Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.1% → 74.9%

Time: 21.9s

Alternatives: 17

Speedup: 1.4×

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: 66.1% 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: 74.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \left(d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\right) \cdot \mathsf{fma}\left(h, {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{-0.5}{\ell}, 1\right)\\ \mathbf{if}\;d \leq -8.6 \cdot 10^{+156}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -3.7 \cdot 10^{-79}:\\ \;\;\;\;\left(t_0 \cdot t_1\right) \cdot \left(1 - \frac{0.125 \cdot \left(D \cdot \left(\left(\left(M \cdot M\right) \cdot \frac{D}{d}\right) \cdot \frac{h}{d}\right)\right)}{\ell}\right)\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq 1.02 \cdot 10^{-91}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(t_1 \cdot \mathsf{fma}\left({\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2}, \frac{-0.5}{\frac{\ell}{h}}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2}\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h)))
        (t_1 (sqrt (/ d l)))
        (t_2
         (*
          (* d (- (pow (* h l) -0.5)))
          (fma h (* (pow (* D (/ (/ M d) 2.0)) 2.0) (/ -0.5 l)) 1.0))))
   (if (<= d -8.6e+156)
     t_2
     (if (<= d -3.7e-79)
       (*
        (* t_0 t_1)
        (- 1.0 (/ (* 0.125 (* D (* (* (* M M) (/ D d)) (/ h d)))) l)))
       (if (<= d -2e-310)
         t_2
         (if (<= d 1.02e-91)
           (*
            (/ (sqrt d) (sqrt h))
            (* t_1 (fma (pow (* M (/ (/ D d) 2.0)) 2.0) (/ -0.5 (/ l h)) 1.0)))
           (*
            (* t_0 (/ (sqrt d) (sqrt l)))
            (- 1.0 (/ (* h (* 0.5 (pow (* M (* (/ D d) 0.5)) 2.0))) l)))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h));
	double t_1 = sqrt((d / l));
	double t_2 = (d * -pow((h * l), -0.5)) * fma(h, (pow((D * ((M / d) / 2.0)), 2.0) * (-0.5 / l)), 1.0);
	double tmp;
	if (d <= -8.6e+156) {
		tmp = t_2;
	} else if (d <= -3.7e-79) {
		tmp = (t_0 * t_1) * (1.0 - ((0.125 * (D * (((M * M) * (D / d)) * (h / d)))) / l));
	} else if (d <= -2e-310) {
		tmp = t_2;
	} else if (d <= 1.02e-91) {
		tmp = (sqrt(d) / sqrt(h)) * (t_1 * fma(pow((M * ((D / d) / 2.0)), 2.0), (-0.5 / (l / h)), 1.0));
	} else {
		tmp = (t_0 * (sqrt(d) / sqrt(l))) * (1.0 - ((h * (0.5 * pow((M * ((D / d) * 0.5)), 2.0))) / l));
	}
	return tmp;
}

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / h))
	t_1 = sqrt(Float64(d / l))
	t_2 = Float64(Float64(d * Float64(-(Float64(h * l) ^ -0.5))) * fma(h, Float64((Float64(D * Float64(Float64(M / d) / 2.0)) ^ 2.0) * Float64(-0.5 / l)), 1.0))
	tmp = 0.0
	if (d <= -8.6e+156)
		tmp = t_2;
	elseif (d <= -3.7e-79)
		tmp = Float64(Float64(t_0 * t_1) * Float64(1.0 - Float64(Float64(0.125 * Float64(D * Float64(Float64(Float64(M * M) * Float64(D / d)) * Float64(h / d)))) / l)));
	elseif (d <= -2e-310)
		tmp = t_2;
	elseif (d <= 1.02e-91)
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(t_1 * fma((Float64(M * Float64(Float64(D / d) / 2.0)) ^ 2.0), Float64(-0.5 / Float64(l / h)), 1.0)));
	else
		tmp = Float64(Float64(t_0 * Float64(sqrt(d) / sqrt(l))) * Float64(1.0 - Float64(Float64(h * Float64(0.5 * (Float64(M * Float64(Float64(D / d) * 0.5)) ^ 2.0))) / l)));
	end
	return tmp
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(d * (-N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision] * N[(h * N[(N[Power[N[(D * N[(N[(M / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.5 / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -8.6e+156], t$95$2, If[LessEqual[d, -3.7e-79], N[(N[(t$95$0 * t$95$1), $MachinePrecision] * N[(1.0 - N[(N[(0.125 * N[(D * N[(N[(N[(M * M), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -2e-310], t$95$2, If[LessEqual[d, 1.02e-91], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(N[Power[N[(M * N[(N[(D / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.5 / N[(l / h), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h * N[(0.5 * N[Power[N[(M * N[(N[(D / d), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -8.5999999999999997e156 or -3.70000000000000018e-79 < d < -1.999999999999994e-310

   1. Initial program 52.0%

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

      1. metadata-eval 52.0%

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

      2. unpow1/2 52.0%

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

      3. metadata-eval 52.0%

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

      4. unpow1/2 52.0%

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

      5. *-commutative 52.0%

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

      6. associate-*l* 52.0%

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

      7. times-frac 52.0%

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

      8. metadata-eval 52.0%

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

   3. Simplified 52.0%

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

   5. Applied egg-rr 18.9%

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

      1. expm1-def 21.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 41.1%

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

      3. *-commutative 41.1%

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

      4. sub-neg 41.1%

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

      5. +-commutative 41.1%

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

      6. distribute-rgt-neg-in 41.1%

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

      7. fma-def 41.1%

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

      8. associate-/l* 41.1%

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

      9. distribute-neg-frac 41.1%

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

      10. metadata-eval 41.1%

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

      11. associate-/l* 41.1%

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

      12. associate-*l/ 41.1%

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

      13. fma-def 41.1%

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

      14. associate-*r* 41.5%

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

   7. Simplified 40.2%

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

   8. Taylor expanded in d around -inf 72.7%

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

      1. mul-1-neg 72.7%

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

      2. *-commutative 72.7%

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

      3. distribute-rgt-neg-in 72.7%

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

      4. unpow-1 72.7%

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

      5. sqr-pow 72.7%

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

      6. rem-sqrt-square 72.7%

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

      7. metadata-eval 72.7%

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

      8. sqr-pow 72.6%

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

      9. fabs-sqr 72.6%

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

      10. sqr-pow 72.7%

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

   10. Simplified 72.7%

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

   if -8.5999999999999997e156 < d < -3.70000000000000018e-79

   1. Initial program 87.9%

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

      1. metadata-eval 87.9%

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

      2. unpow1/2 87.9%

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

      3. metadata-eval 87.9%

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

      4. unpow1/2 87.9%

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

      5. *-commutative 87.9%

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

      6. associate-*l* 87.9%

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

      7. times-frac 88.0%

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

      8. metadata-eval 88.0%

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

   3. Simplified 88.0%

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

      1. associate-*r* 88.0%

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

      2. frac-times 87.9%

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

      3. *-commutative 87.9%

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

      4. metadata-eval 87.9%

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

      5. associate-*r/ 92.0%

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

      6. metadata-eval 92.0%

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

      7. *-commutative 92.0%

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

      8. frac-times 92.1%

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

      9. associate-*l/ 92.1%

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

      10. associate-*r/ 92.1%

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

      11. div-inv 92.1%

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

      12. metadata-eval 92.1%

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

   5. Applied egg-rr 92.1%

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

   6. Taylor expanded in M around 0 69.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}}{\ell}\right) \]
   7. Step-by-step derivation

      1. *-commutative 69.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{{D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{{d}^{2}}}{\ell}\right) \]

      2. *-commutative 69.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{{d}^{2}}}{\ell}\right) \]

      3. associate-*r* 71.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2}}}{\ell}\right) \]

      4. unpow2 71.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}{\color{blue}{d \cdot d}}}{\ell}\right) \]

      5. times-frac 75.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \frac{h}{d}\right)}}{\ell}\right) \]

      6. associate-/l* 79.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}} \cdot \frac{h}{d}\right)}{\ell}\right) \]

      7. unpow2 79.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{\frac{d}{{M}^{2}}} \cdot \frac{h}{d}\right)}{\ell}\right) \]

      8. associate-/l* 83.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\color{blue}{\frac{D}{\frac{\frac{d}{{M}^{2}}}{D}}} \cdot \frac{h}{d}\right)}{\ell}\right) \]

      9. unpow2 83.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\frac{D}{\frac{\frac{d}{\color{blue}{M \cdot M}}}{D}} \cdot \frac{h}{d}\right)}{\ell}\right) \]

      10. associate-/r* 83.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\frac{D}{\frac{\color{blue}{\frac{\frac{d}{M}}{M}}}{D}} \cdot \frac{h}{d}\right)}{\ell}\right) \]

   8. Simplified 83.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{0.125 \cdot \left(\frac{D}{\frac{\frac{\frac{d}{M}}{M}}{D}} \cdot \frac{h}{d}\right)}}{\ell}\right) \]

   9. Taylor expanded in D around 0 69.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}}{\ell}\right) \]
   10. Step-by-step derivation

       1. associate-*r* 71.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2}}}{\ell}\right) \]

       2. unpow2 71.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}{\color{blue}{d \cdot d}}}{\ell}\right) \]

       3. times-frac 75.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \frac{h}{d}\right)}}{\ell}\right) \]

       4. associate-/l* 79.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}} \cdot \frac{h}{d}\right)}{\ell}\right) \]

       5. unpow2 79.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{\frac{d}{{M}^{2}}} \cdot \frac{h}{d}\right)}{\ell}\right) \]

       6. unpow2 79.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{\color{blue}{M \cdot M}}} \cdot \frac{h}{d}\right)}{\ell}\right) \]

       7. associate-*l/ 83.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\color{blue}{\left(\frac{D}{\frac{d}{M \cdot M}} \cdot D\right)} \cdot \frac{h}{d}\right)}{\ell}\right) \]

       8. *-commutative 83.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\color{blue}{\left(D \cdot \frac{D}{\frac{d}{M \cdot M}}\right)} \cdot \frac{h}{d}\right)}{\ell}\right) \]

       9. associate-*l* 83.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \color{blue}{\left(D \cdot \left(\frac{D}{\frac{d}{M \cdot M}} \cdot \frac{h}{d}\right)\right)}}{\ell}\right) \]

       10. unpow2 83.3%

           \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(D \cdot \left(\frac{D}{\frac{d}{\color{blue}{{M}^{2}}}} \cdot \frac{h}{d}\right)\right)}{\ell}\right) \]

       11. associate-/r/ 85.4%

           \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(D \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot {M}^{2}\right)} \cdot \frac{h}{d}\right)\right)}{\ell}\right) \]

       12. *-commutative 85.4%

           \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(D \cdot \left(\color{blue}{\left({M}^{2} \cdot \frac{D}{d}\right)} \cdot \frac{h}{d}\right)\right)}{\ell}\right) \]

       13. unpow2 85.4%

           \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(D \cdot \left(\left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{D}{d}\right) \cdot \frac{h}{d}\right)\right)}{\ell}\right) \]

   11. Simplified 85.4%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \color{blue}{\left(D \cdot \left(\left(\left(M \cdot M\right) \cdot \frac{D}{d}\right) \cdot \frac{h}{d}\right)\right)}}{\ell}\right) \]

   if -1.999999999999994e-310 < d < 1.01999999999999994e-91

   1. Initial program 57.9%

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

      1. associate-*l* 57.9%

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

      2. metadata-eval 57.9%

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

      3. unpow1/2 57.9%

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

      4. metadata-eval 57.9%

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

      5. unpow1/2 57.9%

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

      6. sub-neg 57.9%

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

      7. +-commutative 57.9%

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

      8. *-commutative 57.9%

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

      9. associate-*l* 57.9%

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

      10. distribute-rgt-neg-in 57.9%

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

      11. *-commutative 57.9%

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

   3. Simplified 58.9%

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

      1. sqrt-div 72.0%

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

   5. Applied egg-rr 72.0%

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

   if 1.01999999999999994e-91 < d

   1. Initial program 80.9%

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

      1. metadata-eval 80.9%

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

      2. unpow1/2 80.9%

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

      3. metadata-eval 80.9%

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

      4. unpow1/2 80.9%

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

      5. *-commutative 80.9%

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

      6. associate-*l* 80.9%

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

      7. times-frac 80.9%

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

      8. metadata-eval 80.9%

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

   3. Simplified 80.9%

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

      1. associate-*r* 80.9%

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

      2. frac-times 80.9%

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

      3. *-commutative 80.9%

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

      4. metadata-eval 80.9%

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

      5. associate-*r/ 86.5%

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

      6. metadata-eval 86.5%

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

      7. *-commutative 86.5%

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

      8. frac-times 86.5%

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

      9. associate-*l/ 86.5%

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

      10. associate-*r/ 86.5%

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

      11. div-inv 86.5%

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

      12. metadata-eval 86.5%

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

   5. Applied egg-rr 86.5%

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

      1. sqrt-div 91.4%

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

   7. Applied egg-rr 91.4%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8.6 \cdot 10^{+156}:\\ \;\;\;\;\left(d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\right) \cdot \mathsf{fma}\left(h, {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{-0.5}{\ell}, 1\right)\\ \mathbf{elif}\;d \leq -3.7 \cdot 10^{-79}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(D \cdot \left(\left(\left(M \cdot M\right) \cdot \frac{D}{d}\right) \cdot \frac{h}{d}\right)\right)}{\ell}\right)\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\right) \cdot \mathsf{fma}\left(h, {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{-0.5}{\ell}, 1\right)\\ \mathbf{elif}\;d \leq 1.02 \cdot 10^{-91}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2}, \frac{-0.5}{\frac{\ell}{h}}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2}\right)}{\ell}\right)\\ \end{array} \]

