}{d \cdot 2}}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)} \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{\color{blue}{M \cdot D}}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{\color{blue}{d \cdot 2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\color{blue}{\frac{M \cdot D}{d \cdot 2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \color{blue}{\frac{h}{\ell}}\right) + 1\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right)} + 1\right) \]

      10. *-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)} + 1\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right)} \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) + 1\right) \]

      12. associate-*l* N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\frac{M \cdot D}{d \cdot 2} \cdot \left(\frac{h}{\ell} \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)\right)} + 1\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right), 1\right)} \]

   8. Applied egg-rr 69.6%

      \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right)} \]

   9. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)\right)} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)\right)} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right)\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

       5. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right)\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right)\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

       7. mul-1-neg N/A

          \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

       8. lower-neg.f64 71.3

          \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)}\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right) \]

   11. Simplified 71.3%

       \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\right)} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right) \]

   if -1.999999999999994e-310 < l < 1.75e141

   1. Initial program 79.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      9. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      10. +-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

   4. Applied egg-rr 80.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      2. rem-exp-log N/A

         \[\leadsto \left({\color{blue}{\left(e^{\log \left(\frac{d}{h}\right)}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      5. rem-exp-log N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      7. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(e^{\log \left(\frac{d}{h}\right)} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      8. rem-exp-log N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      9. rem-exp-log N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      10. *-commutative N/A

          \[\leadsto {\color{blue}{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      11. pow-prod-down N/A

          \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      14. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      15. unpow1/2 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      17. clear-num N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{1}{\color{blue}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      19. sqrt-div N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      20. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      21. lift-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

   6. Applied egg-rr 80.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}} \cdot \mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{\color{blue}{M \cdot D}}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{\color{blue}{d \cdot 2}}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \color{blue}{\frac{M \cdot D}{d \cdot 2}}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)} \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{\color{blue}{M \cdot D}}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{\color{blue}{d \cdot 2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\color{blue}{\frac{M \cdot D}{d \cdot 2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \color{blue}{\frac{h}{\ell}}\right) + 1\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right)} + 1\right) \]

      10. *-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)} + 1\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right)} \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) + 1\right) \]

      12. associate-*l* N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\frac{M \cdot D}{d \cdot 2} \cdot \left(\frac{h}{\ell} \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)\right)} + 1\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right), 1\right)} \]

   8. Applied egg-rr 80.7%

      \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right)} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{\ell}}}}{\sqrt{\frac{h}{d}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\color{blue}{\frac{h}{d}}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      3. sqrt-undiv N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{d}{\ell}}{\frac{h}{d}}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      4. div-inv N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{1}{\frac{h}{d}}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell}} \cdot \frac{1}{\frac{h}{d}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{1}{\color{blue}{\frac{h}{d}}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      7. clear-num N/A

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \color{blue}{\frac{d}{h}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      8. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      12. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      14. sqrt-prod N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      15. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      17. lift-/.f64 84.4

          \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right) \]

   10. Applied egg-rr 84.4%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right) \]

   if 1.75e141 < l

   1. Initial program 68.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      9. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      10. +-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

   4. Applied egg-rr 68.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      2. rem-exp-log N/A

         \[\leadsto \left({\color{blue}{\left(e^{\log \left(\frac{d}{h}\right)}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      5. rem-exp-log N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      7. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(e^{\log \left(\frac{d}{h}\right)} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      8. rem-exp-log N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      9. rem-exp-log N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      10. *-commutative N/A

          \[\leadsto {\color{blue}{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      11. pow-prod-down N/A

          \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      14. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      15. unpow1/2 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      17. clear-num N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{1}{\color{blue}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      19. sqrt-div N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      20. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      21. lift-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

   6. Applied egg-rr 70.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}} \cdot \mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

   8. Applied egg-rr 57.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right)} \]

   9. Taylor expanded in h around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{4} \cdot \left(\left(D \cdot M\right) \cdot \sqrt{\frac{1}{\ell}}\right)}, \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right) \cdot \sqrt{\frac{1}{\ell}}}, \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{\ell}} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right)}, \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{\ell}} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right)}, \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{\ell}}} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right), \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

       5. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{\ell}}} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right), \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{\ell}} \cdot \color{blue}{\left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right)}, \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

       7. lower-*.f64 68.1

          \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{\ell}} \cdot \left(-0.25 \cdot \color{blue}{\left(D \cdot M\right)}\right), \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

   11. Simplified 68.1%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{\ell}} \cdot \left(-0.25 \cdot \left(D \cdot M\right)\right)}, \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right) \cdot \left(\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{+141}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(M \cdot D\right) \cdot -0.25\right) \cdot \sqrt{\frac{1}{\ell}}, \left(M \cdot D\right) \cdot \frac{h}{\ell \cdot \left(d \cdot 2\right)}, \frac{d}{\sqrt{h \cdot \ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 65.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{if}\;\ell \leq -2.1 \cdot 10^{-301}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + \frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot 0.5\right)}{\left(d \cdot d\right) \cdot 4}}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{+141}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(M \cdot D\right) \cdot -0.25\right) \cdot \sqrt{\frac{1}{\ell}}, \left(M \cdot D\right) \cdot \frac{h}{\ell \cdot \left(d \cdot 2\right)}, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ d (sqrt (* h l)))))
   (if (<= l -2.1e-301)
     (*
      (* d (sqrt (/ 1.0 (* h l))))
      (+ -1.0 (/ (/ (* (* M (* D (* M D))) (* h 0.5)) (* (* d d) 4.0)) l)))
     (if (<= l 1.75e+141)
       (*
        t_0
        (fma (/ (* M D) (* d 2.0)) (* (/ h l) (* -0.25 (/ (* M D) d))) 1.0))
       (fma
        (* (* (* M D) -0.25) (sqrt (/ 1.0 l)))
        (* (* M D) (/ h (* l (* d 2.0))))
        t_0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = d / sqrt((h * l));
	double tmp;
	if (l <= -2.1e-301) {
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + ((((M * (D * (M * D))) * (h * 0.5)) / ((d * d) * 4.0)) / l));
	} else if (l <= 1.75e+141) {
		tmp = t_0 * fma(((M * D) / (d * 2.0)), ((h / l) * (-0.25 * ((M * D) / d))), 1.0);
	} else {
		tmp = fma((((M * D) * -0.25) * sqrt((1.0 / l))), ((M * D) * (h / (l * (d * 2.0)))), t_0);
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(d / sqrt(Float64(h * l)))
	tmp = 0.0
	if (l <= -2.1e-301)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 + Float64(Float64(Float64(Float64(M * Float64(D * Float64(M * D))) * Float64(h * 0.5)) / Float64(Float64(d * d) * 4.0)) / l)));
	elseif (l <= 1.75e+141)
		tmp = Float64(t_0 * fma(Float64(Float64(M * D) / Float64(d * 2.0)), Float64(Float64(h / l) * Float64(-0.25 * Float64(Float64(M * D) / d))), 1.0));
	else
		tmp = fma(Float64(Float64(Float64(M * D) * -0.25) * sqrt(Float64(1.0 / l))), Float64(Float64(M * D) * Float64(h / Float64(l * Float64(d * 2.0)))), t_0);
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.1e-301], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(N[(N[(N[(M * N[(D * N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h * 0.5), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.75e+141], N[(t$95$0 * N[(N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(-0.25 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(M * D), $MachinePrecision] * -0.25), $MachinePrecision] * N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * N[(h / N[(l * N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.0999999999999999e-301

   1. Initial program 69.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      9. lift--.f64 69.1

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      12. associate-*r/ N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

   4. Applied egg-rr 59.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 - \frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot 0.5\right)}{\left(d \cdot d\right) \cdot 4}}{\ell}\right)} \]

   5. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot \frac{1}{2}\right)}{\left(d \cdot d\right) \cdot 4}}{\ell}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot \frac{1}{2}\right)}{\left(d \cdot d\right) \cdot 4}}{\ell}\right) \]

      2. unpow2 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot \frac{1}{2}\right)}{\left(d \cdot d\right) \cdot 4}}{\ell}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot \frac{1}{2}\right)}{\left(d \cdot d\right) \cdot 4}}{\ell}\right) \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)\right)} \cdot \left(1 - \frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot \frac{1}{2}\right)}{\left(d \cdot d\right) \cdot 4}}{\ell}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)\right)} \cdot \left(1 - \frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot \frac{1}{2}\right)}{\left(d \cdot d\right) \cdot 4}}{\ell}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right)\right) \cdot \left(1 - \frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot \frac{1}{2}\right)}{\left(d \cdot d\right) \cdot 4}}{\ell}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right)\right) \cdot \left(1 - \frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot \frac{1}{2}\right)}{\left(d \cdot d\right) \cdot 4}}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right)\right) \cdot \left(1 - \frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot \frac{1}{2}\right)}{\left(d \cdot d\right) \cdot 4}}{\ell}\right) \]

      9. mul-1-neg N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right) \cdot \left(1 - \frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot \frac{1}{2}\right)}{\left(d \cdot d\right) \cdot 4}}{\ell}\right) \]

      10. lower-neg.f64 64.4

          \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)}\right) \cdot \left(1 - \frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot 0.5\right)}{\left(d \cdot d\right) \cdot 4}}{\ell}\right) \]

   7. Simplified 64.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\right)} \cdot \left(1 - \frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot 0.5\right)}{\left(d \cdot d\right) \cdot 4}}{\ell}\right) \]

   if -2.0999999999999999e-301 < l < 1.75e141

   1. Initial program 79.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      9. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      10. +-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

   4. Applied egg-rr 80.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      2. rem-exp-log N/A

         \[\leadsto \left({\color{blue}{\left(e^{\log \left(\frac{d}{h}\right)}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      5. rem-exp-log N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      7. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(e^{\log \left(\frac{d}{h}\right)} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      8. rem-exp-log N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      9. rem-exp-log N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      10. *-commutative N/A

          \[\leadsto {\color{blue}{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      11. pow-prod-down N/A

          \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      14. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      15. unpow1/2 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      17. clear-num N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{1}{\color{blue}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      19. sqrt-div N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      20. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      21. lift-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

   6. Applied egg-rr 81.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}} \cdot \mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{\color{blue}{M \cdot D}}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{\color{blue}{d \cdot 2}}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \color{blue}{\frac{M \cdot D}{d \cdot 2}}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)} \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{\color{blue}{M \cdot D}}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{\color{blue}{d \cdot 2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\color{blue}{\frac{M \cdot D}{d \cdot 2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \color{blue}{\frac{h}{\ell}}\right) + 1\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right)} + 1\right) \]

      10. *-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)} + 1\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right)} \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) + 1\right) \]

      12. associate-*l* N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\frac{M \cdot D}{d \cdot 2} \cdot \left(\frac{h}{\ell} \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)\right)} + 1\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right), 1\right)} \]