Alternative 2: 74.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \left(d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\right) \cdot \mathsf{fma}\left(h, {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{-0.5}{\ell}, 1\right)\\ \mathbf{if}\;d \leq -6.5 \cdot 10^{+156}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -3 \cdot 10^{-79}:\\ \;\;\;\;\left(t_0 \cdot t_1\right) \cdot \left(1 - \frac{0.125 \cdot \left(D \cdot \left(\left(\left(M \cdot M\right) \cdot \frac{D}{d}\right) \cdot \frac{h}{d}\right)\right)}{\ell}\right)\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq 1.8 \cdot 10^{-93}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(t_1 \cdot \left(1 - 0.5 \cdot \left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2}\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h)))
        (t_1 (sqrt (/ d l)))
        (t_2
         (*
          (* d (- (pow (* h l) -0.5)))
          (fma h (* (pow (* D (/ (/ M d) 2.0)) 2.0) (/ -0.5 l)) 1.0))))
   (if (<= d -6.5e+156)
     t_2
     (if (<= d -3e-79)
       (*
        (* t_0 t_1)
        (- 1.0 (/ (* 0.125 (* D (* (* (* M M) (/ D d)) (/ h d)))) l)))
       (if (<= d -2e-310)
         t_2
         (if (<= d 1.8e-93)
           (*
            (/ (sqrt d) (sqrt h))
            (*
             t_1
             (- 1.0 (* 0.5 (* (pow (* (/ D d) (/ M 2.0)) 2.0) (/ h l))))))
           (*
            (* t_0 (/ (sqrt d) (sqrt l)))
            (- 1.0 (/ (* h (* 0.5 (pow (* M (* (/ D d) 0.5)) 2.0))) l)))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h));
	double t_1 = sqrt((d / l));
	double t_2 = (d * -pow((h * l), -0.5)) * fma(h, (pow((D * ((M / d) / 2.0)), 2.0) * (-0.5 / l)), 1.0);
	double tmp;
	if (d <= -6.5e+156) {
		tmp = t_2;
	} else if (d <= -3e-79) {
		tmp = (t_0 * t_1) * (1.0 - ((0.125 * (D * (((M * M) * (D / d)) * (h / d)))) / l));
	} else if (d <= -2e-310) {
		tmp = t_2;
	} else if (d <= 1.8e-93) {
		tmp = (sqrt(d) / sqrt(h)) * (t_1 * (1.0 - (0.5 * (pow(((D / d) * (M / 2.0)), 2.0) * (h / l)))));
	} else {
		tmp = (t_0 * (sqrt(d) / sqrt(l))) * (1.0 - ((h * (0.5 * pow((M * ((D / d) * 0.5)), 2.0))) / l));
	}
	return tmp;
}

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / h))
	t_1 = sqrt(Float64(d / l))
	t_2 = Float64(Float64(d * Float64(-(Float64(h * l) ^ -0.5))) * fma(h, Float64((Float64(D * Float64(Float64(M / d) / 2.0)) ^ 2.0) * Float64(-0.5 / l)), 1.0))
	tmp = 0.0
	if (d <= -6.5e+156)
		tmp = t_2;
	elseif (d <= -3e-79)
		tmp = Float64(Float64(t_0 * t_1) * Float64(1.0 - Float64(Float64(0.125 * Float64(D * Float64(Float64(Float64(M * M) * Float64(D / d)) * Float64(h / d)))) / l)));
	elseif (d <= -2e-310)
		tmp = t_2;
	elseif (d <= 1.8e-93)
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(t_1 * Float64(1.0 - Float64(0.5 * Float64((Float64(Float64(D / d) * Float64(M / 2.0)) ^ 2.0) * Float64(h / l))))));
	else
		tmp = Float64(Float64(t_0 * Float64(sqrt(d) / sqrt(l))) * Float64(1.0 - Float64(Float64(h * Float64(0.5 * (Float64(M * Float64(Float64(D / d) * 0.5)) ^ 2.0))) / l)));
	end
	return tmp
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(d * (-N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision] * N[(h * N[(N[Power[N[(D * N[(N[(M / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.5 / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -6.5e+156], t$95$2, If[LessEqual[d, -3e-79], N[(N[(t$95$0 * t$95$1), $MachinePrecision] * N[(1.0 - N[(N[(0.125 * N[(D * N[(N[(N[(M * M), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -2e-310], t$95$2, If[LessEqual[d, 1.8e-93], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(1.0 - N[(0.5 * N[(N[Power[N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h * N[(0.5 * N[Power[N[(M * N[(N[(D / d), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -6.50000000000000027e156 or -3e-79 < d < -1.999999999999994e-310

   1. Initial program 52.0%

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

      1. metadata-eval 52.0%

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

      2. unpow1/2 52.0%

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

      3. metadata-eval 52.0%

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

      4. unpow1/2 52.0%

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

      5. *-commutative 52.0%

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

      6. associate-*l* 52.0%

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

      7. times-frac 52.0%

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

      8. metadata-eval 52.0%

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

   3. Simplified 52.0%

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

   5. Applied egg-rr 18.9%

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

      1. expm1-def 21.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 41.1%

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

      3. *-commutative 41.1%

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

      4. sub-neg 41.1%

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

      5. +-commutative 41.1%

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

      6. distribute-rgt-neg-in 41.1%

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

      7. fma-def 41.1%

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

      8. associate-/l* 41.1%

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

      9. distribute-neg-frac 41.1%

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

      10. metadata-eval 41.1%

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

      11. associate-/l* 41.1%

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

      12. associate-*l/ 41.1%

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

      13. fma-def 41.1%

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

      14. associate-*r* 41.5%

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

   7. Simplified 40.2%

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

   8. Taylor expanded in d around -inf 72.7%

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

      1. mul-1-neg 72.7%

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

      2. *-commutative 72.7%

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

      3. distribute-rgt-neg-in 72.7%

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

      4. unpow-1 72.7%

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

      5. sqr-pow 72.7%

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

      6. rem-sqrt-square 72.7%

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

      7. metadata-eval 72.7%

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

      8. sqr-pow 72.6%

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

      9. fabs-sqr 72.6%

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

      10. sqr-pow 72.7%

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

   10. Simplified 72.7%

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

   if -6.50000000000000027e156 < d < -3e-79

   1. Initial program 87.9%

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

      1. metadata-eval 87.9%

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

      2. unpow1/2 87.9%

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

      3. metadata-eval 87.9%

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

      4. unpow1/2 87.9%

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

      5. *-commutative 87.9%

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

      6. associate-*l* 87.9%

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

      7. times-frac 88.0%

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

      8. metadata-eval 88.0%

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

   3. Simplified 88.0%

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

      1. associate-*r* 88.0%

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

      2. frac-times 87.9%

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

      3. *-commutative 87.9%

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

      4. metadata-eval 87.9%

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

      5. associate-*r/ 92.0%

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

      6. metadata-eval 92.0%

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

      7. *-commutative 92.0%

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

      8. frac-times 92.1%

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

      9. associate-*l/ 92.1%

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

      10. associate-*r/ 92.1%

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

      11. div-inv 92.1%

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

      12. metadata-eval 92.1%

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

   5. Applied egg-rr 92.1%

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

   6. Taylor expanded in M around 0 69.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}}{\ell}\right) \]
   7. Step-by-step derivation

      1. *-commutative 69.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{{D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{{d}^{2}}}{\ell}\right) \]

      2. *-commutative 69.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{{d}^{2}}}{\ell}\right) \]

      3. associate-*r* 71.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2}}}{\ell}\right) \]

      4. unpow2 71.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}{\color{blue}{d \cdot d}}}{\ell}\right) \]

      5. times-frac 75.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \frac{h}{d}\right)}}{\ell}\right) \]

      6. associate-/l* 79.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}} \cdot \frac{h}{d}\right)}{\ell}\right) \]

      7. unpow2 79.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{\frac{d}{{M}^{2}}} \cdot \frac{h}{d}\right)}{\ell}\right) \]

      8. associate-/l* 83.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\color{blue}{\frac{D}{\frac{\frac{d}{{M}^{2}}}{D}}} \cdot \frac{h}{d}\right)}{\ell}\right) \]

      9. unpow2 83.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\frac{D}{\frac{\frac{d}{\color{blue}{M \cdot M}}}{D}} \cdot \frac{h}{d}\right)}{\ell}\right) \]

      10. associate-/r* 83.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\frac{D}{\frac{\color{blue}{\frac{\frac{d}{M}}{M}}}{D}} \cdot \frac{h}{d}\right)}{\ell}\right) \]

   8. Simplified 83.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{0.125 \cdot \left(\frac{D}{\frac{\frac{\frac{d}{M}}{M}}{D}} \cdot \frac{h}{d}\right)}}{\ell}\right) \]

   9. Taylor expanded in D around 0 69.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}}{\ell}\right) \]
   10. Step-by-step derivation

       1. associate-*r* 71.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2}}}{\ell}\right) \]

       2. unpow2 71.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}{\color{blue}{d \cdot d}}}{\ell}\right) \]

       3. times-frac 75.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \frac{h}{d}\right)}}{\ell}\right) \]

       4. associate-/l* 79.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}} \cdot \frac{h}{d}\right)}{\ell}\right) \]

       5. unpow2 79.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{\frac{d}{{M}^{2}}} \cdot \frac{h}{d}\right)}{\ell}\right) \]

       6. unpow2 79.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{\color{blue}{M \cdot M}}} \cdot \frac{h}{d}\right)}{\ell}\right) \]

       7. associate-*l/ 83.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\color{blue}{\left(\frac{D}{\frac{d}{M \cdot M}} \cdot D\right)} \cdot \frac{h}{d}\right)}{\ell}\right) \]

       8. *-commutative 83.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\color{blue}{\left(D \cdot \frac{D}{\frac{d}{M \cdot M}}\right)} \cdot \frac{h}{d}\right)}{\ell}\right) \]

       9. associate-*l* 83.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \color{blue}{\left(D \cdot \left(\frac{D}{\frac{d}{M \cdot M}} \cdot \frac{h}{d}\right)\right)}}{\ell}\right) \]

       10. unpow2 83.3%

           \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(D \cdot \left(\frac{D}{\frac{d}{\color{blue}{{M}^{2}}}} \cdot \frac{h}{d}\right)\right)}{\ell}\right) \]

       11. associate-/r/ 85.4%

           \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(D \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot {M}^{2}\right)} \cdot \frac{h}{d}\right)\right)}{\ell}\right) \]

       12. *-commutative 85.4%

           \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(D \cdot \left(\color{blue}{\left({M}^{2} \cdot \frac{D}{d}\right)} \cdot \frac{h}{d}\right)\right)}{\ell}\right) \]

       13. unpow2 85.4%

           \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(D \cdot \left(\left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{D}{d}\right) \cdot \frac{h}{d}\right)\right)}{\ell}\right) \]

   11. Simplified 85.4%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \color{blue}{\left(D \cdot \left(\left(\left(M \cdot M\right) \cdot \frac{D}{d}\right) \cdot \frac{h}{d}\right)\right)}}{\ell}\right) \]

   if -1.999999999999994e-310 < d < 1.8000000000000001e-93

   1. Initial program 57.9%

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

      1. associate-*l* 57.9%

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

      2. metadata-eval 57.9%

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

      3. unpow1/2 57.9%

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

      4. metadata-eval 57.9%

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

      5. unpow1/2 57.9%

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

      6. associate-*l* 57.9%

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

      7. metadata-eval 57.9%

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

      8. times-frac 58.8%

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

   3. Simplified 58.8%

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

      1. sqrt-div 72.0%

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

   5. Applied egg-rr 71.9%

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

   if 1.8000000000000001e-93 < d

   1. Initial program 80.9%

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

      1. metadata-eval 80.9%

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

      2. unpow1/2 80.9%

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

      3. metadata-eval 80.9%

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

      4. unpow1/2 80.9%

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

      5. *-commutative 80.9%

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

      6. associate-*l* 80.9%

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

      7. times-frac 80.9%

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

      8. metadata-eval 80.9%

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

   3. Simplified 80.9%

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

      1. associate-*r* 80.9%

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

      2. frac-times 80.9%

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

      3. *-commutative 80.9%

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

      4. metadata-eval 80.9%

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

      5. associate-*r/ 86.5%

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

      6. metadata-eval 86.5%

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

      7. *-commutative 86.5%

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

      8. frac-times 86.5%

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

      9. associate-*l/ 86.5%

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

      10. associate-*r/ 86.5%

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

      11. div-inv 86.5%

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

      12. metadata-eval 86.5%

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

   5. Applied egg-rr 86.5%

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

      1. sqrt-div 91.4%

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

   7. Applied egg-rr 91.4%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.5 \cdot 10^{+156}:\\ \;\;\;\;\left(d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\right) \cdot \mathsf{fma}\left(h, {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{-0.5}{\ell}, 1\right)\\ \mathbf{elif}\;d \leq -3 \cdot 10^{-79}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(D \cdot \left(\left(\left(M \cdot M\right) \cdot \frac{D}{d}\right) \cdot \frac{h}{d}\right)\right)}{\ell}\right)\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\right) \cdot \mathsf{fma}\left(h, {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{-0.5}{\ell}, 1\right)\\ \mathbf{elif}\;d \leq 1.8 \cdot 10^{-93}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2}\right)}{\ell}\right)\\ \end{array} \]

Alternative 3: 74.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \left(d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\right) \cdot \mathsf{fma}\left(h, {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{-0.5}{\ell}, 1\right)\\ \mathbf{if}\;d \leq -8.6 \cdot 10^{+156}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -3.5 \cdot 10^{-79}:\\ \;\;\;\;\left(t_0 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(D \cdot \left(\left(\left(M \cdot M\right) \cdot \frac{D}{d}\right) \cdot \frac{h}{d}\right)\right)}{\ell}\right)\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{-305}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2}\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h)))
        (t_1
         (*
          (* d (- (pow (* h l) -0.5)))
          (fma h (* (pow (* D (/ (/ M d) 2.0)) 2.0) (/ -0.5 l)) 1.0))))
   (if (<= d -8.6e+156)
     t_1
     (if (<= d -3.5e-79)
       (*
        (* t_0 (sqrt (/ d l)))
        (- 1.0 (/ (* 0.125 (* D (* (* (* M M) (/ D d)) (/ h d)))) l)))
       (if (<= d 1.35e-305)
         t_1
         (*
          (* t_0 (/ (sqrt d) (sqrt l)))
          (- 1.0 (/ (* h (* 0.5 (pow (* M (* (/ D d) 0.5)) 2.0))) l))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h));
	double t_1 = (d * -pow((h * l), -0.5)) * fma(h, (pow((D * ((M / d) / 2.0)), 2.0) * (-0.5 / l)), 1.0);
	double tmp;
	if (d <= -8.6e+156) {
		tmp = t_1;
	} else if (d <= -3.5e-79) {
		tmp = (t_0 * sqrt((d / l))) * (1.0 - ((0.125 * (D * (((M * M) * (D / d)) * (h / d)))) / l));
	} else if (d <= 1.35e-305) {
		tmp = t_1;
	} else {
		tmp = (t_0 * (sqrt(d) / sqrt(l))) * (1.0 - ((h * (0.5 * pow((M * ((D / d) * 0.5)), 2.0))) / l));
	}
	return tmp;
}

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / h))
	t_1 = Float64(Float64(d * Float64(-(Float64(h * l) ^ -0.5))) * fma(h, Float64((Float64(D * Float64(Float64(M / d) / 2.0)) ^ 2.0) * Float64(-0.5 / l)), 1.0))
	tmp = 0.0
	if (d <= -8.6e+156)
		tmp = t_1;
	elseif (d <= -3.5e-79)
		tmp = Float64(Float64(t_0 * sqrt(Float64(d / l))) * Float64(1.0 - Float64(Float64(0.125 * Float64(D * Float64(Float64(Float64(M * M) * Float64(D / d)) * Float64(h / d)))) / l)));
	elseif (d <= 1.35e-305)
		tmp = t_1;
	else
		tmp = Float64(Float64(t_0 * Float64(sqrt(d) / sqrt(l))) * Float64(1.0 - Float64(Float64(h * Float64(0.5 * (Float64(M * Float64(Float64(D / d) * 0.5)) ^ 2.0))) / l)));
	end
	return tmp
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(d * (-N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision] * N[(h * N[(N[Power[N[(D * N[(N[(M / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.5 / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -8.6e+156], t$95$1, If[LessEqual[d, -3.5e-79], N[(N[(t$95$0 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(0.125 * N[(D * N[(N[(N[(M * M), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.35e-305], t$95$1, N[(N[(t$95$0 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h * N[(0.5 * N[Power[N[(M * N[(N[(D / d), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -8.5999999999999997e156 or -3.5000000000000003e-79 < d < 1.35e-305

   1. Initial program 51.4%

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

      1. metadata-eval 51.4%

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

      2. unpow1/2 51.4%

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

      3. metadata-eval 51.4%

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

      4. unpow1/2 51.4%

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

      5. *-commutative 51.4%

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

      6. associate-*l* 51.4%

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

      7. times-frac 51.4%

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

      8. metadata-eval 51.4%

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

   3. Simplified 51.4%

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

   5. Applied egg-rr 18.7%

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

      1. expm1-def 21.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 40.6%

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

      3. *-commutative 40.6%

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

      4. sub-neg 40.6%

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

      5. +-commutative 40.6%

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

      6. distribute-rgt-neg-in 40.6%

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

      7. fma-def 40.6%

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

      8. associate-/l* 40.6%

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

      9. distribute-neg-frac 40.6%

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

      10. metadata-eval 40.6%

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

      11. associate-/l* 40.6%

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

      12. associate-*l/ 40.6%

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

      13. fma-def 40.6%

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

      14. associate-*r* 41.0%

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

   7. Simplified 39.7%

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

   8. Taylor expanded in d around -inf 71.8%

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

      1. mul-1-neg 71.8%

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

      2. *-commutative 71.8%

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

      3. distribute-rgt-neg-in 71.8%

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

      4. unpow-1 71.8%

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

      5. sqr-pow 71.8%

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

      6. rem-sqrt-square 71.8%

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

      7. metadata-eval 71.8%

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

      8. sqr-pow 71.7%

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

      9. fabs-sqr 71.7%

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

      10. sqr-pow 71.8%

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

   10. Simplified 71.8%

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

   if -8.5999999999999997e156 < d < -3.5000000000000003e-79

   1. Initial program 87.9%

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

      1. metadata-eval 87.9%

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

      2. unpow1/2 87.9%

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

      3. metadata-eval 87.9%

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

      4. unpow1/2 87.9%

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

      5. *-commutative 87.9%

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

      6. associate-*l* 87.9%

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

      7. times-frac 88.0%

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

      8. metadata-eval 88.0%

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

   3. Simplified 88.0%

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

      1. associate-*r* 88.0%

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

      2. frac-times 87.9%

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

      3. *-commutative 87.9%

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

      4. metadata-eval 87.9%

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

      5. associate-*r/ 92.0%

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

      6. metadata-eval 92.0%

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

      7. *-commutative 92.0%

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

      8. frac-times 92.1%

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

      9. associate-*l/ 92.1%

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

      10. associate-*r/ 92.1%

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

      11. div-inv 92.1%

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

      12. metadata-eval 92.1%

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

   5. Applied egg-rr 92.1%

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

   6. Taylor expanded in M around 0 69.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}}{\ell}\right) \]
   7. Step-by-step derivation

      1. *-commutative 69.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{{D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{{d}^{2}}}{\ell}\right) \]

      2. *-commutative 69.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{{d}^{2}}}{\ell}\right) \]

      3. associate-*r* 71.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2}}}{\ell}\right) \]

      4. unpow2 71.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}{\color{blue}{d \cdot d}}}{\ell}\right) \]

      5. times-frac 75.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \frac{h}{d}\right)}}{\ell}\right) \]

      6. associate-/l* 79.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}} \cdot \frac{h}{d}\right)}{\ell}\right) \]

      7. unpow2 79.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{\frac{d}{{M}^{2}}} \cdot \frac{h}{d}\right)}{\ell}\right) \]

      8. associate-/l* 83.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\color{blue}{\frac{D}{\frac{\frac{d}{{M}^{2}}}{D}}} \cdot \frac{h}{d}\right)}{\ell}\right) \]

      9. unpow2 83.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\frac{D}{\frac{\frac{d}{\color{blue}{M \cdot M}}}{D}} \cdot \frac{h}{d}\right)}{\ell}\right) \]

      10. associate-/r* 83.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\frac{D}{\frac{\color{blue}{\frac{\frac{d}{M}}{M}}}{D}} \cdot \frac{h}{d}\right)}{\ell}\right) \]

   8. Simplified 83.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{0.125 \cdot \left(\frac{D}{\frac{\frac{\frac{d}{M}}{M}}{D}} \cdot \frac{h}{d}\right)}}{\ell}\right) \]

   9. Taylor expanded in D around 0 69.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}}{\ell}\right) \]
   10. Step-by-step derivation

       1. associate-*r* 71.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2}}}{\ell}\right) \]

       2. unpow2 71.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}{\color{blue}{d \cdot d}}}{\ell}\right) \]

       3. times-frac 75.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \frac{h}{d}\right)}}{\ell}\right) \]

       4. associate-/l* 79.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}} \cdot \frac{h}{d}\right)}{\ell}\right) \]

       5. unpow2 79.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{\frac{d}{{M}^{2}}} \cdot \frac{h}{d}\right)}{\ell}\right) \]

       6. unpow2 79.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{\color{blue}{M \cdot M}}} \cdot \frac{h}{d}\right)}{\ell}\right) \]

       7. associate-*l/ 83.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\color{blue}{\left(\frac{D}{\frac{d}{M \cdot M}} \cdot D\right)} \cdot \frac{h}{d}\right)}{\ell}\right) \]

       8. *-commutative 83.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\color{blue}{\left(D \cdot \frac{D}{\frac{d}{M \cdot M}}\right)} \cdot \frac{h}{d}\right)}{\ell}\right) \]

       9. associate-*l* 83.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \color{blue}{\left(D \cdot \left(\frac{D}{\frac{d}{M \cdot M}} \cdot \frac{h}{d}\right)\right)}}{\ell}\right) \]

       10. unpow2 83.3%

           \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(D \cdot \left(\frac{D}{\frac{d}{\color{blue}{{M}^{2}}}} \cdot \frac{h}{d}\right)\right)}{\ell}\right) \]

       11. associate-/r/ 85.4%

           \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(D \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot {M}^{2}\right)} \cdot \frac{h}{d}\right)\right)}{\ell}\right) \]

       12. *-commutative 85.4%

           \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(D \cdot \left(\color{blue}{\left({M}^{2} \cdot \frac{D}{d}\right)} \cdot \frac{h}{d}\right)\right)}{\ell}\right) \]

       13. unpow2 85.4%

           \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(D \cdot \left(\left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{D}{d}\right) \cdot \frac{h}{d}\right)\right)}{\ell}\right) \]

   11. Simplified 85.4%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \color{blue}{\left(D \cdot \left(\left(\left(M \cdot M\right) \cdot \frac{D}{d}\right) \cdot \frac{h}{d}\right)\right)}}{\ell}\right) \]

   if 1.35e-305 < d

   1. Initial program 74.5%

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

      1. metadata-eval 74.5%

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

      2. unpow1/2 74.5%

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

      3. metadata-eval 74.5%

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

      4. unpow1/2 74.5%

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

      5. *-commutative 74.5%

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

      6. associate-*l* 74.5%

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

      7. times-frac 74.8%

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

      8. metadata-eval 74.8%

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

   3. Simplified 74.8%

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

      1. associate-*r* 74.8%

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

      2. frac-times 74.5%

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

      3. *-commutative 74.5%

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

      4. metadata-eval 74.5%

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

      5. associate-*r/ 78.4%

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

      6. metadata-eval 78.4%

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

      7. *-commutative 78.4%

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

      8. frac-times 78.8%

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

      9. associate-*l/ 78.8%

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

      10. associate-*r/ 78.8%

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

      11. div-inv 78.8%

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

      12. metadata-eval 78.8%

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

   5. Applied egg-rr 78.8%

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

      1. sqrt-div 82.3%

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

   7. Applied egg-rr 82.3%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8.6 \cdot 10^{+156}:\\ \;\;\;\;\left(d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\right) \cdot \mathsf{fma}\left(h, {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{-0.5}{\ell}, 1\right)\\ \mathbf{elif}\;d \leq -3.5 \cdot 10^{-79}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(D \cdot \left(\left(\left(M \cdot M\right) \cdot \frac{D}{d}\right) \cdot \frac{h}{d}\right)\right)}{\ell}\right)\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{-305}:\\ \;\;\;\;\left(d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\right) \cdot \mathsf{fma}\left(h, {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{-0.5}{\ell}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2}\right)}{\ell}\right)\\ \end{array} \]

Alternative 4: 69.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -1.06 \cdot 10^{-224}:\\ \;\;\;\;\left(d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\right) \cdot \mathsf{fma}\left(h, {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{-0.5}{\ell}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2}\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= h -1.06e-224)
   (*
    (* d (- (pow (* h l) -0.5)))
    (fma h (* (pow (* D (/ (/ M d) 2.0)) 2.0) (/ -0.5 l)) 1.0))
   (*
    (* (sqrt (/ d h)) (sqrt (/ d l)))
    (- 1.0 (/ (* h (* 0.5 (pow (* M (* (/ D d) 0.5)) 2.0))) l)))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -1.06e-224) {
		tmp = (d * -pow((h * l), -0.5)) * fma(h, (pow((D * ((M / d) / 2.0)), 2.0) * (-0.5 / l)), 1.0);
	} else {
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - ((h * (0.5 * pow((M * ((D / d) * 0.5)), 2.0))) / l));
	}
	return tmp;
}

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= -1.06e-224)
		tmp = Float64(Float64(d * Float64(-(Float64(h * l) ^ -0.5))) * fma(h, Float64((Float64(D * Float64(Float64(M / d) / 2.0)) ^ 2.0) * Float64(-0.5 / l)), 1.0));
	else
		tmp = Float64(Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))) * Float64(1.0 - Float64(Float64(h * Float64(0.5 * (Float64(M * Float64(Float64(D / d) * 0.5)) ^ 2.0))) / l)));
	end
	return tmp
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[h, -1.06e-224], N[(N[(d * (-N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision] * N[(h * N[(N[Power[N[(D * N[(N[(M / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.5 / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h * N[(0.5 * N[Power[N[(M * N[(N[(D / d), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -1.06000000000000001e-224

   1. Initial program 62.9%

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

      1. metadata-eval 62.9%

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

      2. unpow1/2 62.9%

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

      3. metadata-eval 62.9%

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

      4. unpow1/2 62.9%

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

      5. *-commutative 62.9%

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

      6. associate-*l* 62.9%

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

      7. times-frac 62.9%

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

      8. metadata-eval 62.9%

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

   3. Simplified 62.9%

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

   5. Applied egg-rr 17.3%

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

      1. expm1-def 23.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 51.2%

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

      3. *-commutative 51.2%

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

      4. sub-neg 51.2%

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

      5. +-commutative 51.2%

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

      6. distribute-rgt-neg-in 51.2%

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

      7. fma-def 51.2%

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

      8. associate-/l* 51.2%

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

      9. distribute-neg-frac 51.2%

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

      10. metadata-eval 51.2%

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

      11. associate-/l* 51.2%

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

      12. associate-*l/ 51.3%

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

      13. fma-def 51.3%

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

      14. associate-*r* 53.2%

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

   7. Simplified 52.3%

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

   8. Taylor expanded in d around -inf 72.8%

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

      1. mul-1-neg 72.8%

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

      2. *-commutative 72.8%

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

      3. distribute-rgt-neg-in 72.8%

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

      4. unpow-1 72.8%

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

      5. sqr-pow 72.9%

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

      6. rem-sqrt-square 72.9%

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

      7. metadata-eval 72.9%

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

      8. sqr-pow 72.7%

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

      9. fabs-sqr 72.7%

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

      10. sqr-pow 72.9%

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

   10. Simplified 72.9%

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

   if -1.06000000000000001e-224 < h

   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. Step-by-step derivation

      1. metadata-eval 75.5%

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

      2. unpow1/2 75.5%

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

      3. metadata-eval 75.5%

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

      4. unpow1/2 75.5%

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

      5. *-commutative 75.5%

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

      6. associate-*l* 75.5%

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

      7. times-frac 75.7%

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

      8. metadata-eval 75.7%

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

   3. Simplified 75.7%

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

      1. associate-*r* 75.7%

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

      2. frac-times 75.5%

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

      3. *-commutative 75.5%

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

      4. metadata-eval 75.5%

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

      5. associate-*r/ 79.0%

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

      6. metadata-eval 79.0%

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

      7. *-commutative 79.0%

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

      8. frac-times 79.3%

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

      9. associate-*l/ 79.3%

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

      10. associate-*r/ 79.3%

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

      11. div-inv 79.3%

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

      12. metadata-eval 79.3%

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

   5. Applied egg-rr 79.3%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1.06 \cdot 10^{-224}:\\ \;\;\;\;\left(d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\right) \cdot \mathsf{fma}\left(h, {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{-0.5}{\ell}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2}\right)}{\ell}\right)\\ \end{array} \]

Alternative 5: 71.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\right) \cdot \mathsf{fma}\left(h, {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{-0.5}{\ell}, 1\right)\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{+196}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{D \cdot \frac{h}{d}}{\frac{\frac{d}{M}}{D \cdot M}}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l -3e-309)
   (*
    (* d (- (pow (* h l) -0.5)))
    (fma h (* (pow (* D (/ (/ M d) 2.0)) 2.0) (/ -0.5 l)) 1.0))
   (if (<= l 1.6e+196)
     (*
      (* (sqrt (/ d h)) (sqrt (/ d l)))
      (- 1.0 (/ (* 0.125 (/ (* D (/ h d)) (/ (/ d M) (* D M)))) l)))
     (* d (* (pow h -0.5) (pow l -0.5))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -3e-309) {
		tmp = (d * -pow((h * l), -0.5)) * fma(h, (pow((D * ((M / d) / 2.0)), 2.0) * (-0.5 / l)), 1.0);
	} else if (l <= 1.6e+196) {
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - ((0.125 * ((D * (h / d)) / ((d / M) / (D * M)))) / l));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -3e-309)
		tmp = Float64(Float64(d * Float64(-(Float64(h * l) ^ -0.5))) * fma(h, Float64((Float64(D * Float64(Float64(M / d) / 2.0)) ^ 2.0) * Float64(-0.5 / l)), 1.0));
	elseif (l <= 1.6e+196)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))) * Float64(1.0 - Float64(Float64(0.125 * Float64(Float64(D * Float64(h / d)) / Float64(Float64(d / M) / Float64(D * M)))) / l)));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, -3e-309], N[(N[(d * (-N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision] * N[(h * N[(N[Power[N[(D * N[(N[(M / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.5 / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.6e+196], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(0.125 * N[(N[(D * N[(h / d), $MachinePrecision]), $MachinePrecision] / N[(N[(d / M), $MachinePrecision] / N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.000000000000001e-309