   8. Applied egg-rr 81.2%

      \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right)} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{\ell}}}}{\sqrt{\frac{h}{d}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\color{blue}{\frac{h}{d}}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      3. sqrt-undiv N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{d}{\ell}}{\frac{h}{d}}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      4. div-inv N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{1}{\frac{h}{d}}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell}} \cdot \frac{1}{\frac{h}{d}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{1}{\color{blue}{\frac{h}{d}}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      7. clear-num N/A

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \color{blue}{\frac{d}{h}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      8. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      12. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      14. sqrt-prod N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      15. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      17. lift-/.f64 82.6

          \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right) \]

   10. Applied egg-rr 82.6%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right) \]

   if 1.75e141 < l

   1. Initial program 68.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      9. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      10. +-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

   4. Applied egg-rr 68.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      2. rem-exp-log N/A

         \[\leadsto \left({\color{blue}{\left(e^{\log \left(\frac{d}{h}\right)}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      5. rem-exp-log N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      7. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(e^{\log \left(\frac{d}{h}\right)} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      8. rem-exp-log N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      9. rem-exp-log N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      10. *-commutative N/A

          \[\leadsto {\color{blue}{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      11. pow-prod-down N/A

          \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      14. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      15. unpow1/2 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      17. clear-num N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{1}{\color{blue}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      19. sqrt-div N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      20. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      21. lift-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

   6. Applied egg-rr 70.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}} \cdot \mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

   8. Applied egg-rr 57.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right)} \]

   9. Taylor expanded in h around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{4} \cdot \left(\left(D \cdot M\right) \cdot \sqrt{\frac{1}{\ell}}\right)}, \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right) \cdot \sqrt{\frac{1}{\ell}}}, \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{\ell}} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right)}, \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{\ell}} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right)}, \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{\ell}}} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right), \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

       5. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{\ell}}} \cdot \left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right), \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{\ell}} \cdot \color{blue}{\left(\frac{-1}{4} \cdot \left(D \cdot M\right)\right)}, \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

       7. lower-*.f64 68.1

          \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{\ell}} \cdot \left(-0.25 \cdot \color{blue}{\left(D \cdot M\right)}\right), \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]

   11. Simplified 68.1%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{\ell}} \cdot \left(-0.25 \cdot \left(D \cdot M\right)\right)}, \left(M \cdot D\right) \cdot \frac{h}{\left(d \cdot 2\right) \cdot \ell}, \frac{d}{\sqrt{h \cdot \ell}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.1 \cdot 10^{-301}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + \frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot 0.5\right)}{\left(d \cdot d\right) \cdot 4}}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{+141}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(M \cdot D\right) \cdot -0.25\right) \cdot \sqrt{\frac{1}{\ell}}, \left(M \cdot D\right) \cdot \frac{h}{\ell \cdot \left(d \cdot 2\right)}, \frac{d}{\sqrt{h \cdot \ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 58.6% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;d \leq -3.4 \cdot 10^{+146}:\\ \;\;\;\;\left(-d\right) \cdot t\_0\\ \mathbf{elif}\;d \leq -1.6 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \mathsf{fma}\left(-0.5, \frac{\left(h \cdot \left(M \cdot D\right)\right) \cdot \left(\left(M \cdot D\right) \cdot 0.5\right)}{d \cdot \left(d \cdot \left(2 \cdot \ell\right)\right)}, 1\right)\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{-261}:\\ \;\;\;\;d \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \mathsf{fma}\left(-0.5, \frac{\left(M \cdot D\right) \cdot \left(M \cdot \left(h \cdot D\right)\right)}{\left(d \cdot 2\right) \cdot \left(\ell \cdot \left(d \cdot 2\right)\right)}, 1\right)}{\sqrt{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (* h l)))))
   (if (<= d -3.4e+146)
     (* (- d) t_0)
     (if (<= d -1.6e-162)
       (*
        (sqrt (/ (* d d) (* h l)))
        (fma
         -0.5
         (/ (* (* h (* M D)) (* (* M D) 0.5)) (* d (* d (* 2.0 l))))
         1.0))
       (if (<= d 2.4e-261)
         (* d t_0)
         (/
          (*
           d
           (fma
            -0.5
            (/ (* (* M D) (* M (* h D))) (* (* d 2.0) (* l (* d 2.0))))
            1.0))
          (sqrt (* h l))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((1.0 / (h * l)));
	double tmp;
	if (d <= -3.4e+146) {
		tmp = -d * t_0;
	} else if (d <= -1.6e-162) {
		tmp = sqrt(((d * d) / (h * l))) * fma(-0.5, (((h * (M * D)) * ((M * D) * 0.5)) / (d * (d * (2.0 * l)))), 1.0);
	} else if (d <= 2.4e-261) {
		tmp = d * t_0;
	} else {
		tmp = (d * fma(-0.5, (((M * D) * (M * (h * D))) / ((d * 2.0) * (l * (d * 2.0)))), 1.0)) / sqrt((h * l));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(1.0 / Float64(h * l)))
	tmp = 0.0
	if (d <= -3.4e+146)
		tmp = Float64(Float64(-d) * t_0);
	elseif (d <= -1.6e-162)
		tmp = Float64(sqrt(Float64(Float64(d * d) / Float64(h * l))) * fma(-0.5, Float64(Float64(Float64(h * Float64(M * D)) * Float64(Float64(M * D) * 0.5)) / Float64(d * Float64(d * Float64(2.0 * l)))), 1.0));
	elseif (d <= 2.4e-261)
		tmp = Float64(d * t_0);
	else
		tmp = Float64(Float64(d * fma(-0.5, Float64(Float64(Float64(M * D) * Float64(M * Float64(h * D))) / Float64(Float64(d * 2.0) * Float64(l * Float64(d * 2.0)))), 1.0)) / sqrt(Float64(h * l)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -3.4e+146], N[((-d) * t$95$0), $MachinePrecision], If[LessEqual[d, -1.6e-162], N[(N[Sqrt[N[(N[(d * d), $MachinePrecision] / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.5 * N[(N[(N[(h * N[(M * D), $MachinePrecision]), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.4e-261], N[(d * t$95$0), $MachinePrecision], N[(N[(d * N[(-0.5 * N[(N[(N[(M * D), $MachinePrecision] * N[(M * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(d * 2.0), $MachinePrecision] * N[(l * N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -3.39999999999999991e146

   1. Initial program 79.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 51.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. 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. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      9. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      10. lower-neg.f64 64.8

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   8. Simplified 64.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if -3.39999999999999991e146 < d < -1.59999999999999988e-162

   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-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      9. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      10. +-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

   4. Applied egg-rr 75.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right)} \]

   6. Applied egg-rr 61.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{\left(\left(M \cdot D\right) \cdot 0.5\right) \cdot \left(h \cdot \left(M \cdot D\right)\right)}{d \cdot \left(d \cdot \left(2 \cdot \ell\right)\right)}, 1\right) \cdot \sqrt{\frac{d \cdot d}{h \cdot \ell}}} \]

   if -1.59999999999999988e-162 < d < 2.40000000000000014e-261

   1. Initial program 44.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 12.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in d around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. lower-*.f64 26.6

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   8. Simplified 26.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   if 2.40000000000000014e-261 < d

   1. Initial program 77.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      9. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      10. +-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

   4. Applied egg-rr 78.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      2. rem-exp-log N/A

         \[\leadsto \left({\color{blue}{\left(e^{\log \left(\frac{d}{h}\right)}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      5. rem-exp-log N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      7. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(e^{\log \left(\frac{d}{h}\right)} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      8. rem-exp-log N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      9. rem-exp-log N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      10. *-commutative N/A

          \[\leadsto {\color{blue}{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      11. pow-prod-down N/A

          \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      14. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      15. unpow1/2 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      17. clear-num N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{1}{\color{blue}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      19. sqrt-div N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      20. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      21. lift-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

   6. Applied egg-rr 79.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}} \cdot \mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

   8. Applied egg-rr 67.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5, \frac{\left(M \cdot D\right) \cdot \left(M \cdot \left(D \cdot h\right)\right)}{\left(d \cdot 2\right) \cdot \left(\left(d \cdot 2\right) \cdot \ell\right)}, 1\right) \cdot d}{\sqrt{h \cdot \ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.4 \cdot 10^{+146}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq -1.6 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \mathsf{fma}\left(-0.5, \frac{\left(h \cdot \left(M \cdot D\right)\right) \cdot \left(\left(M \cdot D\right) \cdot 0.5\right)}{d \cdot \left(d \cdot \left(2 \cdot \ell\right)\right)}, 1\right)\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{-261}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \mathsf{fma}\left(-0.5, \frac{\left(M \cdot D\right) \cdot \left(M \cdot \left(h \cdot D\right)\right)}{\left(d \cdot 2\right) \cdot \left(\ell \cdot \left(d \cdot 2\right)\right)}, 1\right)}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 58.6% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;d \leq -3.4 \cdot 10^{+146}:\\ \;\;\;\;\left(-d\right) \cdot t\_0\\ \mathbf{elif}\;d \leq -1.6 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(1 + \frac{\left(h \cdot \left(M \cdot D\right)\right) \cdot \left(D \cdot \left(M \cdot -0.5\right)\right)}{\ell \cdot \left(\left(d \cdot d\right) \cdot 4\right)}\right)\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{-261}:\\ \;\;\;\;d \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \mathsf{fma}\left(-0.5, \frac{\left(M \cdot D\right) \cdot \left(M \cdot \left(h \cdot D\right)\right)}{\left(d \cdot 2\right) \cdot \left(\ell \cdot \left(d \cdot 2\right)\right)}, 1\right)}{\sqrt{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (* h l)))))
   (if (<= d -3.4e+146)
     (* (- d) t_0)
     (if (<= d -1.6e-162)
       (*
        (sqrt (/ (* d d) (* h l)))
        (+ 1.0 (/ (* (* h (* M D)) (* D (* M -0.5))) (* l (* (* d d) 4.0)))))
       (if (<= d 2.4e-261)
         (* d t_0)
         (/
          (*
           d
           (fma
            -0.5
            (/ (* (* M D) (* M (* h D))) (* (* d 2.0) (* l (* d 2.0))))
            1.0))
          (sqrt (* h l))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((1.0 / (h * l)));
	double tmp;
	if (d <= -3.4e+146) {
		tmp = -d * t_0;
	} else if (d <= -1.6e-162) {
		tmp = sqrt(((d * d) / (h * l))) * (1.0 + (((h * (M * D)) * (D * (M * -0.5))) / (l * ((d * d) * 4.0))));
	} else if (d <= 2.4e-261) {
		tmp = d * t_0;
	} else {
		tmp = (d * fma(-0.5, (((M * D) * (M * (h * D))) / ((d * 2.0) * (l * (d * 2.0)))), 1.0)) / sqrt((h * l));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(1.0 / Float64(h * l)))
	tmp = 0.0
	if (d <= -3.4e+146)
		tmp = Float64(Float64(-d) * t_0);
	elseif (d <= -1.6e-162)
		tmp = Float64(sqrt(Float64(Float64(d * d) / Float64(h * l))) * Float64(1.0 + Float64(Float64(Float64(h * Float64(M * D)) * Float64(D * Float64(M * -0.5))) / Float64(l * Float64(Float64(d * d) * 4.0)))));
	elseif (d <= 2.4e-261)
		tmp = Float64(d * t_0);
	else
		tmp = Float64(Float64(d * fma(-0.5, Float64(Float64(Float64(M * D) * Float64(M * Float64(h * D))) / Float64(Float64(d * 2.0) * Float64(l * Float64(d * 2.0)))), 1.0)) / sqrt(Float64(h * l)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -3.4e+146], N[((-d) * t$95$0), $MachinePrecision], If[LessEqual[d, -1.6e-162], N[(N[Sqrt[N[(N[(d * d), $MachinePrecision] / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(N[(h * N[(M * D), $MachinePrecision]), $MachinePrecision] * N[(D * N[(M * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(N[(d * d), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.4e-261], N[(d * t$95$0), $MachinePrecision], N[(N[(d * N[(-0.5 * N[(N[(N[(M * D), $MachinePrecision] * N[(M * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(d * 2.0), $MachinePrecision] * N[(l * N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -3.39999999999999991e146

   1. Initial program 79.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 51.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. 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. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      9. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      10. lower-neg.f64 64.8

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   8. Simplified 64.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if -3.39999999999999991e146 < d < -1.59999999999999988e-162

   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-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      9. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      10. +-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

   4. Applied egg-rr 75.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      2. rem-exp-log N/A

         \[\leadsto \left({\color{blue}{\left(e^{\log \left(\frac{d}{h}\right)}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      5. rem-exp-log N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      7. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(e^{\log \left(\frac{d}{h}\right)} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      8. rem-exp-log N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      9. rem-exp-log N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      10. lift-/.f64 N/A

          \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      11. metadata-eval N/A

          \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      12. unpow1/2 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      16. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      18. lower-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      19. lower-*.f64 61.9

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

   6. Applied egg-rr 61.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}}} \cdot \mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{\color{blue}{M \cdot D}}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{\color{blue}{d \cdot 2}}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \color{blue}{\frac{M \cdot D}{d \cdot 2}}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)} \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{\color{blue}{M \cdot D}}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{\color{blue}{d \cdot 2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\color{blue}{\frac{M \cdot D}{d \cdot 2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \color{blue}{\frac{h}{\ell}}\right) + 1\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right)} + 1\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right)} + 1\right) \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \frac{h}{\ell}} + 1\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \color{blue}{\mathsf{fma}\left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell}, 1\right)} \]

   8. Applied egg-rr 58.9%

      \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{\left(-0.5 \cdot \left(M \cdot D\right)\right) \cdot \left(M \cdot D\right)}{\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right)} \]

   10. Applied egg-rr 61.9%

       \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \color{blue}{\left(\frac{\left(D \cdot \left(M \cdot -0.5\right)\right) \cdot \left(\left(M \cdot D\right) \cdot h\right)}{\ell \cdot \left(\left(d \cdot d\right) \cdot 4\right)} + 1\right)} \]

   if -1.59999999999999988e-162 < d < 2.40000000000000014e-261

   1. Initial program 44.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 12.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in d around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. lower-*.f64 26.6

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   8. Simplified 26.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   if 2.40000000000000014e-261 < d

   1. Initial program 77.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      9. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      10. +-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

   4. Applied egg-rr 78.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      2. rem-exp-log N/A

         \[\leadsto \left({\color{blue}{\left(e^{\log \left(\frac{d}{h}\right)}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      5. rem-exp-log N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      7. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(e^{\log \left(\frac{d}{h}\right)} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      8. rem-exp-log N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      9. rem-exp-log N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      10. *-commutative N/A

          \[\leadsto {\color{blue}{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      11. pow-prod-down N/A

          \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      14. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      15. unpow1/2 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      17. clear-num N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{1}{\color{blue}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      19. sqrt-div N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      20. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      21. lift-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

   6. Applied egg-rr 79.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}} \cdot \mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

   8. Applied egg-rr 67.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5, \frac{\left(M \cdot D\right) \cdot \left(M \cdot \left(D \cdot h\right)\right)}{\left(d \cdot 2\right) \cdot \left(\left(d \cdot 2\right) \cdot \ell\right)}, 1\right) \cdot d}{\sqrt{h \cdot \ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.4 \cdot 10^{+146}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq -1.6 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(1 + \frac{\left(h \cdot \left(M \cdot D\right)\right) \cdot \left(D \cdot \left(M \cdot -0.5\right)\right)}{\ell \cdot \left(\left(d \cdot d\right) \cdot 4\right)}\right)\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{-261}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \mathsf{fma}\left(-0.5, \frac{\left(M \cdot D\right) \cdot \left(M \cdot \left(h \cdot D\right)\right)}{\left(d \cdot 2\right) \cdot \left(\ell \cdot \left(d \cdot 2\right)\right)}, 1\right)}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 57.2% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;d \leq -2.6 \cdot 10^{+146}:\\ \;\;\;\;\left(-d\right) \cdot t\_0\\ \mathbf{elif}\;d \leq -1.6 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \mathsf{fma}\left(M \cdot \left(D \cdot \left(D \cdot \left(M \cdot -0.5\right)\right)\right), \frac{h}{\ell \cdot \left(\left(d \cdot d\right) \cdot 4\right)}, 1\right)\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{-261}:\\ \;\;\;\;d \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \mathsf{fma}\left(-0.5, \frac{\left(M \cdot D\right) \cdot \left(M \cdot \left(h \cdot D\right)\right)}{\left(d \cdot 2\right) \cdot \left(\ell \cdot \left(d \cdot 2\right)\right)}, 1\right)}{\sqrt{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (* h l)))))
   (if (<= d -2.6e+146)
     (* (- d) t_0)
     (if (<= d -1.6e-162)
       (*
        (sqrt (/ (* d d) (* h l)))
        (fma (* M (* D (* D (* M -0.5)))) (/ h (* l (* (* d d) 4.0))) 1.0))
       (if (<= d 2.4e-261)
         (* d t_0)
         (/
          (*
           d
           (fma
            -0.5
            (/ (* (* M D) (* M (* h D))) (* (* d 2.0) (* l (* d 2.0))))
            1.0))
          (sqrt (* h l))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((1.0 / (h * l)));
	double tmp;
	if (d <= -2.6e+146) {
		tmp = -d * t_0;
	} else if (d <= -1.6e-162) {
		tmp = sqrt(((d * d) / (h * l))) * fma((M * (D * (D * (M * -0.5)))), (h / (l * ((d * d) * 4.0))), 1.0);
	} else if (d <= 2.4e-261) {
		tmp = d * t_0;
	} else {
		tmp = (d * fma(-0.5, (((M * D) * (M * (h * D))) / ((d * 2.0) * (l * (d * 2.0)))), 1.0)) / sqrt((h * l));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(1.0 / Float64(h * l)))
	tmp = 0.0
	if (d <= -2.6e+146)
		tmp = Float64(Float64(-d) * t_0);
	elseif (d <= -1.6e-162)
		tmp = Float64(sqrt(Float64(Float64(d * d) / Float64(h * l))) * fma(Float64(M * Float64(D * Float64(D * Float64(M * -0.5)))), Float64(h / Float64(l * Float64(Float64(d * d) * 4.0))), 1.0));
	elseif (d <= 2.4e-261)
		tmp = Float64(d * t_0);
	else
		tmp = Float64(Float64(d * fma(-0.5, Float64(Float64(Float64(M * D) * Float64(M * Float64(h * D))) / Float64(Float64(d * 2.0) * Float64(l * Float64(d * 2.0)))), 1.0)) / sqrt(Float64(h * l)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -2.6e+146], N[((-d) * t$95$0), $MachinePrecision], If[LessEqual[d, -1.6e-162], N[(N[Sqrt[N[(N[(d * d), $MachinePrecision] / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(M * N[(D * N[(D * N[(M * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / N[(l * N[(N[(d * d), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.4e-261], N[(d * t$95$0), $MachinePrecision], N[(N[(d * N[(-0.5 * N[(N[(N[(M * D), $MachinePrecision] * N[(M * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(d * 2.0), $MachinePrecision] * N[(l * N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.60000000000000014e146

   1. Initial program 79.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 51.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. 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. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      9. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      10. lower-neg.f64 64.8

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   8. Simplified 64.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if -2.60000000000000014e146 < d < -1.59999999999999988e-162

   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-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      9. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      10. +-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

   4. Applied egg-rr 75.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      2. rem-exp-log N/A

         \[\leadsto \left({\color{blue}{\left(e^{\log \left(\frac{d}{h}\right)}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      5. rem-exp-log N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      7. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(e^{\log \left(\frac{d}{h}\right)} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      8. rem-exp-log N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      9. rem-exp-log N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      10. lift-/.f64 N/A

          \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      11. metadata-eval N/A

          \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      12. unpow1/2 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      16. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      18. lower-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      19. lower-*.f64 61.9

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

   6. Applied egg-rr 61.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}}} \cdot \mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{\color{blue}{M \cdot D}}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{\color{blue}{d \cdot 2}}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \color{blue}{\frac{M \cdot D}{d \cdot 2}}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)} \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{\color{blue}{M \cdot D}}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{\color{blue}{d \cdot 2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\color{blue}{\frac{M \cdot D}{d \cdot 2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \color{blue}{\frac{h}{\ell}}\right) + 1\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right)} + 1\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right)} + 1\right) \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \frac{h}{\ell}} + 1\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \color{blue}{\mathsf{fma}\left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell}, 1\right)} \]