   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. Step-by-step derivation

      1. metadata-eval 65.6%

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

      2. unpow1/2 65.6%

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

      3. metadata-eval 65.6%

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

      4. unpow1/2 65.6%

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

      5. *-commutative 65.6%

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

      6. associate-*l* 65.6%

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

      7. times-frac 65.6%

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

      8. metadata-eval 65.6%

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

   3. Simplified 65.6%

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

   5. Applied egg-rr 19.5%

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

      1. expm1-def 25.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 52.8%

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

      3. *-commutative 52.8%

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

      4. sub-neg 52.8%

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

      5. +-commutative 52.8%

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

      6. distribute-rgt-neg-in 52.8%

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

      7. fma-def 52.8%

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

      8. associate-/l* 52.8%

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

      9. distribute-neg-frac 52.8%

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

      10. metadata-eval 52.8%

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

      11. associate-/l* 52.8%

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

      12. associate-*l/ 52.8%

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

      13. fma-def 52.8%

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

      14. associate-*r* 54.6%

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

   7. Simplified 53.8%

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

   8. Taylor expanded in d around -inf 73.1%

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

      1. mul-1-neg 73.1%

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

      2. *-commutative 73.1%

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

      3. distribute-rgt-neg-in 73.1%

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

      4. unpow-1 73.1%

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

      5. sqr-pow 73.2%

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

      6. rem-sqrt-square 73.2%

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

      7. metadata-eval 73.2%

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

      8. sqr-pow 73.0%

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

      9. fabs-sqr 73.0%

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

      10. sqr-pow 73.2%

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

   10. Simplified 73.2%

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

   if -3.000000000000001e-309 < l < 1.59999999999999996e196

   1. Initial program 78.5%

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

      1. metadata-eval 78.5%

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

      2. unpow1/2 78.5%

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

      3. metadata-eval 78.5%

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

      4. unpow1/2 78.5%

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

      5. *-commutative 78.5%

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

      6. associate-*l* 78.5%

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

      7. times-frac 78.9%

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

      8. metadata-eval 78.9%

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

   3. Simplified 78.9%

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

      1. associate-*r* 78.9%

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

      2. frac-times 78.5%

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

      3. *-commutative 78.5%

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

      4. metadata-eval 78.5%

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

      5. associate-*r/ 83.2%

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

      6. metadata-eval 83.2%

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

      7. *-commutative 83.2%

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

      8. frac-times 83.6%

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

      9. associate-*l/ 83.6%

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

      10. associate-*r/ 83.6%

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

      11. div-inv 83.6%

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

      12. metadata-eval 83.6%

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

   5. Applied egg-rr 83.6%

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

   6. Taylor expanded in M around 0 56.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}}{\ell}\right) \]
   7. Step-by-step derivation

      1. *-commutative 56.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{{D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{{d}^{2}}}{\ell}\right) \]

      2. *-commutative 56.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{{d}^{2}}}{\ell}\right) \]

      3. associate-*r* 56.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2}}}{\ell}\right) \]

      4. unpow2 56.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}{\color{blue}{d \cdot d}}}{\ell}\right) \]

      5. times-frac 65.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \frac{h}{d}\right)}}{\ell}\right) \]

      6. associate-/l* 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}} \cdot \frac{h}{d}\right)}{\ell}\right) \]

      7. unpow2 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{\frac{d}{{M}^{2}}} \cdot \frac{h}{d}\right)}{\ell}\right) \]

      8. associate-/l* 74.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\color{blue}{\frac{D}{\frac{\frac{d}{{M}^{2}}}{D}}} \cdot \frac{h}{d}\right)}{\ell}\right) \]

      9. unpow2 74.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\frac{D}{\frac{\frac{d}{\color{blue}{M \cdot M}}}{D}} \cdot \frac{h}{d}\right)}{\ell}\right) \]

      10. associate-/r* 76.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\frac{D}{\frac{\color{blue}{\frac{\frac{d}{M}}{M}}}{D}} \cdot \frac{h}{d}\right)}{\ell}\right) \]

   8. Simplified 76.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{0.125 \cdot \left(\frac{D}{\frac{\frac{\frac{d}{M}}{M}}{D}} \cdot \frac{h}{d}\right)}}{\ell}\right) \]
   9. Step-by-step derivation

      1. associate-*l/ 76.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \color{blue}{\frac{D \cdot \frac{h}{d}}{\frac{\frac{\frac{d}{M}}{M}}{D}}}}{\ell}\right) \]

      2. associate-/l/ 79.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{D \cdot \frac{h}{d}}{\color{blue}{\frac{\frac{d}{M}}{D \cdot M}}}}{\ell}\right) \]

   10. Applied egg-rr 79.5%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \color{blue}{\frac{D \cdot \frac{h}{d}}{\frac{\frac{d}{M}}{D \cdot M}}}}{\ell}\right) \]

   if 1.59999999999999996e196 < l

   1. Initial program 50.5%

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

      1. metadata-eval 50.5%

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

      2. unpow1/2 50.5%

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

      3. metadata-eval 50.5%

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

      4. unpow1/2 50.5%

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

      5. *-commutative 50.5%

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

      6. associate-*l* 50.5%

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

      7. times-frac 50.5%

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

      8. metadata-eval 50.5%

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

   3. Simplified 50.5%

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

   4. Taylor expanded in d around inf 54.7%

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

      1. *-un-lft-identity 54.7%

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

      2. *-commutative 54.7%

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

   6. Applied egg-rr 54.7%

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

      1. *-lft-identity 54.7%

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

      2. unpow-1 54.7%

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

      3. sqr-pow 54.7%

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

      4. rem-sqrt-square 54.7%

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

      5. metadata-eval 54.7%

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

      6. sqr-pow 54.6%

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

      7. fabs-sqr 54.6%

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

      8. sqr-pow 54.7%

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

   8. Simplified 54.7%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]
   9. Step-by-step derivation

      1. unpow-prod-down 76.2%

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

   10. Applied egg-rr 76.2%

       \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\right) \cdot \mathsf{fma}\left(h, {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{-0.5}{\ell}, 1\right)\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{+196}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{D \cdot \frac{h}{d}}{\frac{\frac{d}{M}}{D \cdot M}}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 6: 67.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.22 \cdot 10^{-101}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \left(-1 - \frac{-0.5}{\frac{\frac{\ell}{h}}{{\left(D \cdot \frac{0.5}{\frac{d}{M}}\right)}^{2}}}\right)\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{-186}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\frac{\ell}{h \cdot {\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}^{2}}}\right)\\ \mathbf{elif}\;\ell \leq 7.3 \cdot 10^{+195}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(D \cdot \left(\left(\left(M \cdot M\right) \cdot \frac{D}{d}\right) \cdot \frac{h}{d}\right)\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1.22e-101)
   (*
    (* d (sqrt (/ (/ 1.0 h) l)))
    (- -1.0 (/ -0.5 (/ (/ l h) (pow (* D (/ 0.5 (/ d M))) 2.0)))))
   (if (<= l 1.45e-186)
     (*
      (sqrt (* (/ d h) (/ d l)))
      (+ 1.0 (/ -0.5 (/ l (* h (pow (* D (* (/ M d) 0.5)) 2.0))))))
     (if (<= l 7.3e+195)
       (*
        (* (sqrt (/ d h)) (sqrt (/ d l)))
        (- 1.0 (/ (* 0.125 (* D (* (* (* M M) (/ D d)) (/ h d)))) l)))
       (* d (* (pow h -0.5) (pow l -0.5)))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.22e-101) {
		tmp = (d * sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 / ((l / h) / pow((D * (0.5 / (d / M))), 2.0))));
	} else if (l <= 1.45e-186) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 / (l / (h * pow((D * ((M / d) * 0.5)), 2.0)))));
	} else if (l <= 7.3e+195) {
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - ((0.125 * (D * (((M * M) * (D / d)) * (h / d)))) / l));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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 <= (-1.22d-101)) then
        tmp = (d * sqrt(((1.0d0 / h) / l))) * ((-1.0d0) - ((-0.5d0) / ((l / h) / ((d_1 * (0.5d0 / (d / m))) ** 2.0d0))))
    else if (l <= 1.45d-186) then
        tmp = sqrt(((d / h) * (d / l))) * (1.0d0 + ((-0.5d0) / (l / (h * ((d_1 * ((m / d) * 0.5d0)) ** 2.0d0)))))
    else if (l <= 7.3d+195) then
        tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0d0 - ((0.125d0 * (d_1 * (((m * m) * (d_1 / d)) * (h / d)))) / l))
    else
        tmp = d * ((h ** (-0.5d0)) * (l ** (-0.5d0)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.22e-101) {
		tmp = (d * Math.sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 / ((l / h) / Math.pow((D * (0.5 / (d / M))), 2.0))));
	} else if (l <= 1.45e-186) {
		tmp = Math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 / (l / (h * Math.pow((D * ((M / d) * 0.5)), 2.0)))));
	} else if (l <= 7.3e+195) {
		tmp = (Math.sqrt((d / h)) * Math.sqrt((d / l))) * (1.0 - ((0.125 * (D * (((M * M) * (D / d)) * (h / d)))) / l));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= -1.22e-101:
		tmp = (d * math.sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 / ((l / h) / math.pow((D * (0.5 / (d / M))), 2.0))))
	elif l <= 1.45e-186:
		tmp = math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 / (l / (h * math.pow((D * ((M / d) * 0.5)), 2.0)))))
	elif l <= 7.3e+195:
		tmp = (math.sqrt((d / h)) * math.sqrt((d / l))) * (1.0 - ((0.125 * (D * (((M * M) * (D / d)) * (h / d)))) / l))
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1.22e-101)
		tmp = Float64(Float64(d * sqrt(Float64(Float64(1.0 / h) / l))) * Float64(-1.0 - Float64(-0.5 / Float64(Float64(l / h) / (Float64(D * Float64(0.5 / Float64(d / M))) ^ 2.0)))));
	elseif (l <= 1.45e-186)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 + Float64(-0.5 / Float64(l / Float64(h * (Float64(D * Float64(Float64(M / d) * 0.5)) ^ 2.0))))));
	elseif (l <= 7.3e+195)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))) * Float64(1.0 - Float64(Float64(0.125 * Float64(D * Float64(Float64(Float64(M * M) * Float64(D / d)) * Float64(h / d)))) / l)));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -1.22e-101)
		tmp = (d * sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 / ((l / h) / ((D * (0.5 / (d / M))) ^ 2.0))));
	elseif (l <= 1.45e-186)
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 / (l / (h * ((D * ((M / d) * 0.5)) ^ 2.0)))));
	elseif (l <= 7.3e+195)
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - ((0.125 * (D * (((M * M) * (D / d)) * (h / d)))) / l));
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1.22e-101], N[(N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(-0.5 / N[(N[(l / h), $MachinePrecision] / N[Power[N[(D * N[(0.5 / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.45e-186], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 / N[(l / N[(h * N[Power[N[(D * N[(N[(M / d), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 7.3e+195], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(0.125 * N[(D * N[(N[(N[(M * M), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.2199999999999999e-101

   1. Initial program 60.6%

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

      1. metadata-eval 60.6%

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

      2. unpow1/2 60.6%

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

      3. metadata-eval 60.6%

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

      4. unpow1/2 60.6%

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

      5. *-commutative 60.6%

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

      6. associate-*l* 60.6%

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

      7. times-frac 60.6%

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

      8. metadata-eval 60.6%

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

   3. Simplified 60.6%

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

      1. associate-*r* 60.6%

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

      2. frac-times 60.6%

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

      3. *-commutative 60.6%

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

      4. metadata-eval 60.6%

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

      5. associate-*r/ 60.5%

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

      6. metadata-eval 60.5%

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

      7. *-commutative 60.5%

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

      8. frac-times 60.5%

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

      9. associate-*l/ 60.5%

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

      10. associate-*r/ 60.5%

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

      11. div-inv 60.5%

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

      12. metadata-eval 60.5%

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

   5. Applied egg-rr 60.5%

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

      1. pow1 60.5%

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

      2. *-commutative 60.5%

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

      3. associate-/l* 60.5%

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

      4. *-commutative 60.5%

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

      5. associate-*l/ 60.5%

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

      6. sqrt-unprod 44.4%

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

      7. *-commutative 44.4%

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

      8. frac-times 33.6%

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

      9. *-commutative 33.6%

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

   7. Applied egg-rr 33.6%

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

      1. unpow1 33.6%

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

      2. *-commutative 33.6%

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

      3. times-frac 44.4%

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

      4. sub-neg 44.4%

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

      5. associate-/l* 44.4%

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

      6. distribute-neg-frac 44.4%

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

      7. metadata-eval 44.4%

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

   9. Simplified 43.2%

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

   10. Taylor expanded in d around -inf 68.1%

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

       1. mul-1-neg 68.1%

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

       2. *-commutative 68.1%

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

       3. distribute-rgt-neg-in 68.1%

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

       4. *-commutative 68.1%

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

       5. associate-/r* 68.1%

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

   12. Simplified 68.1%

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

   if -1.2199999999999999e-101 < l < 1.4500000000000001e-186

   1. Initial program 77.5%

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

      1. metadata-eval 77.5%

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

      2. unpow1/2 77.5%

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

      3. metadata-eval 77.5%

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

      4. unpow1/2 77.5%

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

      5. *-commutative 77.5%

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

      6. associate-*l* 77.5%

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

      7. times-frac 78.0%

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

      8. metadata-eval 78.0%

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

   3. Simplified 78.0%

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

      1. associate-*r* 78.0%

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

      2. frac-times 77.5%

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

      3. *-commutative 77.5%

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

      4. metadata-eval 77.5%

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

      5. associate-*r/ 84.8%

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

      6. metadata-eval 84.8%

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

      7. *-commutative 84.8%

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

      8. frac-times 85.3%

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

      9. associate-*l/ 85.3%

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

      10. associate-*r/ 85.3%

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

      11. div-inv 85.3%

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

      12. metadata-eval 85.3%

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

   5. Applied egg-rr 85.3%

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

      1. pow1 85.3%

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

      2. *-commutative 85.3%

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

      3. associate-/l* 78.1%

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

      4. *-commutative 78.1%

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

      5. associate-*l/ 78.1%

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

      6. sqrt-unprod 73.0%

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

      7. *-commutative 73.0%

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

      8. frac-times 48.8%

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

      9. *-commutative 48.8%

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

   7. Applied egg-rr 48.8%

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

      1. unpow1 48.8%

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

      2. *-commutative 48.8%

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

      3. times-frac 73.0%

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

      4. sub-neg 73.0%

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

      5. associate-/l* 73.0%

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

      6. distribute-neg-frac 73.0%

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

      7. metadata-eval 73.0%

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

   9. Simplified 73.0%

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

       1. *-un-lft-identity 73.0%

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

       2. associate-/l/ 80.3%

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

       3. div-inv 80.3%

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

       4. clear-num 80.3%

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

   11. Applied egg-rr 80.3%

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

       1. *-lft-identity 80.3%

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

       2. *-commutative 80.3%

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

   13. Simplified 80.3%

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

   if 1.4500000000000001e-186 < l < 7.3000000000000004e195

   1. Initial program 77.5%

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

      1. metadata-eval 77.5%

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

      2. unpow1/2 77.5%

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

      3. metadata-eval 77.5%

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

      4. unpow1/2 77.5%

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

      5. *-commutative 77.5%

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

      6. associate-*l* 77.5%

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

      7. times-frac 77.5%

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

      8. metadata-eval 77.5%

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

   3. Simplified 77.5%

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

      1. associate-*r* 77.5%

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

      2. frac-times 77.5%

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

      3. *-commutative 77.5%

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

      4. metadata-eval 77.5%

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

      5. associate-*r/ 80.0%

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

      6. metadata-eval 80.0%

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

      7. *-commutative 80.0%

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

      8. frac-times 80.1%

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

      9. associate-*l/ 80.1%

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

      10. associate-*r/ 80.1%

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

      11. div-inv 80.1%

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

      12. metadata-eval 80.1%

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

   5. Applied egg-rr 80.1%

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

   6. Taylor expanded in M around 0 63.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}}{\ell}\right) \]
   7. Step-by-step derivation