   8. Applied egg-rr 58.9%

      \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{\left(-0.5 \cdot \left(M \cdot D\right)\right) \cdot \left(M \cdot D\right)}{\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\frac{\left(\frac{-1}{2} \cdot \color{blue}{\left(M \cdot D\right)}\right) \cdot \left(M \cdot D\right)}{\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)} \cdot \frac{h}{\ell} + 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\frac{\color{blue}{\left(\frac{-1}{2} \cdot \left(M \cdot D\right)\right)} \cdot \left(M \cdot D\right)}{\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)} \cdot \frac{h}{\ell} + 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\frac{\left(\frac{-1}{2} \cdot \left(M \cdot D\right)\right) \cdot \color{blue}{\left(M \cdot D\right)}}{\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)} \cdot \frac{h}{\ell} + 1\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\frac{\color{blue}{\left(\frac{-1}{2} \cdot \left(M \cdot D\right)\right) \cdot \left(M \cdot D\right)}}{\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)} \cdot \frac{h}{\ell} + 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\frac{\left(\frac{-1}{2} \cdot \left(M \cdot D\right)\right) \cdot \left(M \cdot D\right)}{\color{blue}{\left(d \cdot 2\right)} \cdot \left(d \cdot 2\right)} \cdot \frac{h}{\ell} + 1\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\frac{\left(\frac{-1}{2} \cdot \left(M \cdot D\right)\right) \cdot \left(M \cdot D\right)}{\left(d \cdot 2\right) \cdot \color{blue}{\left(d \cdot 2\right)}} \cdot \frac{h}{\ell} + 1\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\frac{\left(\frac{-1}{2} \cdot \left(M \cdot D\right)\right) \cdot \left(M \cdot D\right)}{\color{blue}{\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)}} \cdot \frac{h}{\ell} + 1\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\color{blue}{\frac{\left(\frac{-1}{2} \cdot \left(M \cdot D\right)\right) \cdot \left(M \cdot D\right)}{\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)}} \cdot \frac{h}{\ell} + 1\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\frac{\left(\frac{-1}{2} \cdot \left(M \cdot D\right)\right) \cdot \left(M \cdot D\right)}{\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)} \cdot \color{blue}{\frac{h}{\ell}} + 1\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\color{blue}{\frac{\left(\frac{-1}{2} \cdot \left(M \cdot D\right)\right) \cdot \left(M \cdot D\right)}{\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)}} \cdot \frac{h}{\ell} + 1\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\frac{\left(\frac{-1}{2} \cdot \left(M \cdot D\right)\right) \cdot \left(M \cdot D\right)}{\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)} \cdot \color{blue}{\frac{h}{\ell}} + 1\right) \]

      12. frac-times N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\color{blue}{\frac{\left(\left(\frac{-1}{2} \cdot \left(M \cdot D\right)\right) \cdot \left(M \cdot D\right)\right) \cdot h}{\left(\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)\right) \cdot \ell}} + 1\right) \]

      13. associate-/l* N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\left(\frac{-1}{2} \cdot \left(M \cdot D\right)\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{h}{\left(\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)\right) \cdot \ell}} + 1\right) \]

   10. Applied egg-rr 57.4%

       \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \color{blue}{\mathsf{fma}\left(M \cdot \left(D \cdot \left(D \cdot \left(M \cdot -0.5\right)\right)\right), \frac{h}{\ell \cdot \left(\left(d \cdot d\right) \cdot 4\right)}, 1\right)} \]

   if -1.59999999999999988e-162 < d < 2.40000000000000014e-261

   1. Initial program 44.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 12.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in d around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. lower-*.f64 26.6

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   8. Simplified 26.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   if 2.40000000000000014e-261 < d

   1. Initial program 77.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      9. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      10. +-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

   4. Applied egg-rr 78.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      2. rem-exp-log N/A

         \[\leadsto \left({\color{blue}{\left(e^{\log \left(\frac{d}{h}\right)}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      5. rem-exp-log N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      7. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(e^{\log \left(\frac{d}{h}\right)} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      8. rem-exp-log N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      9. rem-exp-log N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      10. *-commutative N/A

          \[\leadsto {\color{blue}{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      11. pow-prod-down N/A

          \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      14. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      15. unpow1/2 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      17. clear-num N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{1}{\color{blue}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      19. sqrt-div N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      20. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      21. lift-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

   6. Applied egg-rr 79.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}} \cdot \mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

   8. Applied egg-rr 67.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5, \frac{\left(M \cdot D\right) \cdot \left(M \cdot \left(D \cdot h\right)\right)}{\left(d \cdot 2\right) \cdot \left(\left(d \cdot 2\right) \cdot \ell\right)}, 1\right) \cdot d}{\sqrt{h \cdot \ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.6 \cdot 10^{+146}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq -1.6 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \mathsf{fma}\left(M \cdot \left(D \cdot \left(D \cdot \left(M \cdot -0.5\right)\right)\right), \frac{h}{\ell \cdot \left(\left(d \cdot d\right) \cdot 4\right)}, 1\right)\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{-261}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \mathsf{fma}\left(-0.5, \frac{\left(M \cdot D\right) \cdot \left(M \cdot \left(h \cdot D\right)\right)}{\left(d \cdot 2\right) \cdot \left(\ell \cdot \left(d \cdot 2\right)\right)}, 1\right)}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 56.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;d \leq -2.6 \cdot 10^{+146}:\\ \;\;\;\;\left(-d\right) \cdot t\_0\\ \mathbf{elif}\;d \leq -1.6 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \mathsf{fma}\left(M \cdot \left(D \cdot \left(D \cdot \left(M \cdot -0.5\right)\right)\right), \frac{h}{\ell \cdot \left(\left(d \cdot d\right) \cdot 4\right)}, 1\right)\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{-261}:\\ \;\;\;\;d \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(-0.5, \frac{\left(M \cdot D\right) \cdot \left(M \cdot \left(h \cdot D\right)\right)}{\left(d \cdot 2\right) \cdot \left(\ell \cdot \left(d \cdot 2\right)\right)}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (* h l)))))
   (if (<= d -2.6e+146)
     (* (- d) t_0)
     (if (<= d -1.6e-162)
       (*
        (sqrt (/ (* d d) (* h l)))
        (fma (* M (* D (* D (* M -0.5)))) (/ h (* l (* (* d d) 4.0))) 1.0))
       (if (<= d 2.4e-261)
         (* d t_0)
         (*
          (/ d (sqrt (* h l)))
          (fma
           -0.5
           (/ (* (* M D) (* M (* h D))) (* (* d 2.0) (* l (* d 2.0))))
           1.0)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((1.0 / (h * l)));
	double tmp;
	if (d <= -2.6e+146) {
		tmp = -d * t_0;
	} else if (d <= -1.6e-162) {
		tmp = sqrt(((d * d) / (h * l))) * fma((M * (D * (D * (M * -0.5)))), (h / (l * ((d * d) * 4.0))), 1.0);
	} else if (d <= 2.4e-261) {
		tmp = d * t_0;
	} else {
		tmp = (d / sqrt((h * l))) * fma(-0.5, (((M * D) * (M * (h * D))) / ((d * 2.0) * (l * (d * 2.0)))), 1.0);
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(1.0 / Float64(h * l)))
	tmp = 0.0
	if (d <= -2.6e+146)
		tmp = Float64(Float64(-d) * t_0);
	elseif (d <= -1.6e-162)
		tmp = Float64(sqrt(Float64(Float64(d * d) / Float64(h * l))) * fma(Float64(M * Float64(D * Float64(D * Float64(M * -0.5)))), Float64(h / Float64(l * Float64(Float64(d * d) * 4.0))), 1.0));
	elseif (d <= 2.4e-261)
		tmp = Float64(d * t_0);
	else
		tmp = Float64(Float64(d / sqrt(Float64(h * l))) * fma(-0.5, Float64(Float64(Float64(M * D) * Float64(M * Float64(h * D))) / Float64(Float64(d * 2.0) * Float64(l * Float64(d * 2.0)))), 1.0));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -2.6e+146], N[((-d) * t$95$0), $MachinePrecision], If[LessEqual[d, -1.6e-162], N[(N[Sqrt[N[(N[(d * d), $MachinePrecision] / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(M * N[(D * N[(D * N[(M * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / N[(l * N[(N[(d * d), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.4e-261], N[(d * t$95$0), $MachinePrecision], N[(N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 * N[(N[(N[(M * D), $MachinePrecision] * N[(M * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(d * 2.0), $MachinePrecision] * N[(l * N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.60000000000000014e146

   1. Initial program 79.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 51.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. 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. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      9. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      10. lower-neg.f64 64.8

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   8. Simplified 64.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if -2.60000000000000014e146 < d < -1.59999999999999988e-162

   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-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      9. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      10. +-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

   4. Applied egg-rr 75.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      2. rem-exp-log N/A

         \[\leadsto \left({\color{blue}{\left(e^{\log \left(\frac{d}{h}\right)}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      5. rem-exp-log N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      7. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(e^{\log \left(\frac{d}{h}\right)} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      8. rem-exp-log N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      9. rem-exp-log N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      10. lift-/.f64 N/A

          \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      11. metadata-eval N/A

          \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      12. unpow1/2 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      16. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      18. lower-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      19. lower-*.f64 61.9

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

   6. Applied egg-rr 61.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}}} \cdot \mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{\color{blue}{M \cdot D}}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{\color{blue}{d \cdot 2}}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \color{blue}{\frac{M \cdot D}{d \cdot 2}}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)} \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{\color{blue}{M \cdot D}}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{\color{blue}{d \cdot 2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\color{blue}{\frac{M \cdot D}{d \cdot 2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \color{blue}{\frac{h}{\ell}}\right) + 1\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right)} + 1\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right)} + 1\right) \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \frac{h}{\ell}} + 1\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \color{blue}{\mathsf{fma}\left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell}, 1\right)} \]

   8. Applied egg-rr 58.9%

      \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{\left(-0.5 \cdot \left(M \cdot D\right)\right) \cdot \left(M \cdot D\right)}{\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)}, \frac{h}{\ell}, 1\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\frac{\left(\frac{-1}{2} \cdot \color{blue}{\left(M \cdot D\right)}\right) \cdot \left(M \cdot D\right)}{\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)} \cdot \frac{h}{\ell} + 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\frac{\color{blue}{\left(\frac{-1}{2} \cdot \left(M \cdot D\right)\right)} \cdot \left(M \cdot D\right)}{\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)} \cdot \frac{h}{\ell} + 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\frac{\left(\frac{-1}{2} \cdot \left(M \cdot D\right)\right) \cdot \color{blue}{\left(M \cdot D\right)}}{\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)} \cdot \frac{h}{\ell} + 1\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\frac{\color{blue}{\left(\frac{-1}{2} \cdot \left(M \cdot D\right)\right) \cdot \left(M \cdot D\right)}}{\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)} \cdot \frac{h}{\ell} + 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\frac{\left(\frac{-1}{2} \cdot \left(M \cdot D\right)\right) \cdot \left(M \cdot D\right)}{\color{blue}{\left(d \cdot 2\right)} \cdot \left(d \cdot 2\right)} \cdot \frac{h}{\ell} + 1\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\frac{\left(\frac{-1}{2} \cdot \left(M \cdot D\right)\right) \cdot \left(M \cdot D\right)}{\left(d \cdot 2\right) \cdot \color{blue}{\left(d \cdot 2\right)}} \cdot \frac{h}{\ell} + 1\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\frac{\left(\frac{-1}{2} \cdot \left(M \cdot D\right)\right) \cdot \left(M \cdot D\right)}{\color{blue}{\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)}} \cdot \frac{h}{\ell} + 1\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\color{blue}{\frac{\left(\frac{-1}{2} \cdot \left(M \cdot D\right)\right) \cdot \left(M \cdot D\right)}{\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)}} \cdot \frac{h}{\ell} + 1\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\frac{\left(\frac{-1}{2} \cdot \left(M \cdot D\right)\right) \cdot \left(M \cdot D\right)}{\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)} \cdot \color{blue}{\frac{h}{\ell}} + 1\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\color{blue}{\frac{\left(\frac{-1}{2} \cdot \left(M \cdot D\right)\right) \cdot \left(M \cdot D\right)}{\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)}} \cdot \frac{h}{\ell} + 1\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\frac{\left(\frac{-1}{2} \cdot \left(M \cdot D\right)\right) \cdot \left(M \cdot D\right)}{\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)} \cdot \color{blue}{\frac{h}{\ell}} + 1\right) \]

      12. frac-times N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\color{blue}{\frac{\left(\left(\frac{-1}{2} \cdot \left(M \cdot D\right)\right) \cdot \left(M \cdot D\right)\right) \cdot h}{\left(\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)\right) \cdot \ell}} + 1\right) \]

      13. associate-/l* N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\left(\frac{-1}{2} \cdot \left(M \cdot D\right)\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{h}{\left(\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)\right) \cdot \ell}} + 1\right) \]

   10. Applied egg-rr 57.4%

       \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \color{blue}{\mathsf{fma}\left(M \cdot \left(D \cdot \left(D \cdot \left(M \cdot -0.5\right)\right)\right), \frac{h}{\ell \cdot \left(\left(d \cdot d\right) \cdot 4\right)}, 1\right)} \]

   if -1.59999999999999988e-162 < d < 2.40000000000000014e-261

   1. Initial program 44.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 12.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in d around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. lower-*.f64 26.6

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   8. Simplified 26.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   if 2.40000000000000014e-261 < d

   1. Initial program 77.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      9. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      10. +-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

   4. Applied egg-rr 78.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      2. rem-exp-log N/A

         \[\leadsto \left({\color{blue}{\left(e^{\log \left(\frac{d}{h}\right)}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      5. rem-exp-log N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      7. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(e^{\log \left(\frac{d}{h}\right)} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      8. rem-exp-log N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      9. rem-exp-log N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      10. *-commutative N/A

          \[\leadsto {\color{blue}{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      11. pow-prod-down N/A

          \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      14. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      15. unpow1/2 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      17. clear-num N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{1}{\color{blue}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      19. sqrt-div N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      20. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      21. lift-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

   6. Applied egg-rr 79.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}} \cdot \mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

   8. Applied egg-rr 66.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{\left(M \cdot D\right) \cdot \left(M \cdot \left(D \cdot h\right)\right)}{\left(d \cdot 2\right) \cdot \left(\left(d \cdot 2\right) \cdot \ell\right)}, 1\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.6 \cdot 10^{+146}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq -1.6 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \mathsf{fma}\left(M \cdot \left(D \cdot \left(D \cdot \left(M \cdot -0.5\right)\right)\right), \frac{h}{\ell \cdot \left(\left(d \cdot d\right) \cdot 4\right)}, 1\right)\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{-261}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(-0.5, \frac{\left(M \cdot D\right) \cdot \left(M \cdot \left(h \cdot D\right)\right)}{\left(d \cdot 2\right) \cdot \left(\ell \cdot \left(d \cdot 2\right)\right)}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 64.8% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.1 \cdot 10^{-301}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + \frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot 0.5\right)}{\left(d \cdot d\right) \cdot 4}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -2.1e-301)
   (*
    (* d (sqrt (/ 1.0 (* h l))))
    (+ -1.0 (/ (/ (* (* M (* D (* M D))) (* h 0.5)) (* (* d d) 4.0)) l)))
   (*
    (/ d (sqrt (* h l)))
    (fma (/ (* M D) (* d 2.0)) (* (/ h l) (* -0.25 (/ (* M D) d))) 1.0))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -2.1e-301) {
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + ((((M * (D * (M * D))) * (h * 0.5)) / ((d * d) * 4.0)) / l));
	} else {
		tmp = (d / sqrt((h * l))) * fma(((M * D) / (d * 2.0)), ((h / l) * (-0.25 * ((M * D) / d))), 1.0);
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -2.1e-301)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 + Float64(Float64(Float64(Float64(M * Float64(D * Float64(M * D))) * Float64(h * 0.5)) / Float64(Float64(d * d) * 4.0)) / l)));
	else
		tmp = Float64(Float64(d / sqrt(Float64(h * l))) * fma(Float64(Float64(M * D) / Float64(d * 2.0)), Float64(Float64(h / l) * Float64(-0.25 * Float64(Float64(M * D) / d))), 1.0));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -2.1e-301], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(N[(N[(N[(M * N[(D * N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h * 0.5), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(-0.25 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -2.0999999999999999e-301

   1. Initial program 69.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      9. lift--.f64 69.1

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      12. associate-*r/ N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

   4. Applied egg-rr 59.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 - \frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot 0.5\right)}{\left(d \cdot d\right) \cdot 4}}{\ell}\right)} \]

   5. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot \frac{1}{2}\right)}{\left(d \cdot d\right) \cdot 4}}{\ell}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot \frac{1}{2}\right)}{\left(d \cdot d\right) \cdot 4}}{\ell}\right) \]

      2. unpow2 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot \frac{1}{2}\right)}{\left(d \cdot d\right) \cdot 4}}{\ell}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot \frac{1}{2}\right)}{\left(d \cdot d\right) \cdot 4}}{\ell}\right) \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)\right)} \cdot \left(1 - \frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot \frac{1}{2}\right)}{\left(d \cdot d\right) \cdot 4}}{\ell}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)\right)} \cdot \left(1 - \frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot \frac{1}{2}\right)}{\left(d \cdot d\right) \cdot 4}}{\ell}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right)\right) \cdot \left(1 - \frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot \frac{1}{2}\right)}{\left(d \cdot d\right) \cdot 4}}{\ell}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right)\right) \cdot \left(1 - \frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot \frac{1}{2}\right)}{\left(d \cdot d\right) \cdot 4}}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right)\right) \cdot \left(1 - \frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot \frac{1}{2}\right)}{\left(d \cdot d\right) \cdot 4}}{\ell}\right) \]

      9. mul-1-neg N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right) \cdot \left(1 - \frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot \frac{1}{2}\right)}{\left(d \cdot d\right) \cdot 4}}{\ell}\right) \]

      10. lower-neg.f64 64.4

          \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)}\right) \cdot \left(1 - \frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot 0.5\right)}{\left(d \cdot d\right) \cdot 4}}{\ell}\right) \]

   7. Simplified 64.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\right)} \cdot \left(1 - \frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot 0.5\right)}{\left(d \cdot d\right) \cdot 4}}{\ell}\right) \]

   if -2.0999999999999999e-301 < l

   1. Initial program 76.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      9. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      10. +-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

   4. Applied egg-rr 77.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      2. rem-exp-log N/A

         \[\leadsto \left({\color{blue}{\left(e^{\log \left(\frac{d}{h}\right)}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      5. rem-exp-log N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      7. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(e^{\log \left(\frac{d}{h}\right)} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      8. rem-exp-log N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      9. rem-exp-log N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      10. *-commutative N/A

          \[\leadsto {\color{blue}{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      11. pow-prod-down N/A

          \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      14. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      15. unpow1/2 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      17. clear-num N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{1}{\color{blue}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      19. sqrt-div N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      20. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      21. lift-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

   6. Applied egg-rr 78.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}} \cdot \mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{\color{blue}{M \cdot D}}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{\color{blue}{d \cdot 2}}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \color{blue}{\frac{M \cdot D}{d \cdot 2}}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)} \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{\color{blue}{M \cdot D}}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{\color{blue}{d \cdot 2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\color{blue}{\frac{M \cdot D}{d \cdot 2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \color{blue}{\frac{h}{\ell}}\right) + 1\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right)} + 1\right) \]

      10. *-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)} + 1\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right)} \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) + 1\right) \]

      12. associate-*l* N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\frac{M \cdot D}{d \cdot 2} \cdot \left(\frac{h}{\ell} \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)\right)} + 1\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right), 1\right)} \]

   8. Applied egg-rr 78.2%

      \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right)} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{\ell}}}}{\sqrt{\frac{h}{d}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\color{blue}{\frac{h}{d}}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      3. sqrt-undiv N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{d}{\ell}}{\frac{h}{d}}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      4. div-inv N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{1}{\frac{h}{d}}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell}} \cdot \frac{1}{\frac{h}{d}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{1}{\color{blue}{\frac{h}{d}}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      7. clear-num N/A

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \color{blue}{\frac{d}{h}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      8. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      12. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      14. sqrt-prod N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      15. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{4} \cdot \frac{M \cdot D}{d}\right), 1\right) \]

      17. lift-/.f64 75.3

          \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right) \]

   10. Applied egg-rr 75.3%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.1 \cdot 10^{-301}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + \frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot 0.5\right)}{\left(d \cdot d\right) \cdot 4}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 54.7% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.82 \cdot 10^{-178}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\sqrt{\frac{1}{\ell} \cdot \frac{1}{\ell}}}\\ \mathbf{elif}\;\ell \leq 7.8 \cdot 10^{+121}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(M \cdot \left(D \cdot \left(D \cdot \left(M \cdot -0.5\right)\right)\right), \frac{h}{\ell \cdot \left(\left(d \cdot d\right) \cdot 4\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1.82e-178)
   (* (- d) (sqrt (/ 1.0 (* h l))))
   (if (<= l -2e-310)
     (* d (sqrt (sqrt (* (/ 1.0 l) (/ 1.0 l)))))
     (if (<= l 7.8e+121)
       (*
        (/ d (sqrt (* h l)))
        (fma (* M (* D (* D (* M -0.5)))) (/ h (* l (* (* d d) 4.0))) 1.0))
       (/ (/ d (sqrt h)) (sqrt l))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.82e-178) {
		tmp = -d * sqrt((1.0 / (h * l)));
	} else if (l <= -2e-310) {
		tmp = d * sqrt(sqrt(((1.0 / l) * (1.0 / l))));
	} else if (l <= 7.8e+121) {
		tmp = (d / sqrt((h * l))) * fma((M * (D * (D * (M * -0.5)))), (h / (l * ((d * d) * 4.0))), 1.0);
	} else {
		tmp = (d / sqrt(h)) / sqrt(l);
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1.82e-178)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l))));
	elseif (l <= -2e-310)
		tmp = Float64(d * sqrt(sqrt(Float64(Float64(1.0 / l) * Float64(1.0 / l)))));
	elseif (l <= 7.8e+121)
		tmp = Float64(Float64(d / sqrt(Float64(h * l))) * fma(Float64(M * Float64(D * Float64(D * Float64(M * -0.5)))), Float64(h / Float64(l * Float64(Float64(d * d) * 4.0))), 1.0));
	else
		tmp = Float64(Float64(d / sqrt(h)) / sqrt(l));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1.82e-178], N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-310], N[(d * N[Sqrt[N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 7.8e+121], N[(N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(M * N[(D * N[(D * N[(M * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / N[(l * N[(N[(d * d), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.8199999999999999e-178