      1. *-commutative 63.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{{D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{{d}^{2}}}{\ell}\right) \]

      2. *-commutative 63.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{{d}^{2}}}{\ell}\right) \]

      3. associate-*r* 62.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2}}}{\ell}\right) \]

      4. unpow2 62.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}{\color{blue}{d \cdot d}}}{\ell}\right) \]

      5. times-frac 67.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \frac{h}{d}\right)}}{\ell}\right) \]

      6. associate-/l* 68.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}} \cdot \frac{h}{d}\right)}{\ell}\right) \]

      7. unpow2 68.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{\frac{d}{{M}^{2}}} \cdot \frac{h}{d}\right)}{\ell}\right) \]

      8. associate-/l* 75.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\color{blue}{\frac{D}{\frac{\frac{d}{{M}^{2}}}{D}}} \cdot \frac{h}{d}\right)}{\ell}\right) \]

      9. unpow2 75.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\frac{D}{\frac{\frac{d}{\color{blue}{M \cdot M}}}{D}} \cdot \frac{h}{d}\right)}{\ell}\right) \]

      10. associate-/r* 76.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\frac{D}{\frac{\color{blue}{\frac{\frac{d}{M}}{M}}}{D}} \cdot \frac{h}{d}\right)}{\ell}\right) \]

   8. Simplified 76.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{0.125 \cdot \left(\frac{D}{\frac{\frac{\frac{d}{M}}{M}}{D}} \cdot \frac{h}{d}\right)}}{\ell}\right) \]

   9. Taylor expanded in D around 0 63.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}}{\ell}\right) \]
   10. Step-by-step derivation

       1. associate-*r* 62.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2}}}{\ell}\right) \]

       2. unpow2 62.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}{\color{blue}{d \cdot d}}}{\ell}\right) \]

       3. times-frac 67.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \frac{h}{d}\right)}}{\ell}\right) \]

       4. associate-/l* 68.5%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}} \cdot \frac{h}{d}\right)}{\ell}\right) \]

       5. unpow2 68.5%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{\frac{d}{{M}^{2}}} \cdot \frac{h}{d}\right)}{\ell}\right) \]

       6. unpow2 68.5%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{\color{blue}{M \cdot M}}} \cdot \frac{h}{d}\right)}{\ell}\right) \]

       7. associate-*l/ 75.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\color{blue}{\left(\frac{D}{\frac{d}{M \cdot M}} \cdot D\right)} \cdot \frac{h}{d}\right)}{\ell}\right) \]

       8. *-commutative 75.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\color{blue}{\left(D \cdot \frac{D}{\frac{d}{M \cdot M}}\right)} \cdot \frac{h}{d}\right)}{\ell}\right) \]

       9. associate-*l* 76.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \color{blue}{\left(D \cdot \left(\frac{D}{\frac{d}{M \cdot M}} \cdot \frac{h}{d}\right)\right)}}{\ell}\right) \]

       10. unpow2 76.2%

           \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(D \cdot \left(\frac{D}{\frac{d}{\color{blue}{{M}^{2}}}} \cdot \frac{h}{d}\right)\right)}{\ell}\right) \]

       11. associate-/r/ 76.2%

           \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(D \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot {M}^{2}\right)} \cdot \frac{h}{d}\right)\right)}{\ell}\right) \]

       12. *-commutative 76.2%

           \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(D \cdot \left(\color{blue}{\left({M}^{2} \cdot \frac{D}{d}\right)} \cdot \frac{h}{d}\right)\right)}{\ell}\right) \]

       13. unpow2 76.2%

           \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(D \cdot \left(\left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{D}{d}\right) \cdot \frac{h}{d}\right)\right)}{\ell}\right) \]

   11. Simplified 76.2%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \color{blue}{\left(D \cdot \left(\left(\left(M \cdot M\right) \cdot \frac{D}{d}\right) \cdot \frac{h}{d}\right)\right)}}{\ell}\right) \]

   if 7.3000000000000004e195 < l

   1. Initial program 50.5%

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

      1. metadata-eval 50.5%

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

      2. unpow1/2 50.5%

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

      3. metadata-eval 50.5%

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

      4. unpow1/2 50.5%

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

      5. *-commutative 50.5%

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

      6. associate-*l* 50.5%

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

      7. times-frac 50.5%

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

      8. metadata-eval 50.5%

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

   3. Simplified 50.5%

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

   4. Taylor expanded in d around inf 54.7%

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

      1. *-un-lft-identity 54.7%

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

      2. *-commutative 54.7%

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

   6. Applied egg-rr 54.7%

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

      1. *-lft-identity 54.7%

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

      2. unpow-1 54.7%

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

      3. sqr-pow 54.7%

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

      4. rem-sqrt-square 54.7%

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

      5. metadata-eval 54.7%

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

      6. sqr-pow 54.6%

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

      7. fabs-sqr 54.6%

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

      8. sqr-pow 54.7%

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

   8. Simplified 54.7%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]
   9. Step-by-step derivation

      1. unpow-prod-down 76.2%

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

   10. Applied egg-rr 76.2%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.22 \cdot 10^{-101}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \left(-1 - \frac{-0.5}{\frac{\frac{\ell}{h}}{{\left(D \cdot \frac{0.5}{\frac{d}{M}}\right)}^{2}}}\right)\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{-186}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\frac{\ell}{h \cdot {\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}^{2}}}\right)\\ \mathbf{elif}\;\ell \leq 7.3 \cdot 10^{+195}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(D \cdot \left(\left(\left(M \cdot M\right) \cdot \frac{D}{d}\right) \cdot \frac{h}{d}\right)\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 7: 68.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.75 \cdot 10^{-96}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \left(-1 - \frac{-0.5}{\frac{\frac{\ell}{h}}{{\left(D \cdot \frac{0.5}{\frac{d}{M}}\right)}^{2}}}\right)\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+195}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{D \cdot \frac{h}{d}}{\frac{\frac{d}{M}}{D \cdot M}}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1.75e-96)
   (*
    (* d (sqrt (/ (/ 1.0 h) l)))
    (- -1.0 (/ -0.5 (/ (/ l h) (pow (* D (/ 0.5 (/ d M))) 2.0)))))
   (if (<= l 5e+195)
     (*
      (* (sqrt (/ d h)) (sqrt (/ d l)))
      (- 1.0 (/ (* 0.125 (/ (* D (/ h d)) (/ (/ d M) (* D M)))) l)))
     (* d (* (pow h -0.5) (pow l -0.5))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.75e-96) {
		tmp = (d * sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 / ((l / h) / pow((D * (0.5 / (d / M))), 2.0))));
	} else if (l <= 5e+195) {
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - ((0.125 * ((D * (h / d)) / ((d / M) / (D * M)))) / l));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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 <= (-1.75d-96)) then
        tmp = (d * sqrt(((1.0d0 / h) / l))) * ((-1.0d0) - ((-0.5d0) / ((l / h) / ((d_1 * (0.5d0 / (d / m))) ** 2.0d0))))
    else if (l <= 5d+195) then
        tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0d0 - ((0.125d0 * ((d_1 * (h / d)) / ((d / m) / (d_1 * m)))) / l))
    else
        tmp = d * ((h ** (-0.5d0)) * (l ** (-0.5d0)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.75e-96) {
		tmp = (d * Math.sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 / ((l / h) / Math.pow((D * (0.5 / (d / M))), 2.0))));
	} else if (l <= 5e+195) {
		tmp = (Math.sqrt((d / h)) * Math.sqrt((d / l))) * (1.0 - ((0.125 * ((D * (h / d)) / ((d / M) / (D * M)))) / l));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= -1.75e-96:
		tmp = (d * math.sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 / ((l / h) / math.pow((D * (0.5 / (d / M))), 2.0))))
	elif l <= 5e+195:
		tmp = (math.sqrt((d / h)) * math.sqrt((d / l))) * (1.0 - ((0.125 * ((D * (h / d)) / ((d / M) / (D * M)))) / l))
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1.75e-96)
		tmp = Float64(Float64(d * sqrt(Float64(Float64(1.0 / h) / l))) * Float64(-1.0 - Float64(-0.5 / Float64(Float64(l / h) / (Float64(D * Float64(0.5 / Float64(d / M))) ^ 2.0)))));
	elseif (l <= 5e+195)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))) * Float64(1.0 - Float64(Float64(0.125 * Float64(Float64(D * Float64(h / d)) / Float64(Float64(d / M) / Float64(D * M)))) / l)));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -1.75e-96)
		tmp = (d * sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 / ((l / h) / ((D * (0.5 / (d / M))) ^ 2.0))));
	elseif (l <= 5e+195)
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - ((0.125 * ((D * (h / d)) / ((d / M) / (D * M)))) / l));
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1.75e-96], N[(N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(-0.5 / N[(N[(l / h), $MachinePrecision] / N[Power[N[(D * N[(0.5 / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5e+195], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(0.125 * N[(N[(D * N[(h / d), $MachinePrecision]), $MachinePrecision] / N[(N[(d / M), $MachinePrecision] / N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.7499999999999999e-96

   1. Initial program 60.8%

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

      1. metadata-eval 60.8%

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

      2. unpow1/2 60.8%

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

      3. metadata-eval 60.8%

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

      4. unpow1/2 60.8%

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

      5. *-commutative 60.8%

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

      6. associate-*l* 60.8%

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

      7. times-frac 60.8%

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

      8. metadata-eval 60.8%

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

   3. Simplified 60.8%

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

      1. associate-*r* 60.8%

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

      2. frac-times 60.8%

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

      3. *-commutative 60.8%

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

      4. metadata-eval 60.8%

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

      5. associate-*r/ 60.6%

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

      6. metadata-eval 60.6%

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

      7. *-commutative 60.6%

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

      8. frac-times 60.7%

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

      9. associate-*l/ 60.7%

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

      10. associate-*r/ 60.7%

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

      11. div-inv 60.7%

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

      12. metadata-eval 60.7%

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

   5. Applied egg-rr 60.7%

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

      1. pow1 60.7%

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

      2. *-commutative 60.7%

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

      3. associate-/l* 60.7%

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

      4. *-commutative 60.7%

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

      5. associate-*l/ 60.7%

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

      6. sqrt-unprod 44.2%

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

      7. *-commutative 44.2%

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

      8. frac-times 33.2%

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

      9. *-commutative 33.2%

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

   7. Applied egg-rr 33.2%

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

      1. unpow1 33.2%

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

      2. *-commutative 33.2%

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

      3. times-frac 44.2%

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

      4. sub-neg 44.2%

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

      5. associate-/l* 44.3%

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

      6. distribute-neg-frac 44.3%

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

      7. metadata-eval 44.3%

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

   9. Simplified 43.0%

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

   10. Taylor expanded in d around -inf 68.4%

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

       1. mul-1-neg 68.4%

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

       2. *-commutative 68.4%

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

       3. distribute-rgt-neg-in 68.4%

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

       4. *-commutative 68.4%

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

       5. associate-/r* 68.5%

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

   12. Simplified 68.5%

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

   if -1.7499999999999999e-96 < l < 4.9999999999999998e195

   1. Initial program 77.2%

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

      1. metadata-eval 77.2%

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

      2. unpow1/2 77.2%

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

      3. metadata-eval 77.2%

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

      4. unpow1/2 77.2%

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

      5. *-commutative 77.2%

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

      6. associate-*l* 77.2%

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

      7. times-frac 77.4%

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

      8. metadata-eval 77.4%

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

   3. Simplified 77.4%

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

      1. associate-*r* 77.4%

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

      2. frac-times 77.2%

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

      3. *-commutative 77.2%

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

      4. metadata-eval 77.2%

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

      5. associate-*r/ 81.9%

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

      6. metadata-eval 81.9%

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

      7. *-commutative 81.9%

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

      8. frac-times 82.2%

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

      9. associate-*l/ 82.2%

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

      10. associate-*r/ 82.2%

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

      11. div-inv 82.2%

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

      12. metadata-eval 82.2%

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

   5. Applied egg-rr 82.2%

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

   6. Taylor expanded in M around 0 58.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}}{\ell}\right) \]
   7. Step-by-step derivation

      1. *-commutative 58.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{{D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{{d}^{2}}}{\ell}\right) \]

      2. *-commutative 58.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{{d}^{2}}}{\ell}\right) \]

      3. associate-*r* 58.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2}}}{\ell}\right) \]

      4. unpow2 58.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}{\color{blue}{d \cdot d}}}{\ell}\right) \]

      5. times-frac 67.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \frac{h}{d}\right)}}{\ell}\right) \]

      6. associate-/l* 68.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}} \cdot \frac{h}{d}\right)}{\ell}\right) \]

      7. unpow2 68.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{\frac{d}{{M}^{2}}} \cdot \frac{h}{d}\right)}{\ell}\right) \]

      8. associate-/l* 75.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\color{blue}{\frac{D}{\frac{\frac{d}{{M}^{2}}}{D}}} \cdot \frac{h}{d}\right)}{\ell}\right) \]

      9. unpow2 75.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\frac{D}{\frac{\frac{d}{\color{blue}{M \cdot M}}}{D}} \cdot \frac{h}{d}\right)}{\ell}\right) \]