   1. Initial program 67.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 35.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. 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. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      9. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      10. lower-neg.f64 39.5

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   8. Simplified 39.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if -1.8199999999999999e-178 < l < -1.999999999999994e-310

   1. Initial program 75.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 22.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in h around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{\ell}}} \]

      3. lower-/.f64 0.0

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{\ell}}} \]

   8. Simplified 0.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell}}} \]
   9. Step-by-step derivation

      1. lift-/.f64 0.0

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{\ell}}} \]

      2. rem-square-sqrt N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\sqrt{\frac{1}{\ell}} \cdot \sqrt{\frac{1}{\ell}}}} \]

      3. sqrt-unprod N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\sqrt{\frac{1}{\ell} \cdot \frac{1}{\ell}}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\sqrt{\frac{1}{\ell} \cdot \frac{1}{\ell}}}} \]

      5. lower-*.f64 65.0

         \[\leadsto d \cdot \sqrt{\sqrt{\color{blue}{\frac{1}{\ell} \cdot \frac{1}{\ell}}}} \]

   10. Applied egg-rr 65.0%

       \[\leadsto d \cdot \sqrt{\color{blue}{\sqrt{\frac{1}{\ell} \cdot \frac{1}{\ell}}}} \]

   if -1.999999999999994e-310 < l < 7.79999999999999967e121

   1. Initial program 80.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      9. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      10. +-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

   4. Applied egg-rr 81.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      2. rem-exp-log N/A

         \[\leadsto \left({\color{blue}{\left(e^{\log \left(\frac{d}{h}\right)}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      5. rem-exp-log N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      7. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(e^{\log \left(\frac{d}{h}\right)} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      8. rem-exp-log N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot e^{\log \left(\frac{d}{\ell}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      9. rem-exp-log N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      10. *-commutative N/A

          \[\leadsto {\color{blue}{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      11. pow-prod-down N/A

          \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      14. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      15. unpow1/2 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      17. clear-num N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{1}{\color{blue}{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      19. sqrt-div N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      20. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

      21. lift-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{h}{d}}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]

   6. Applied egg-rr 82.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}} \cdot \mathsf{fma}\left(-0.5 \cdot \frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}, 1\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{\color{blue}{M \cdot D}}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{\color{blue}{d \cdot 2}}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \color{blue}{\frac{M \cdot D}{d \cdot 2}}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)} \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{\color{blue}{M \cdot D}}{d \cdot 2} \cdot \frac{h}{\ell}\right) + 1\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{\color{blue}{d \cdot 2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\color{blue}{\frac{M \cdot D}{d \cdot 2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \color{blue}{\frac{h}{\ell}}\right) + 1\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right)} + 1\right) \]

      10. *-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right) \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)} + 1\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{h}{\ell}\right)} \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right) + 1\right) \]

      12. associate-*l* N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\frac{M \cdot D}{d \cdot 2} \cdot \left(\frac{h}{\ell} \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right)\right)} + 1\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(\frac{-1}{2} \cdot \frac{M \cdot D}{d \cdot 2}\right), 1\right)} \]

   8. Applied egg-rr 82.3%

      \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{h}{\ell} \cdot \left(-0.25 \cdot \frac{M \cdot D}{d}\right), 1\right)} \]

   10. Applied egg-rr 63.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(M \cdot \left(D \cdot \left(D \cdot \left(M \cdot -0.5\right)\right)\right), \frac{h}{\ell \cdot \left(\left(d \cdot d\right) \cdot 4\right)}, 1\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]

   if 7.79999999999999967e121 < l

   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. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 51.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      3. pow-to-exp N/A

         \[\leadsto \left(\color{blue}{e^{\log \left(\frac{d}{h}\right) \cdot \frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      4. lift-/.f64 N/A

         \[\leadsto \left(e^{\log \left(\frac{d}{h}\right) \cdot \frac{1}{2}} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      5. lift-/.f64 N/A

         \[\leadsto \left(e^{\log \left(\frac{d}{h}\right) \cdot \frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot 1 \]

      6. pow-to-exp N/A

         \[\leadsto \left(e^{\log \left(\frac{d}{h}\right) \cdot \frac{1}{2}} \cdot \color{blue}{e^{\log \left(\frac{d}{\ell}\right) \cdot \frac{1}{2}}}\right) \cdot 1 \]

      7. pow-to-exp N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot e^{\log \left(\frac{d}{\ell}\right) \cdot \frac{1}{2}}\right) \cdot 1 \]

      8. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot e^{\log \left(\frac{d}{\ell}\right) \cdot \frac{1}{2}}\right) \cdot 1 \]

      9. pow-to-exp N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot 1 \]

      10. lift-pow.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot 1 \]

      11. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot 1 \]

      12. *-lft-identity N/A

          \[\leadsto \color{blue}{\left(1 \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\right)} \cdot 1 \]

      13. *-lft-identity N/A

          \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot 1 \]

      14. *-rgt-identity 51.1

          \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

      15. lift-*.f64 N/A

          \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

      16. lift-pow.f64 N/A

          \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \]

      17. lift-pow.f64 N/A

          \[\leadsto {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

   7. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h} \cdot \left(-d\right)}}{\sqrt{-\ell}}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{h}} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{h} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\sqrt{\color{blue}{\mathsf{neg}\left(\ell\right)}}} \]

      5. sqrt-undiv N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}{\mathsf{neg}\left(\ell\right)}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\mathsf{neg}\left(\ell\right)}} \]

      7. lift-neg.f64 N/A

         \[\leadsto \sqrt{\frac{\frac{d}{h} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)}}{\mathsf{neg}\left(\ell\right)}} \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{neg}\left(\frac{d}{h} \cdot d\right)}}{\mathsf{neg}\left(\ell\right)}} \]

      9. lift-neg.f64 N/A

         \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\frac{d}{h} \cdot d\right)}{\color{blue}{\mathsf{neg}\left(\ell\right)}}} \]

      10. frac-2neg N/A

          \[\leadsto \sqrt{\color{blue}{\frac{\frac{d}{h} \cdot d}{\ell}}} \]

      11. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h} \cdot d}}{\sqrt{\ell}}} \]

      12. pow1/2 N/A

          \[\leadsto \frac{\color{blue}{{\left(\frac{d}{h} \cdot d\right)}^{\frac{1}{2}}}}{\sqrt{\ell}} \]

      13. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{{\left(\frac{d}{h} \cdot d\right)}^{\frac{1}{2}}}{\sqrt{\ell}}} \]

   9. Applied egg-rr 62.1%

      \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.82 \cdot 10^{-178}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\sqrt{\frac{1}{\ell} \cdot \frac{1}{\ell}}}\\ \mathbf{elif}\;\ell \leq 7.8 \cdot 10^{+121}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(M \cdot \left(D \cdot \left(D \cdot \left(M \cdot -0.5\right)\right)\right), \frac{h}{\ell \cdot \left(\left(d \cdot d\right) \cdot 4\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 47.4% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.82 \cdot 10^{-178}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\sqrt{\frac{1}{\ell} \cdot \frac{1}{\ell}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1.82e-178)
   (* (- d) (sqrt (/ 1.0 (* h l))))
   (if (<= l -2e-310)
     (* d (sqrt (sqrt (* (/ 1.0 l) (/ 1.0 l)))))
     (/ (/ d (sqrt h)) (sqrt l)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.82e-178) {
		tmp = -d * sqrt((1.0 / (h * l)));
	} else if (l <= -2e-310) {
		tmp = d * sqrt(sqrt(((1.0 / l) * (1.0 / l))));
	} else {
		tmp = (d / sqrt(h)) / sqrt(l);
	}
	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 <= (-1.82d-178)) then
        tmp = -d * sqrt((1.0d0 / (h * l)))
    else if (l <= (-2d-310)) then
        tmp = d * sqrt(sqrt(((1.0d0 / l) * (1.0d0 / l))))
    else
        tmp = (d / sqrt(h)) / sqrt(l)
    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 <= -1.82e-178) {
		tmp = -d * Math.sqrt((1.0 / (h * l)));
	} else if (l <= -2e-310) {
		tmp = d * Math.sqrt(Math.sqrt(((1.0 / l) * (1.0 / l))));
	} else {
		tmp = (d / Math.sqrt(h)) / Math.sqrt(l);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -1.82e-178:
		tmp = -d * math.sqrt((1.0 / (h * l)))
	elif l <= -2e-310:
		tmp = d * math.sqrt(math.sqrt(((1.0 / l) * (1.0 / l))))
	else:
		tmp = (d / math.sqrt(h)) / math.sqrt(l)
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1.82e-178)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l))));
	elseif (l <= -2e-310)
		tmp = Float64(d * sqrt(sqrt(Float64(Float64(1.0 / l) * Float64(1.0 / l)))));
	else
		tmp = Float64(Float64(d / sqrt(h)) / sqrt(l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -1.82e-178)
		tmp = -d * sqrt((1.0 / (h * l)));
	elseif (l <= -2e-310)
		tmp = d * sqrt(sqrt(((1.0 / l) * (1.0 / l))));
	else
		tmp = (d / sqrt(h)) / sqrt(l);
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1.82e-178], N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-310], N[(d * N[Sqrt[N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.8199999999999999e-178

   1. Initial program 67.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 35.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. 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. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      9. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      10. lower-neg.f64 39.5

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   8. Simplified 39.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if -1.8199999999999999e-178 < l < -1.999999999999994e-310

   1. Initial program 75.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 22.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in h around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{\ell}}} \]

      3. lower-/.f64 0.0

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{\ell}}} \]

   8. Simplified 0.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell}}} \]
   9. Step-by-step derivation

      1. lift-/.f64 0.0

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{\ell}}} \]

      2. rem-square-sqrt N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\sqrt{\frac{1}{\ell}} \cdot \sqrt{\frac{1}{\ell}}}} \]

      3. sqrt-unprod N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\sqrt{\frac{1}{\ell} \cdot \frac{1}{\ell}}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\sqrt{\frac{1}{\ell} \cdot \frac{1}{\ell}}}} \]

      5. lower-*.f64 65.0

         \[\leadsto d \cdot \sqrt{\sqrt{\color{blue}{\frac{1}{\ell} \cdot \frac{1}{\ell}}}} \]

   10. Applied egg-rr 65.0%

       \[\leadsto d \cdot \sqrt{\color{blue}{\sqrt{\frac{1}{\ell} \cdot \frac{1}{\ell}}}} \]

   if -1.999999999999994e-310 < l

   1. Initial program 76.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 37.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      3. pow-to-exp N/A

         \[\leadsto \left(\color{blue}{e^{\log \left(\frac{d}{h}\right) \cdot \frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      4. lift-/.f64 N/A

         \[\leadsto \left(e^{\log \left(\frac{d}{h}\right) \cdot \frac{1}{2}} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      5. lift-/.f64 N/A

         \[\leadsto \left(e^{\log \left(\frac{d}{h}\right) \cdot \frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot 1 \]

      6. pow-to-exp N/A

         \[\leadsto \left(e^{\log \left(\frac{d}{h}\right) \cdot \frac{1}{2}} \cdot \color{blue}{e^{\log \left(\frac{d}{\ell}\right) \cdot \frac{1}{2}}}\right) \cdot 1 \]

      7. pow-to-exp N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot e^{\log \left(\frac{d}{\ell}\right) \cdot \frac{1}{2}}\right) \cdot 1 \]

      8. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot e^{\log \left(\frac{d}{\ell}\right) \cdot \frac{1}{2}}\right) \cdot 1 \]

      9. pow-to-exp N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot 1 \]

      10. lift-pow.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot 1 \]

      11. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot 1 \]

      12. *-lft-identity N/A

          \[\leadsto \color{blue}{\left(1 \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\right)} \cdot 1 \]

      13. *-lft-identity N/A

          \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot 1 \]

      14. *-rgt-identity 37.3

          \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

      15. lift-*.f64 N/A

          \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

      16. lift-pow.f64 N/A

          \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \]

      17. lift-pow.f64 N/A

          \[\leadsto {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

   7. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h} \cdot \left(-d\right)}}{\sqrt{-\ell}}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{h}} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{h} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\sqrt{\color{blue}{\mathsf{neg}\left(\ell\right)}}} \]

      5. sqrt-undiv N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}{\mathsf{neg}\left(\ell\right)}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\mathsf{neg}\left(\ell\right)}} \]

      7. lift-neg.f64 N/A

         \[\leadsto \sqrt{\frac{\frac{d}{h} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)}}{\mathsf{neg}\left(\ell\right)}} \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{neg}\left(\frac{d}{h} \cdot d\right)}}{\mathsf{neg}\left(\ell\right)}} \]

      9. lift-neg.f64 N/A

         \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\frac{d}{h} \cdot d\right)}{\color{blue}{\mathsf{neg}\left(\ell\right)}}} \]

      10. frac-2neg N/A

          \[\leadsto \sqrt{\color{blue}{\frac{\frac{d}{h} \cdot d}{\ell}}} \]

      11. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h} \cdot d}}{\sqrt{\ell}}} \]

      12. pow1/2 N/A

          \[\leadsto \frac{\color{blue}{{\left(\frac{d}{h} \cdot d\right)}^{\frac{1}{2}}}}{\sqrt{\ell}} \]

      13. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{{\left(\frac{d}{h} \cdot d\right)}^{\frac{1}{2}}}{\sqrt{\ell}}} \]

   9. Applied egg-rr 43.6%

      \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.82 \cdot 10^{-178}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\sqrt{\frac{1}{\ell} \cdot \frac{1}{\ell}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 42.6% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq 1.3 \cdot 10^{-233}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= h 1.3e-233)
   (* (- d) (sqrt (/ 1.0 (* h l))))
   (/ (/ d (sqrt h)) (sqrt l))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= 1.3e-233) {
		tmp = -d * sqrt((1.0 / (h * l)));
	} else {
		tmp = (d / sqrt(h)) / sqrt(l);
	}
	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 (h <= 1.3d-233) then
        tmp = -d * sqrt((1.0d0 / (h * l)))
    else
        tmp = (d / sqrt(h)) / sqrt(l)
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= 1.3e-233) {
		tmp = -d * Math.sqrt((1.0 / (h * l)));
	} else {
		tmp = (d / Math.sqrt(h)) / Math.sqrt(l);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if h <= 1.3e-233:
		tmp = -d * math.sqrt((1.0 / (h * l)))
	else:
		tmp = (d / math.sqrt(h)) / math.sqrt(l)
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= 1.3e-233)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l))));
	else
		tmp = Float64(Float64(d / sqrt(h)) / sqrt(l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= 1.3e-233)
		tmp = -d * sqrt((1.0 / (h * l)));
	else
		tmp = (d / sqrt(h)) / sqrt(l);
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[h, 1.3e-233], N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 1.2999999999999999e-233

   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 M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 31.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. 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. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      9. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      10. lower-neg.f64 36.4

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   8. Simplified 36.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if 1.2999999999999999e-233 < h

   1. Initial program 75.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 39.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      3. pow-to-exp N/A

         \[\leadsto \left(\color{blue}{e^{\log \left(\frac{d}{h}\right) \cdot \frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      4. lift-/.f64 N/A

         \[\leadsto \left(e^{\log \left(\frac{d}{h}\right) \cdot \frac{1}{2}} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      5. lift-/.f64 N/A

         \[\leadsto \left(e^{\log \left(\frac{d}{h}\right) \cdot \frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot 1 \]

      6. pow-to-exp N/A

         \[\leadsto \left(e^{\log \left(\frac{d}{h}\right) \cdot \frac{1}{2}} \cdot \color{blue}{e^{\log \left(\frac{d}{\ell}\right) \cdot \frac{1}{2}}}\right) \cdot 1 \]

      7. pow-to-exp N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot e^{\log \left(\frac{d}{\ell}\right) \cdot \frac{1}{2}}\right) \cdot 1 \]

      8. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot e^{\log \left(\frac{d}{\ell}\right) \cdot \frac{1}{2}}\right) \cdot 1 \]

      9. pow-to-exp N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot 1 \]

      10. lift-pow.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot 1 \]

      11. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot 1 \]

      12. *-lft-identity N/A

          \[\leadsto \color{blue}{\left(1 \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\right)} \cdot 1 \]

      13. *-lft-identity N/A

          \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot 1 \]

      14. *-rgt-identity 39.6

          \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

      15. lift-*.f64 N/A

          \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

      16. lift-pow.f64 N/A

          \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \]

      17. lift-pow.f64 N/A

          \[\leadsto {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

   7. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h} \cdot \left(-d\right)}}{\sqrt{-\ell}}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{h}} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{h} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\sqrt{\color{blue}{\mathsf{neg}\left(\ell\right)}}} \]

      5. sqrt-undiv N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}{\mathsf{neg}\left(\ell\right)}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\mathsf{neg}\left(\ell\right)}} \]

      7. lift-neg.f64 N/A

         \[\leadsto \sqrt{\frac{\frac{d}{h} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)}}{\mathsf{neg}\left(\ell\right)}} \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{neg}\left(\frac{d}{h} \cdot d\right)}}{\mathsf{neg}\left(\ell\right)}} \]

      9. lift-neg.f64 N/A

         \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\frac{d}{h} \cdot d\right)}{\color{blue}{\mathsf{neg}\left(\ell\right)}}} \]

      10. frac-2neg N/A

          \[\leadsto \sqrt{\color{blue}{\frac{\frac{d}{h} \cdot d}{\ell}}} \]

      11. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h} \cdot d}}{\sqrt{\ell}}} \]

      12. pow1/2 N/A

          \[\leadsto \frac{\color{blue}{{\left(\frac{d}{h} \cdot d\right)}^{\frac{1}{2}}}}{\sqrt{\ell}} \]

      13. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{{\left(\frac{d}{h} \cdot d\right)}^{\frac{1}{2}}}{\sqrt{\ell}}} \]

   9. Applied egg-rr 45.9%

      \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 1.3 \cdot 10^{-233}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 41.2% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.02 \cdot 10^{-149}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{h \cdot \ell}}{d}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -1.02e-149)
   (* (- d) (sqrt (/ 1.0 (* h l))))
   (/ 1.0 (/ (sqrt (* h l)) d))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -1.02e-149) {
		tmp = -d * sqrt((1.0 / (h * l)));
	} else {
		tmp = 1.0 / (sqrt((h * l)) / 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) :: tmp
    if (d <= (-1.02d-149)) then
        tmp = -d * sqrt((1.0d0 / (h * l)))
    else
        tmp = 1.0d0 / (sqrt((h * l)) / d)
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -1.02e-149) {
		tmp = -d * Math.sqrt((1.0 / (h * l)));
	} else {
		tmp = 1.0 / (Math.sqrt((h * l)) / d);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= -1.02e-149:
		tmp = -d * math.sqrt((1.0 / (h * l)))
	else:
		tmp = 1.0 / (math.sqrt((h * l)) / d)
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -1.02e-149)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l))));
	else
		tmp = Float64(1.0 / Float64(sqrt(Float64(h * l)) / d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -1.02e-149)
		tmp = -d * sqrt((1.0 / (h * l)));
	else
		tmp = 1.0 / (sqrt((h * l)) / d);
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -1.02e-149], N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.0200000000000001e-149

   1. Initial program 77.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 40.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. 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. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      9. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      10. lower-neg.f64 43.3

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   8. Simplified 43.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if -1.0200000000000001e-149 < d

   1. Initial program 69.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 31.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      3. pow-to-exp N/A

         \[\leadsto \left(\color{blue}{e^{\log \left(\frac{d}{h}\right) \cdot \frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      4. lift-/.f64 N/A

         \[\leadsto \left(e^{\log \left(\frac{d}{h}\right) \cdot \frac{1}{2}} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      5. lift-/.f64 N/A

         \[\leadsto \left(e^{\log \left(\frac{d}{h}\right) \cdot \frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot 1 \]

      6. pow-to-exp N/A

         \[\leadsto \left(e^{\log \left(\frac{d}{h}\right) \cdot \frac{1}{2}} \cdot \color{blue}{e^{\log \left(\frac{d}{\ell}\right) \cdot \frac{1}{2}}}\right) \cdot 1 \]

      7. pow-to-exp N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot e^{\log \left(\frac{d}{\ell}\right) \cdot \frac{1}{2}}\right) \cdot 1 \]

      8. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot e^{\log \left(\frac{d}{\ell}\right) \cdot \frac{1}{2}}\right) \cdot 1 \]

      9. pow-to-exp N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot 1 \]

      10. lift-pow.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot 1 \]

      11. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot 1 \]

      12. *-lft-identity N/A

          \[\leadsto \color{blue}{\left(1 \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\right)} \cdot 1 \]

      13. *-lft-identity N/A

          \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot 1 \]

      14. *-rgt-identity 31.8

          \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

      15. lift-*.f64 N/A

          \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

      16. lift-pow.f64 N/A

          \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \]