      10. associate-/r* 76.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\frac{D}{\frac{\color{blue}{\frac{\frac{d}{M}}{M}}}{D}} \cdot \frac{h}{d}\right)}{\ell}\right) \]

   8. Simplified 76.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{0.125 \cdot \left(\frac{D}{\frac{\frac{\frac{d}{M}}{M}}{D}} \cdot \frac{h}{d}\right)}}{\ell}\right) \]
   9. Step-by-step derivation

      1. associate-*l/ 75.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \color{blue}{\frac{D \cdot \frac{h}{d}}{\frac{\frac{\frac{d}{M}}{M}}{D}}}}{\ell}\right) \]

      2. associate-/l/ 78.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{D \cdot \frac{h}{d}}{\color{blue}{\frac{\frac{d}{M}}{D \cdot M}}}}{\ell}\right) \]

   10. Applied egg-rr 78.0%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \color{blue}{\frac{D \cdot \frac{h}{d}}{\frac{\frac{d}{M}}{D \cdot M}}}}{\ell}\right) \]

   if 4.9999999999999998e195 < l

   1. Initial program 50.5%

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

      1. metadata-eval 50.5%

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

      2. unpow1/2 50.5%

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

      3. metadata-eval 50.5%

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

      4. unpow1/2 50.5%

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

      5. *-commutative 50.5%

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

      6. associate-*l* 50.5%

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

      7. times-frac 50.5%

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

      8. metadata-eval 50.5%

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

   3. Simplified 50.5%

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

   4. Taylor expanded in d around inf 54.7%

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

      1. *-un-lft-identity 54.7%

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

      2. *-commutative 54.7%

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

   6. Applied egg-rr 54.7%

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

      1. *-lft-identity 54.7%

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

      2. unpow-1 54.7%

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

      3. sqr-pow 54.7%

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

      4. rem-sqrt-square 54.7%

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

      5. metadata-eval 54.7%

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

      6. sqr-pow 54.6%

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

      7. fabs-sqr 54.6%

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

      8. sqr-pow 54.7%

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

   8. Simplified 54.7%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]
   9. Step-by-step derivation

      1. unpow-prod-down 76.2%

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

   10. Applied egg-rr 76.2%

       \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.75 \cdot 10^{-96}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \left(-1 - \frac{-0.5}{\frac{\frac{\ell}{h}}{{\left(D \cdot \frac{0.5}{\frac{d}{M}}\right)}^{2}}}\right)\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+195}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{D \cdot \frac{h}{d}}{\frac{\frac{d}{M}}{D \cdot M}}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 8: 58.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;M \leq 9.5 \cdot 10^{-260}:\\ \;\;\;\;t_0 \cdot t_1\\ \mathbf{elif}\;M \leq 210000000000:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\frac{\ell}{h \cdot {\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(-0.125 \cdot \frac{D \cdot \left(\frac{M}{d} \cdot \left(h \cdot \frac{M}{d}\right)\right)}{\frac{\ell}{D}}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h))) (t_1 (sqrt (/ d l))))
   (if (<= M 9.5e-260)
     (* t_0 t_1)
     (if (<= M 210000000000.0)
       (*
        (sqrt (* (/ d h) (/ d l)))
        (+ 1.0 (/ -0.5 (/ l (* h (pow (* D (* (/ M d) 0.5)) 2.0))))))
       (*
        t_0
        (* t_1 (* -0.125 (/ (* D (* (/ M d) (* h (/ M d)))) (/ l D)))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h));
	double t_1 = sqrt((d / l));
	double tmp;
	if (M <= 9.5e-260) {
		tmp = t_0 * t_1;
	} else if (M <= 210000000000.0) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 / (l / (h * pow((D * ((M / d) * 0.5)), 2.0)))));
	} else {
		tmp = t_0 * (t_1 * (-0.125 * ((D * ((M / d) * (h * (M / d)))) / (l / D))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((d / h))
    t_1 = sqrt((d / l))
    if (m <= 9.5d-260) then
        tmp = t_0 * t_1
    else if (m <= 210000000000.0d0) then
        tmp = sqrt(((d / h) * (d / l))) * (1.0d0 + ((-0.5d0) / (l / (h * ((d_1 * ((m / d) * 0.5d0)) ** 2.0d0)))))
    else
        tmp = t_0 * (t_1 * ((-0.125d0) * ((d_1 * ((m / d) * (h * (m / d)))) / (l / d_1))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((d / h));
	double t_1 = Math.sqrt((d / l));
	double tmp;
	if (M <= 9.5e-260) {
		tmp = t_0 * t_1;
	} else if (M <= 210000000000.0) {
		tmp = Math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 / (l / (h * Math.pow((D * ((M / d) * 0.5)), 2.0)))));
	} else {
		tmp = t_0 * (t_1 * (-0.125 * ((D * ((M / d) * (h * (M / d)))) / (l / D))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.sqrt((d / h))
	t_1 = math.sqrt((d / l))
	tmp = 0
	if M <= 9.5e-260:
		tmp = t_0 * t_1
	elif M <= 210000000000.0:
		tmp = math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 / (l / (h * math.pow((D * ((M / d) * 0.5)), 2.0)))))
	else:
		tmp = t_0 * (t_1 * (-0.125 * ((D * ((M / d) * (h * (M / d)))) / (l / D))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / h))
	t_1 = sqrt(Float64(d / l))
	tmp = 0.0
	if (M <= 9.5e-260)
		tmp = Float64(t_0 * t_1);
	elseif (M <= 210000000000.0)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 + Float64(-0.5 / Float64(l / Float64(h * (Float64(D * Float64(Float64(M / d) * 0.5)) ^ 2.0))))));
	else
		tmp = Float64(t_0 * Float64(t_1 * Float64(-0.125 * Float64(Float64(D * Float64(Float64(M / d) * Float64(h * Float64(M / d)))) / Float64(l / D)))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / h));
	t_1 = sqrt((d / l));
	tmp = 0.0;
	if (M <= 9.5e-260)
		tmp = t_0 * t_1;
	elseif (M <= 210000000000.0)
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 / (l / (h * ((D * ((M / d) * 0.5)) ^ 2.0)))));
	else
		tmp = t_0 * (t_1 * (-0.125 * ((D * ((M / d) * (h * (M / d)))) / (l / D))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[M, 9.5e-260], N[(t$95$0 * t$95$1), $MachinePrecision], If[LessEqual[M, 210000000000.0], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 / N[(l / N[(h * N[Power[N[(D * N[(N[(M / d), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(t$95$1 * N[(-0.125 * N[(N[(D * N[(N[(M / d), $MachinePrecision] * N[(h * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if M < 9.5000000000000001e-260

   1. Initial program 68.4%

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

      1. associate-*l* 68.4%

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

      2. metadata-eval 68.4%

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

      3. unpow1/2 68.4%

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

      4. metadata-eval 68.4%

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

      5. unpow1/2 68.4%

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

      6. sub-neg 68.4%

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

      7. +-commutative 68.4%

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

      8. *-commutative 68.4%

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

      9. distribute-rgt-neg-in 68.4%

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

      10. fma-def 68.4%

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

   3. Simplified 67.9%

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

   4. Taylor expanded in h around 0 39.3%

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

   if 9.5000000000000001e-260 < M < 2.1e11

   1. Initial program 73.9%

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

      1. metadata-eval 73.9%

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

      2. unpow1/2 73.9%

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

      3. metadata-eval 73.9%

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

      4. unpow1/2 73.9%

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

      5. *-commutative 73.9%

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

      6. associate-*l* 73.9%

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

      7. times-frac 74.0%

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

      8. metadata-eval 74.0%

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

   3. Simplified 74.0%

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

      1. associate-*r* 74.0%

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

      2. frac-times 73.9%

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

      3. *-commutative 73.9%

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

      4. metadata-eval 73.9%

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

      5. associate-*r/ 74.1%

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

      6. metadata-eval 74.1%

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

      7. *-commutative 74.1%

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

      8. frac-times 74.1%

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

      9. associate-*l/ 74.1%

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

      10. associate-*r/ 74.1%

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

      11. div-inv 74.1%

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

      12. metadata-eval 74.1%

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

   5. Applied egg-rr 74.1%

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

      1. pow1 74.1%

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

      2. *-commutative 74.1%

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

      3. associate-/l* 73.9%

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

      4. *-commutative 73.9%

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

      5. associate-*l/ 73.9%

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

      6. sqrt-unprod 63.8%

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

      7. *-commutative 63.8%

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

      8. frac-times 44.4%

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

      9. *-commutative 44.4%

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

   7. Applied egg-rr 44.4%

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

      1. unpow1 44.4%

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

      2. *-commutative 44.4%

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

      3. times-frac 63.8%

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

      4. sub-neg 63.8%

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

      5. associate-/l* 63.8%

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

      6. distribute-neg-frac 63.8%

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

      7. metadata-eval 63.8%

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

   9. Simplified 63.8%

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

       1. *-un-lft-identity 63.8%

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

       2. associate-/l/ 63.9%

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

       3. div-inv 63.9%

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

       4. clear-num 63.9%

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

   11. Applied egg-rr 63.9%

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

       1. *-lft-identity 63.9%

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

       2. *-commutative 63.9%

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

   13. Simplified 63.9%

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

   if 2.1e11 < M

   1. Initial program 69.4%

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

      1. associate-*l* 69.4%

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

      2. metadata-eval 69.4%

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

      3. unpow1/2 69.4%

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

      4. metadata-eval 69.4%

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

      5. unpow1/2 69.4%

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

      6. sub-neg 69.4%

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

      7. +-commutative 69.4%

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

      8. *-commutative 69.4%

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

      9. distribute-rgt-neg-in 69.4%

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

      10. fma-def 69.4%

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

   3. Simplified 69.4%

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

   4. Taylor expanded in h around inf 29.8%

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

      1. associate-*r/ 29.8%

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

      2. *-commutative 29.8%

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

      3. associate-*r/ 29.8%

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

      4. times-frac 31.7%

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

      5. unpow2 31.7%

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

      6. associate-/l* 34.0%

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

      7. associate-/l* 34.1%

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

      8. unpow2 34.1%

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

      9. unpow2 34.1%

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

      10. associate-*l/ 36.0%

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

      11. *-commutative 36.0%

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

      12. times-frac 38.1%

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

   6. Simplified 38.1%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(-0.125 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{M}{\frac{d}{h}}\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. associate-*l/ 36.1%

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

      2. associate-/r/ 38.3%

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

   8. Applied egg-rr 38.3%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \color{blue}{\frac{D \cdot \left(\frac{M}{d} \cdot \left(\frac{M}{d} \cdot h\right)\right)}{\frac{\ell}{D}}}\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 9.5 \cdot 10^{-260}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;M \leq 210000000000:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\frac{\ell}{h \cdot {\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \frac{D \cdot \left(\frac{M}{d} \cdot \left(h \cdot \frac{M}{d}\right)\right)}{\frac{\ell}{D}}\right)\right)\\ \end{array} \]

Alternative 9: 64.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.2 \cdot 10^{-101}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \left(-1 - \frac{-0.5}{\frac{\frac{\ell}{h}}{{\left(D \cdot \frac{0.5}{\frac{d}{M}}\right)}^{2}}}\right)\\ \mathbf{elif}\;\ell \leq 1.85 \cdot 10^{+198}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\frac{\ell}{h \cdot {\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1.2e-101)
   (*
    (* d (sqrt (/ (/ 1.0 h) l)))
    (- -1.0 (/ -0.5 (/ (/ l h) (pow (* D (/ 0.5 (/ d M))) 2.0)))))
   (if (<= l 1.85e+198)
     (*
      (sqrt (* (/ d h) (/ d l)))
      (+ 1.0 (/ -0.5 (/ l (* h (pow (* D (* (/ M d) 0.5)) 2.0))))))
     (* d (* (pow h -0.5) (pow l -0.5))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.2e-101) {
		tmp = (d * sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 / ((l / h) / pow((D * (0.5 / (d / M))), 2.0))));
	} else if (l <= 1.85e+198) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 / (l / (h * pow((D * ((M / d) * 0.5)), 2.0)))));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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 <= (-1.2d-101)) then
        tmp = (d * sqrt(((1.0d0 / h) / l))) * ((-1.0d0) - ((-0.5d0) / ((l / h) / ((d_1 * (0.5d0 / (d / m))) ** 2.0d0))))
    else if (l <= 1.85d+198) then
        tmp = sqrt(((d / h) * (d / l))) * (1.0d0 + ((-0.5d0) / (l / (h * ((d_1 * ((m / d) * 0.5d0)) ** 2.0d0)))))
    else
        tmp = d * ((h ** (-0.5d0)) * (l ** (-0.5d0)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.2e-101) {
		tmp = (d * Math.sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 / ((l / h) / Math.pow((D * (0.5 / (d / M))), 2.0))));
	} else if (l <= 1.85e+198) {
		tmp = Math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 / (l / (h * Math.pow((D * ((M / d) * 0.5)), 2.0)))));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= -1.2e-101:
		tmp = (d * math.sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 / ((l / h) / math.pow((D * (0.5 / (d / M))), 2.0))))
	elif l <= 1.85e+198:
		tmp = math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 / (l / (h * math.pow((D * ((M / d) * 0.5)), 2.0)))))
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1.2e-101)
		tmp = Float64(Float64(d * sqrt(Float64(Float64(1.0 / h) / l))) * Float64(-1.0 - Float64(-0.5 / Float64(Float64(l / h) / (Float64(D * Float64(0.5 / Float64(d / M))) ^ 2.0)))));
	elseif (l <= 1.85e+198)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 + Float64(-0.5 / Float64(l / Float64(h * (Float64(D * Float64(Float64(M / d) * 0.5)) ^ 2.0))))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -1.2e-101)
		tmp = (d * sqrt(((1.0 / h) / l))) * (-1.0 - (-0.5 / ((l / h) / ((D * (0.5 / (d / M))) ^ 2.0))));
	elseif (l <= 1.85e+198)
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 / (l / (h * ((D * ((M / d) * 0.5)) ^ 2.0)))));
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1.2e-101], N[(N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(-0.5 / N[(N[(l / h), $MachinePrecision] / N[Power[N[(D * N[(0.5 / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.85e+198], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 / N[(l / N[(h * N[Power[N[(D * N[(N[(M / d), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.2e-101

   1. Initial program 60.6%

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

      1. metadata-eval 60.6%

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

      2. unpow1/2 60.6%

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

      3. metadata-eval 60.6%

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

      4. unpow1/2 60.6%

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

      5. *-commutative 60.6%

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

      6. associate-*l* 60.6%

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

      7. times-frac 60.6%

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

      8. metadata-eval 60.6%

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

   3. Simplified 60.6%

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

      1. associate-*r* 60.6%

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

      2. frac-times 60.6%

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

      3. *-commutative 60.6%

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

      4. metadata-eval 60.6%

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

      5. associate-*r/ 60.5%

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

      6. metadata-eval 60.5%

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

      7. *-commutative 60.5%

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

      8. frac-times 60.5%

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

      9. associate-*l/ 60.5%

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

      10. associate-*r/ 60.5%

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

      11. div-inv 60.5%

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

      12. metadata-eval 60.5%

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

   5. Applied egg-rr 60.5%

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

      1. pow1 60.5%

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

      2. *-commutative 60.5%

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

      3. associate-/l* 60.5%

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

      4. *-commutative 60.5%

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

      5. associate-*l/ 60.5%

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

      6. sqrt-unprod 44.4%

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

      7. *-commutative 44.4%

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

      8. frac-times 33.6%

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

      9. *-commutative 33.6%

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

   7. Applied egg-rr 33.6%

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

      1. unpow1 33.6%

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

      2. *-commutative 33.6%

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

      3. times-frac 44.4%

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

      4. sub-neg 44.4%

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

      5. associate-/l* 44.4%

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

      6. distribute-neg-frac 44.4%

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

      7. metadata-eval 44.4%

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

   9. Simplified 43.2%

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

   10. Taylor expanded in d around -inf 68.1%

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

       1. mul-1-neg 68.1%

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

       2. *-commutative 68.1%

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

       3. distribute-rgt-neg-in 68.1%

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

       4. *-commutative 68.1%

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

       5. associate-/r* 68.1%

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

   12. Simplified 68.1%

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

   if -1.2e-101 < l < 1.8499999999999999e198

   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. Step-by-step derivation

      1. metadata-eval 77.6%

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

      2. unpow1/2 77.6%

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

      3. metadata-eval 77.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 77.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 77.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 77.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 77.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 77.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 77.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 77.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 77.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 77.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 77.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 82.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 82.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 82.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 82.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. associate-*l/ 82.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. associate-*r/ 82.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      11. div-inv 82.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{1}{2}\right)}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      12. metadata-eval 82.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \left(\frac{D}{d} \cdot \color{blue}{0.5}\right)\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 82.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. pow1 82.7%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1}} \]

      2. *-commutative 82.7%

         \[\leadsto {\color{blue}{\left(\left(1 - \frac{\left({\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}}^{1} \]

      3. associate-/l* 78.2%

         \[\leadsto {\left(\left(1 - \color{blue}{\frac{{\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}^{1} \]

      4. *-commutative 78.2%

         \[\leadsto {\left(\left(1 - \frac{\color{blue}{0.5 \cdot {\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2}}}{\frac{\ell}{h}}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}^{1} \]

      5. associate-*l/ 78.2%

         \[\leadsto {\left(\left(1 - \frac{0.5 \cdot {\left(M \cdot \color{blue}{\frac{D \cdot 0.5}{d}}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}^{1} \]

      6. sqrt-unprod 68.9%

         \[\leadsto {\left(\left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}}\right)}^{1} \]

      7. *-commutative 68.9%

         \[\leadsto {\left(\left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}}\right)}^{1} \]

      8. frac-times 50.4%

         \[\leadsto {\left(\left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \sqrt{\color{blue}{\frac{d \cdot d}{\ell \cdot h}}}\right)}^{1} \]

      9. *-commutative 50.4%

         \[\leadsto {\left(\left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}}\right)}^{1} \]

   7. Applied egg-rr 50.4%

      \[\leadsto \color{blue}{{\left(\left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \sqrt{\frac{d \cdot d}{h \cdot \ell}}\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 50.4%

         \[\leadsto \color{blue}{\left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \sqrt{\frac{d \cdot d}{h \cdot \ell}}} \]

      2. *-commutative 50.4%

         \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right)} \]

      3. times-frac 68.9%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \]

      4. sub-neg 68.9%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right)\right)} \]

      5. associate-/l* 68.9%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \left(-\color{blue}{\frac{0.5}{\frac{\frac{\ell}{h}}{{\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}}}\right)\right) \]

      6. distribute-neg-frac 68.9%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \color{blue}{\frac{-0.5}{\frac{\frac{\ell}{h}}{{\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}}}\right) \]

      7. metadata-eval 68.9%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{\color{blue}{-0.5}}{\frac{\frac{\ell}{h}}{{\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}}\right) \]

   9. Simplified 68.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\frac{\frac{\ell}{h}}{{\left(D \cdot \frac{0.5}{\frac{d}{M}}\right)}^{2}}}\right)} \]
   10. Step-by-step derivation

       1. *-un-lft-identity 68.9%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\color{blue}{1 \cdot \frac{\frac{\ell}{h}}{{\left(D \cdot \frac{0.5}{\frac{d}{M}}\right)}^{2}}}}\right) \]

       2. associate-/l/ 73.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{1 \cdot \color{blue}{\frac{\ell}{{\left(D \cdot \frac{0.5}{\frac{d}{M}}\right)}^{2} \cdot h}}}\right) \]

       3. div-inv 73.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{1 \cdot \frac{\ell}{{\left(D \cdot \color{blue}{\left(0.5 \cdot \frac{1}{\frac{d}{M}}\right)}\right)}^{2} \cdot h}}\right) \]

       4. clear-num 73.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{1 \cdot \frac{\ell}{{\left(D \cdot \left(0.5 \cdot \color{blue}{\frac{M}{d}}\right)\right)}^{2} \cdot h}}\right) \]

   11. Applied egg-rr 73.4%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\color{blue}{1 \cdot \frac{\ell}{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot h}}}\right) \]
   12. Step-by-step derivation

       1. *-lft-identity 73.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\color{blue}{\frac{\ell}{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot h}}}\right) \]

       2. *-commutative 73.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\frac{\ell}{\color{blue}{h \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}}}\right) \]

   13. Simplified 73.4%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\color{blue}{\frac{\ell}{h \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}}}\right) \]

   if 1.8499999999999999e198 < l

   1. Initial program 48.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 48.1%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 48.1%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 48.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 48.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 48.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 48.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 48.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 48.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 48.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 52.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 52.6%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. *-commutative 52.6%

         \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

   6. Applied egg-rr 52.6%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
   7. Step-by-step derivation

      1. *-lft-identity 52.6%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. unpow-1 52.6%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      3. sqr-pow 52.6%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      4. rem-sqrt-square 52.6%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      5. metadata-eval 52.6%

         \[\leadsto \left|{\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right| \cdot d \]

      6. sqr-pow 52.4%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)}}\right| \cdot d \]

      7. fabs-sqr 52.4%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)}\right)} \cdot d \]

      8. sqr-pow 52.6%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]

   8. Simplified 52.6%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]
   9. Step-by-step derivation

      1. unpow-prod-down 75.1%

         \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]

   10. Applied egg-rr 75.1%

       \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.2 \cdot 10^{-101}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \left(-1 - \frac{-0.5}{\frac{\frac{\ell}{h}}{{\left(D \cdot \frac{0.5}{\frac{d}{M}}\right)}^{2}}}\right)\\ \mathbf{elif}\;\ell \leq 1.85 \cdot 10^{+198}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\frac{\ell}{h \cdot {\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 10: 57.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 2.7 \cdot 10^{-259}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\frac{\ell}{h \cdot {\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}^{2}}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= M 2.7e-259)
   (* (sqrt (/ d h)) (sqrt (/ d l)))
   (*
    (sqrt (* (/ d h) (/ d l)))
    (+ 1.0 (/ -0.5 (/ l (* h (pow (* D (* (/ M d) 0.5)) 2.0))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (M <= 2.7e-259) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 / (l / (h * pow((D * ((M / d) * 0.5)), 2.0)))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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 (m <= 2.7d-259) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else
        tmp = sqrt(((d / h) * (d / l))) * (1.0d0 + ((-0.5d0) / (l / (h * ((d_1 * ((m / d) * 0.5d0)) ** 2.0d0)))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (M <= 2.7e-259) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else {
		tmp = Math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 / (l / (h * Math.pow((D * ((M / d) * 0.5)), 2.0)))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if M <= 2.7e-259:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	else:
		tmp = math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 / (l / (h * math.pow((D * ((M / d) * 0.5)), 2.0)))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (M <= 2.7e-259)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	else
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 + Float64(-0.5 / Float64(l / Float64(h * (Float64(D * Float64(Float64(M / d) * 0.5)) ^ 2.0))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (M <= 2.7e-259)
		tmp = sqrt((d / h)) * sqrt((d / l));
	else
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 / (l / (h * ((D * ((M / d) * 0.5)) ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[M, 2.7e-259], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 / N[(l / N[(h * N[Power[N[(D * N[(N[(M / d), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.69999999999999984e-259

   1. Initial program 68.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 68.4%

         \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      2. metadata-eval 68.4%

         \[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. unpow1/2 68.4%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval 68.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. unpow1/2 68.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. sub-neg 68.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      7. +-commutative 68.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right) \]

      8. *-commutative 68.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) + 1\right)\right) \]

      9. distribute-rgt-neg-in 68.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\frac{h}{\ell} \cdot \left(-\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} + 1\right)\right) \]

      10. fma-def 68.4%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{h}{\ell}, -\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}, 1\right)}\right) \]

   3. Simplified 67.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2} \cdot -0.5, 1\right)\right)} \]

   4. Taylor expanded in h around 0 39.3%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]

   if 2.69999999999999984e-259 < M

   1. Initial program 71.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 71.8%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 71.8%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 71.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 71.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 71.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 71.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 71.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 71.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 71.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 71.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 71.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 71.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 71.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 73.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 73.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 73.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. associate-*l/ 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. associate-*r/ 73.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      11. div-inv 73.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{1}{2}\right)}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      12. metadata-eval 73.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \left(\frac{D}{d} \cdot \color{blue}{0.5}\right)\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 73.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. pow1 73.9%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1}} \]

      2. *-commutative 73.9%

         \[\leadsto {\color{blue}{\left(\left(1 - \frac{\left({\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}}^{1} \]

      3. associate-/l* 71.8%

         \[\leadsto {\left(\left(1 - \color{blue}{\frac{{\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}^{1} \]

      4. *-commutative 71.8%

         \[\leadsto {\left(\left(1 - \frac{\color{blue}{0.5 \cdot {\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2}}}{\frac{\ell}{h}}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}^{1} \]

      5. associate-*l/ 71.8%

         \[\leadsto {\left(\left(1 - \frac{0.5 \cdot {\left(M \cdot \color{blue}{\frac{D \cdot 0.5}{d}}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}^{1} \]

      6. sqrt-unprod 62.8%

         \[\leadsto {\left(\left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}}\right)}^{1} \]

      7. *-commutative 62.8%

         \[\leadsto {\left(\left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}}\right)}^{1} \]

      8. frac-times 45.5%

         \[\leadsto {\left(\left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \sqrt{\color{blue}{\frac{d \cdot d}{\ell \cdot h}}}\right)}^{1} \]

      9. *-commutative 45.5%

         \[\leadsto {\left(\left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}}\right)}^{1} \]

   7. Applied egg-rr 45.5%

      \[\leadsto \color{blue}{{\left(\left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \sqrt{\frac{d \cdot d}{h \cdot \ell}}\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 45.5%

         \[\leadsto \color{blue}{\left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \sqrt{\frac{d \cdot d}{h \cdot \ell}}} \]

      2. *-commutative 45.5%

         \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right)} \]

      3. times-frac 62.8%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \]

      4. sub-neg 62.8%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right)\right)} \]

      5. associate-/l* 62.8%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \left(-\color{blue}{\frac{0.5}{\frac{\frac{\ell}{h}}{{\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}}}\right)\right) \]

      6. distribute-neg-frac 62.8%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \color{blue}{\frac{-0.5}{\frac{\frac{\ell}{h}}{{\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}}}\right) \]

      7. metadata-eval 62.8%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{\color{blue}{-0.5}}{\frac{\frac{\ell}{h}}{{\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}}\right) \]

   9. Simplified 62.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\frac{\frac{\ell}{h}}{{\left(D \cdot \frac{0.5}{\frac{d}{M}}\right)}^{2}}}\right)} \]
   10. Step-by-step derivation

       1. *-un-lft-identity 62.8%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\color{blue}{1 \cdot \frac{\frac{\ell}{h}}{{\left(D \cdot \frac{0.5}{\frac{d}{M}}\right)}^{2}}}}\right) \]

       2. associate-/l/ 64.9%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{1 \cdot \color{blue}{\frac{\ell}{{\left(D \cdot \frac{0.5}{\frac{d}{M}}\right)}^{2} \cdot h}}}\right) \]

       3. div-inv 64.9%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{1 \cdot \frac{\ell}{{\left(D \cdot \color{blue}{\left(0.5 \cdot \frac{1}{\frac{d}{M}}\right)}\right)}^{2} \cdot h}}\right) \]

       4. clear-num 64.9%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{1 \cdot \frac{\ell}{{\left(D \cdot \left(0.5 \cdot \color{blue}{\frac{M}{d}}\right)\right)}^{2} \cdot h}}\right) \]

   11. Applied egg-rr 64.9%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\color{blue}{1 \cdot \frac{\ell}{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot h}}}\right) \]
   12. Step-by-step derivation

       1. *-lft-identity 64.9%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\color{blue}{\frac{\ell}{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot h}}}\right) \]

       2. *-commutative 64.9%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\frac{\ell}{\color{blue}{h \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}}}\right) \]

   13. Simplified 64.9%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\color{blue}{\frac{\ell}{h \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.7 \cdot 10^{-259}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\frac{\ell}{h \cdot {\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}^{2}}}\right)\\ \end{array} \]