      17. lift-pow.f64 N/A

          \[\leadsto {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

   7. Applied egg-rr 1.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h} \cdot \left(-d\right)}}{\sqrt{-\ell}}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{h}} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{h} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\sqrt{\color{blue}{\mathsf{neg}\left(\ell\right)}}} \]

      5. sqrt-undiv N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}{\mathsf{neg}\left(\ell\right)}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\mathsf{neg}\left(\ell\right)}} \]

      7. associate-*r/ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h} \cdot \frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}} \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}} \]

      9. lift-neg.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{\color{blue}{\mathsf{neg}\left(d\right)}}{\mathsf{neg}\left(\ell\right)}} \]

      10. lift-neg.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{\mathsf{neg}\left(d\right)}{\color{blue}{\mathsf{neg}\left(\ell\right)}}} \]

      11. frac-2neg N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \]

      12. times-frac N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \]

      15. clear-num N/A

          \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{h \cdot \ell}{d \cdot d}}}} \]

      16. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{\frac{\color{blue}{h \cdot \ell}}{d \cdot d}}} \]

      17. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{\frac{h \cdot \ell}{\color{blue}{d \cdot d}}}} \]

      18. frac-times N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\frac{h}{d} \cdot \frac{\ell}{d}}}} \]

      19. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\frac{h}{d}} \cdot \frac{\ell}{d}}} \]

      20. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{1}{\frac{h}{d} \cdot \color{blue}{\frac{\ell}{d}}}} \]

      21. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\frac{h}{d} \cdot \frac{\ell}{d}}}} \]

   9. Applied egg-rr 34.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{h \cdot \ell}}{d}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.02 \cdot 10^{-149}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{h \cdot \ell}}{d}}\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 41.2% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.02 \cdot 10^{-149}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -1.02e-149) (* (- d) (sqrt (/ 1.0 (* h l)))) (/ d (sqrt (* h l)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -1.02e-149) {
		tmp = -d * sqrt((1.0 / (h * l)));
	} else {
		tmp = d / sqrt((h * l));
	}
	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 (d <= (-1.02d-149)) then
        tmp = -d * sqrt((1.0d0 / (h * l)))
    else
        tmp = d / sqrt((h * l))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -1.02e-149) {
		tmp = -d * Math.sqrt((1.0 / (h * l)));
	} else {
		tmp = d / Math.sqrt((h * l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= -1.02e-149:
		tmp = -d * math.sqrt((1.0 / (h * l)))
	else:
		tmp = d / math.sqrt((h * l))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -1.02e-149)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l))));
	else
		tmp = Float64(d / sqrt(Float64(h * l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -1.02e-149)
		tmp = -d * sqrt((1.0 / (h * l)));
	else
		tmp = d / sqrt((h * l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -1.02e-149], N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.0200000000000001e-149

   1. Initial program 77.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 40.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. 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. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      9. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      10. lower-neg.f64 43.3

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   8. Simplified 43.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if -1.0200000000000001e-149 < d

   1. Initial program 69.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 31.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      3. pow-to-exp N/A

         \[\leadsto \left(\color{blue}{e^{\log \left(\frac{d}{h}\right) \cdot \frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      4. lift-/.f64 N/A

         \[\leadsto \left(e^{\log \left(\frac{d}{h}\right) \cdot \frac{1}{2}} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      5. lift-/.f64 N/A

         \[\leadsto \left(e^{\log \left(\frac{d}{h}\right) \cdot \frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot 1 \]

      6. pow-to-exp N/A

         \[\leadsto \left(e^{\log \left(\frac{d}{h}\right) \cdot \frac{1}{2}} \cdot \color{blue}{e^{\log \left(\frac{d}{\ell}\right) \cdot \frac{1}{2}}}\right) \cdot 1 \]

      7. pow-to-exp N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot e^{\log \left(\frac{d}{\ell}\right) \cdot \frac{1}{2}}\right) \cdot 1 \]

      8. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot e^{\log \left(\frac{d}{\ell}\right) \cdot \frac{1}{2}}\right) \cdot 1 \]

      9. pow-to-exp N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot 1 \]

      10. lift-pow.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot 1 \]

      11. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot 1 \]

      12. *-lft-identity N/A

          \[\leadsto \color{blue}{\left(1 \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\right)} \cdot 1 \]

      13. *-lft-identity N/A

          \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot 1 \]

      14. *-rgt-identity 31.8

          \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

      15. lift-*.f64 N/A

          \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

      16. lift-pow.f64 N/A

          \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \]

      17. lift-pow.f64 N/A

          \[\leadsto {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

   7. Applied egg-rr 1.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h} \cdot \left(-d\right)}}{\sqrt{-\ell}}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{h}} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{h} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\sqrt{\color{blue}{\mathsf{neg}\left(\ell\right)}}} \]

      5. sqrt-undiv N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}{\mathsf{neg}\left(\ell\right)}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\mathsf{neg}\left(\ell\right)}} \]

      7. associate-*r/ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h} \cdot \frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}} \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}} \]

      9. lift-neg.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{\color{blue}{\mathsf{neg}\left(d\right)}}{\mathsf{neg}\left(\ell\right)}} \]

      10. lift-neg.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{\mathsf{neg}\left(d\right)}{\color{blue}{\mathsf{neg}\left(\ell\right)}}} \]

      11. frac-2neg N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \]

      12. times-frac N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \]

      15. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \]

      17. sqrt-prod N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}} \]

      18. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

      19. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      20. lower-sqrt.f64 34.7

          \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

   9. Applied egg-rr 34.7%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.02 \cdot 10^{-149}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 26.8% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \frac{d}{\sqrt{h \cdot \ell}} \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (/ d (sqrt (* h l))))

C

double code(double d, double h, double l, double M, double D) {
	return d / sqrt((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 / sqrt((h * l))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return d / Math.sqrt((h * l));
}

Python

def code(d, h, l, M, D):
	return d / math.sqrt((h * l))

Julia

function code(d, h, l, M, D)
	return Float64(d / sqrt(Float64(h * l)))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = d / sqrt((h * l));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

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. Taylor expanded in M around 0

   \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
4. Step-by-step derivation

5. Simplified 35.1%

   \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

   2. lift-/.f64 N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

   3. pow-to-exp N/A

      \[\leadsto \left(\color{blue}{e^{\log \left(\frac{d}{h}\right) \cdot \frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

   4. lift-/.f64 N/A

      \[\leadsto \left(e^{\log \left(\frac{d}{h}\right) \cdot \frac{1}{2}} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

   5. lift-/.f64 N/A

      \[\leadsto \left(e^{\log \left(\frac{d}{h}\right) \cdot \frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot 1 \]

   6. pow-to-exp N/A

      \[\leadsto \left(e^{\log \left(\frac{d}{h}\right) \cdot \frac{1}{2}} \cdot \color{blue}{e^{\log \left(\frac{d}{\ell}\right) \cdot \frac{1}{2}}}\right) \cdot 1 \]

   7. pow-to-exp N/A

      \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot e^{\log \left(\frac{d}{\ell}\right) \cdot \frac{1}{2}}\right) \cdot 1 \]

   8. lift-pow.f64 N/A

      \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot e^{\log \left(\frac{d}{\ell}\right) \cdot \frac{1}{2}}\right) \cdot 1 \]

   9. pow-to-exp N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot 1 \]

   10. lift-pow.f64 N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot 1 \]

   11. lift-*.f64 N/A

       \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot 1 \]

   12. *-lft-identity N/A

       \[\leadsto \color{blue}{\left(1 \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\right)} \cdot 1 \]

   13. *-lft-identity N/A

       \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot 1 \]

   14. *-rgt-identity 35.1

       \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

   15. lift-*.f64 N/A

       \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

   16. lift-pow.f64 N/A

       \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \]

   17. lift-pow.f64 N/A

       \[\leadsto {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

7. Applied egg-rr 15.2%

   \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h} \cdot \left(-d\right)}}{\sqrt{-\ell}}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{h}} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \]

   2. lift-neg.f64 N/A

      \[\leadsto \frac{\sqrt{\frac{d}{h} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \]

   4. lift-neg.f64 N/A

      \[\leadsto \frac{\sqrt{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\sqrt{\color{blue}{\mathsf{neg}\left(\ell\right)}}} \]

   5. sqrt-undiv N/A

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}{\mathsf{neg}\left(\ell\right)}}} \]

   6. lift-*.f64 N/A

      \[\leadsto \sqrt{\frac{\color{blue}{\frac{d}{h} \cdot \left(\mathsf{neg}\left(d\right)\right)}}{\mathsf{neg}\left(\ell\right)}} \]

   7. associate-*r/ N/A

      \[\leadsto \sqrt{\color{blue}{\frac{d}{h} \cdot \frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}} \]

   8. lift-/.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}} \]

   9. lift-neg.f64 N/A

      \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{\color{blue}{\mathsf{neg}\left(d\right)}}{\mathsf{neg}\left(\ell\right)}} \]

   10. lift-neg.f64 N/A

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{\mathsf{neg}\left(d\right)}{\color{blue}{\mathsf{neg}\left(\ell\right)}}} \]

   11. frac-2neg N/A

       \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \]

   12. times-frac N/A

       \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \]

   13. lift-*.f64 N/A

       \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \]

   14. lift-*.f64 N/A

       \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \]

   15. sqrt-div N/A

       \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \]

   16. lift-*.f64 N/A

       \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \]

   17. sqrt-prod N/A

       \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}} \]

   18. rem-square-sqrt N/A

       \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   19. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

   20. lower-sqrt.f64 24.9

       \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

9. Applied egg-rr 24.9%

   \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
10. Add Preprocessing

Alternative 29: 4.2% accurate, 18.8× speedup

Math

\[\begin{array}{l} \\ \frac{d}{\sqrt{\ell}} \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (/ d (sqrt l)))

C

double code(double d, double h, double l, double M, double D) {
	return d / sqrt(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 / sqrt(l)
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return d / Math.sqrt(l);
}

Python

def code(d, h, l, M, D):
	return d / math.sqrt(l)

Julia

function code(d, h, l, M, D)
	return Float64(d / sqrt(l))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = d / sqrt(l);
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]

Derivation

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. Taylor expanded in M around 0

   \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
4. Step-by-step derivation

5. Simplified 35.1%

   \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

6. Taylor expanded in h around inf

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell}}} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell}}} \]

   2. lower-sqrt.f64 N/A

      \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{\ell}}} \]

   3. lower-/.f64 3.1

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{\ell}}} \]

8. Simplified 3.1%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell}}} \]
9. Step-by-step derivation

   1. sqrt-div N/A

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell}}} \]

   2. metadata-eval N/A

      \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{\ell}} \]

   3. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell}}} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell}}} \]

   5. lower-sqrt.f64 3.1

      \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell}}} \]

10. Applied egg-rr 3.1%

    \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell}}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(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)))))