Alternative 11: 58.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ t_1 := t_0 \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(\frac{D}{\ell} \cdot \left(M \cdot \left(\frac{M}{d} \cdot \frac{h}{d}\right)\right)\right)\right)\right)\\ \mathbf{if}\;d \leq -1 \cdot 10^{+106}:\\ \;\;\;\;t_0 \cdot \left(1 + \frac{-0.5}{4 \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\ell}{M \cdot \left(h \cdot M\right)}\right)}\right)\\ \mathbf{elif}\;d \leq -7.4 \cdot 10^{-283}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 8 \cdot 10^{-215}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d}\right)\\ \mathbf{elif}\;d \leq 2.75 \cdot 10^{+138}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (* (/ d h) (/ d l))))
        (t_1
         (*
          t_0
          (+ 1.0 (* -0.125 (* D (* (/ D l) (* M (* (/ M d) (/ h d))))))))))
   (if (<= d -1e+106)
     (*
      t_0
      (+ 1.0 (/ -0.5 (* 4.0 (* (* (/ d D) (/ d D)) (/ l (* M (* h M))))))))
     (if (<= d -7.4e-283)
       t_1
       (if (<= d 8e-215)
         (* (sqrt (/ h (pow l 3.0))) (* -0.125 (/ (* (* D M) (* D M)) d)))
         (if (<= d 2.75e+138) t_1 (* d (* (pow h -0.5) (pow l -0.5)))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(((d / h) * (d / l)));
	double t_1 = t_0 * (1.0 + (-0.125 * (D * ((D / l) * (M * ((M / d) * (h / d)))))));
	double tmp;
	if (d <= -1e+106) {
		tmp = t_0 * (1.0 + (-0.5 / (4.0 * (((d / D) * (d / D)) * (l / (M * (h * M)))))));
	} else if (d <= -7.4e-283) {
		tmp = t_1;
	} else if (d <= 8e-215) {
		tmp = sqrt((h / pow(l, 3.0))) * (-0.125 * (((D * M) * (D * M)) / d));
	} else if (d <= 2.75e+138) {
		tmp = t_1;
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(((d / h) * (d / l)))
    t_1 = t_0 * (1.0d0 + ((-0.125d0) * (d_1 * ((d_1 / l) * (m * ((m / d) * (h / d)))))))
    if (d <= (-1d+106)) then
        tmp = t_0 * (1.0d0 + ((-0.5d0) / (4.0d0 * (((d / d_1) * (d / d_1)) * (l / (m * (h * m)))))))
    else if (d <= (-7.4d-283)) then
        tmp = t_1
    else if (d <= 8d-215) then
        tmp = sqrt((h / (l ** 3.0d0))) * ((-0.125d0) * (((d_1 * m) * (d_1 * m)) / d))
    else if (d <= 2.75d+138) then
        tmp = t_1
    else
        tmp = d * ((h ** (-0.5d0)) * (l ** (-0.5d0)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt(((d / h) * (d / l)));
	double t_1 = t_0 * (1.0 + (-0.125 * (D * ((D / l) * (M * ((M / d) * (h / d)))))));
	double tmp;
	if (d <= -1e+106) {
		tmp = t_0 * (1.0 + (-0.5 / (4.0 * (((d / D) * (d / D)) * (l / (M * (h * M)))))));
	} else if (d <= -7.4e-283) {
		tmp = t_1;
	} else if (d <= 8e-215) {
		tmp = Math.sqrt((h / Math.pow(l, 3.0))) * (-0.125 * (((D * M) * (D * M)) / d));
	} else if (d <= 2.75e+138) {
		tmp = t_1;
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.sqrt(((d / h) * (d / l)))
	t_1 = t_0 * (1.0 + (-0.125 * (D * ((D / l) * (M * ((M / d) * (h / d)))))))
	tmp = 0
	if d <= -1e+106:
		tmp = t_0 * (1.0 + (-0.5 / (4.0 * (((d / D) * (d / D)) * (l / (M * (h * M)))))))
	elif d <= -7.4e-283:
		tmp = t_1
	elif d <= 8e-215:
		tmp = math.sqrt((h / math.pow(l, 3.0))) * (-0.125 * (((D * M) * (D * M)) / d))
	elif d <= 2.75e+138:
		tmp = t_1
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(Float64(d / h) * Float64(d / l)))
	t_1 = Float64(t_0 * Float64(1.0 + Float64(-0.125 * Float64(D * Float64(Float64(D / l) * Float64(M * Float64(Float64(M / d) * Float64(h / d))))))))
	tmp = 0.0
	if (d <= -1e+106)
		tmp = Float64(t_0 * Float64(1.0 + Float64(-0.5 / Float64(4.0 * Float64(Float64(Float64(d / D) * Float64(d / D)) * Float64(l / Float64(M * Float64(h * M))))))));
	elseif (d <= -7.4e-283)
		tmp = t_1;
	elseif (d <= 8e-215)
		tmp = Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(-0.125 * Float64(Float64(Float64(D * M) * Float64(D * M)) / d)));
	elseif (d <= 2.75e+138)
		tmp = t_1;
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(((d / h) * (d / l)));
	t_1 = t_0 * (1.0 + (-0.125 * (D * ((D / l) * (M * ((M / d) * (h / d)))))));
	tmp = 0.0;
	if (d <= -1e+106)
		tmp = t_0 * (1.0 + (-0.5 / (4.0 * (((d / D) * (d / D)) * (l / (M * (h * M)))))));
	elseif (d <= -7.4e-283)
		tmp = t_1;
	elseif (d <= 8e-215)
		tmp = sqrt((h / (l ^ 3.0))) * (-0.125 * (((D * M) * (D * M)) / d));
	elseif (d <= 2.75e+138)
		tmp = t_1;
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(1.0 + N[(-0.125 * N[(D * N[(N[(D / l), $MachinePrecision] * N[(M * N[(N[(M / d), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1e+106], N[(t$95$0 * N[(1.0 + N[(-0.5 / N[(4.0 * N[(N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] * N[(l / N[(M * N[(h * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -7.4e-283], t$95$1, If[LessEqual[d, 8e-215], N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.125 * N[(N[(N[(D * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.75e+138], t$95$1, N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.00000000000000009e106

   1. Initial program 71.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 71.8%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 71.8%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 71.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 71.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 71.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 71.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 71.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 71.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 71.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 71.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 71.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 71.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 71.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 72.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 72.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 72.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 72.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. associate-*l/ 72.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. associate-*r/ 72.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      11. div-inv 72.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{1}{2}\right)}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      12. metadata-eval 72.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \left(\frac{D}{d} \cdot \color{blue}{0.5}\right)\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 72.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. pow1 72.3%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1}} \]

      2. *-commutative 72.3%

         \[\leadsto {\color{blue}{\left(\left(1 - \frac{\left({\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}}^{1} \]

      3. associate-/l* 71.8%

         \[\leadsto {\left(\left(1 - \color{blue}{\frac{{\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}^{1} \]

      4. *-commutative 71.8%

         \[\leadsto {\left(\left(1 - \frac{\color{blue}{0.5 \cdot {\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2}}}{\frac{\ell}{h}}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}^{1} \]

      5. associate-*l/ 71.8%

         \[\leadsto {\left(\left(1 - \frac{0.5 \cdot {\left(M \cdot \color{blue}{\frac{D \cdot 0.5}{d}}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}^{1} \]

      6. sqrt-unprod 57.2%

         \[\leadsto {\left(\left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}}\right)}^{1} \]

      7. *-commutative 57.2%

         \[\leadsto {\left(\left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}}\right)}^{1} \]

      8. frac-times 39.1%

         \[\leadsto {\left(\left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \sqrt{\color{blue}{\frac{d \cdot d}{\ell \cdot h}}}\right)}^{1} \]

      9. *-commutative 39.1%

         \[\leadsto {\left(\left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}}\right)}^{1} \]

   7. Applied egg-rr 39.1%

      \[\leadsto \color{blue}{{\left(\left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \sqrt{\frac{d \cdot d}{h \cdot \ell}}\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 39.1%

         \[\leadsto \color{blue}{\left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \sqrt{\frac{d \cdot d}{h \cdot \ell}}} \]

      2. *-commutative 39.1%

         \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right)} \]

      3. times-frac 57.2%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \]

      4. sub-neg 57.2%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right)\right)} \]

      5. associate-/l* 57.2%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \left(-\color{blue}{\frac{0.5}{\frac{\frac{\ell}{h}}{{\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}}}\right)\right) \]

      6. distribute-neg-frac 57.2%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \color{blue}{\frac{-0.5}{\frac{\frac{\ell}{h}}{{\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}}}\right) \]

      7. metadata-eval 57.2%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{\color{blue}{-0.5}}{\frac{\frac{\ell}{h}}{{\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}}\right) \]

   9. Simplified 57.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\frac{\frac{\ell}{h}}{{\left(D \cdot \frac{0.5}{\frac{d}{M}}\right)}^{2}}}\right)} \]

   10. Taylor expanded in l around 0 31.4%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\color{blue}{4 \cdot \frac{{d}^{2} \cdot \ell}{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}}}\right) \]
   11. Step-by-step derivation

       1. associate-/r* 35.8%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{4 \cdot \color{blue}{\frac{\frac{{d}^{2} \cdot \ell}{{D}^{2}}}{h \cdot {M}^{2}}}}\right) \]

       2. *-commutative 35.8%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{4 \cdot \frac{\frac{{d}^{2} \cdot \ell}{{D}^{2}}}{\color{blue}{{M}^{2} \cdot h}}}\right) \]

       3. associate-/r* 31.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{4 \cdot \color{blue}{\frac{{d}^{2} \cdot \ell}{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}}}\right) \]

       4. times-frac 33.1%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{4 \cdot \color{blue}{\left(\frac{{d}^{2}}{{D}^{2}} \cdot \frac{\ell}{{M}^{2} \cdot h}\right)}}\right) \]

       5. unpow2 33.1%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{4 \cdot \left(\frac{\color{blue}{d \cdot d}}{{D}^{2}} \cdot \frac{\ell}{{M}^{2} \cdot h}\right)}\right) \]

       6. unpow2 33.1%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{4 \cdot \left(\frac{d \cdot d}{\color{blue}{D \cdot D}} \cdot \frac{\ell}{{M}^{2} \cdot h}\right)}\right) \]

       7. times-frac 45.0%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{4 \cdot \left(\color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D}\right)} \cdot \frac{\ell}{{M}^{2} \cdot h}\right)}\right) \]

       8. unpow2 45.0%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{4 \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\ell}{\color{blue}{\left(M \cdot M\right)} \cdot h}\right)}\right) \]

       9. associate-*l* 47.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{4 \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\ell}{\color{blue}{M \cdot \left(M \cdot h\right)}}\right)}\right) \]

   12. Simplified 47.4%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\color{blue}{4 \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\ell}{M \cdot \left(M \cdot h\right)}\right)}}\right) \]

   if -1.00000000000000009e106 < d < -7.4000000000000001e-283 or 8.00000000000000033e-215 < d < 2.7499999999999999e138

   1. Initial program 73.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 73.6%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 73.6%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 73.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 73.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 73.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 73.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 73.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 73.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 73.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. pow1/2 73.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{0.5}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      2. sqr-pow 73.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{0.5}{2}\right)}\right)}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      3. metadata-eval 73.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.25}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{0.5}{2}\right)}\right)\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval 73.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.25}}\right)\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied egg-rr 73.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   6. Taylor expanded in M around 0 55.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   7. Step-by-step derivation

      1. *-commutative 55.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}} \cdot 0.125}\right) \]

      2. times-frac 56.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \cdot 0.125\right) \]

      3. unpow2 56.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right) \cdot 0.125\right) \]

      4. associate-/l* 60.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{\frac{\ell}{D}}} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right) \cdot 0.125\right) \]

      5. associate-/l* 58.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \left(\frac{D}{\frac{\ell}{D}} \cdot \color{blue}{\frac{{M}^{2}}{\frac{{d}^{2}}{h}}}\right) \cdot 0.125\right) \]

      6. unpow2 58.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{\color{blue}{M \cdot M}}{\frac{{d}^{2}}{h}}\right) \cdot 0.125\right) \]

      7. unpow2 58.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{M \cdot M}{\frac{\color{blue}{d \cdot d}}{h}}\right) \cdot 0.125\right) \]

      8. associate-*l/ 60.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{M \cdot M}{\color{blue}{\frac{d}{h} \cdot d}}\right) \cdot 0.125\right) \]

      9. *-commutative 60.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{M \cdot M}{\color{blue}{d \cdot \frac{d}{h}}}\right) \cdot 0.125\right) \]

      10. times-frac 71.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \left(\frac{D}{\frac{\ell}{D}} \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{M}{\frac{d}{h}}\right)}\right) \cdot 0.125\right) \]

   8. Simplified 71.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{M}{\frac{d}{h}}\right)\right) \cdot 0.125}\right) \]
   9. Step-by-step derivation

      1. pow1 71.0%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{M}{\frac{d}{h}}\right)\right) \cdot 0.125\right)\right)}^{1}} \]

   10. Applied egg-rr 51.9%

       \[\leadsto \color{blue}{{\left(\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{\ell} \cdot D\right) \cdot \left(\left(\frac{M}{d} \cdot \left(M \cdot \frac{h}{d}\right)\right) \cdot 0.125\right)\right)\right)}^{1}} \]
   11. Step-by-step derivation

       1. unpow1 51.9%

          \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{\ell} \cdot D\right) \cdot \left(\left(\frac{M}{d} \cdot \left(M \cdot \frac{h}{d}\right)\right) \cdot 0.125\right)\right)} \]

       2. times-frac 62.6%

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{D}{\ell} \cdot D\right) \cdot \left(\left(\frac{M}{d} \cdot \left(M \cdot \frac{h}{d}\right)\right) \cdot 0.125\right)\right) \]

       3. sub-neg 62.6%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{D}{\ell} \cdot D\right) \cdot \left(\left(\frac{M}{d} \cdot \left(M \cdot \frac{h}{d}\right)\right) \cdot 0.125\right)\right)\right)} \]

       4. associate-*r* 62.6%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \left(-\color{blue}{\left(\left(\frac{D}{\ell} \cdot D\right) \cdot \left(\frac{M}{d} \cdot \left(M \cdot \frac{h}{d}\right)\right)\right) \cdot 0.125}\right)\right) \]

       5. distribute-rgt-neg-in 62.6%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \color{blue}{\left(\left(\frac{D}{\ell} \cdot D\right) \cdot \left(\frac{M}{d} \cdot \left(M \cdot \frac{h}{d}\right)\right)\right) \cdot \left(-0.125\right)}\right) \]

   12. Simplified 63.4%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \left(D \cdot \left(\frac{D}{\ell} \cdot \left(M \cdot \left(\frac{h}{d} \cdot \frac{M}{d}\right)\right)\right)\right) \cdot -0.125\right)} \]

   if -7.4000000000000001e-283 < d < 8.00000000000000033e-215

   1. Initial program 40.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 40.0%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 40.0%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 40.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 40.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 40.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 40.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 41.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 41.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 41.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 41.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 40.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 40.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 40.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 40.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 40.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 40.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 41.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. associate-*l/ 41.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. associate-*r/ 41.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      11. div-inv 41.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{1}{2}\right)}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      12. metadata-eval 41.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \left(\frac{D}{d} \cdot \color{blue}{0.5}\right)\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 41.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]

   6. Taylor expanded in d around 0 35.2%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 35.2%

         \[\leadsto \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative 35.2%

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

      3. unpow2 35.2%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{d}\right) \]

      4. unpow2 35.2%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}}{d}\right) \]

      5. unswap-sqr 52.1%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{\color{blue}{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}}{d}\right) \]

   8. Simplified 52.1%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d}\right)} \]

   if 2.7499999999999999e138 < d

   1. Initial program 75.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 75.7%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 75.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 75.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 75.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 75.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 75.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 75.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 75.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 75.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 67.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 67.2%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. *-commutative 67.2%

         \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

   6. Applied egg-rr 67.2%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
   7. Step-by-step derivation

      1. *-lft-identity 67.2%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. unpow-1 67.2%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      3. sqr-pow 67.2%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      4. rem-sqrt-square 67.2%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      5. metadata-eval 67.2%

         \[\leadsto \left|{\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right| \cdot d \]

      6. sqr-pow 66.9%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)}}\right| \cdot d \]

      7. fabs-sqr 66.9%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)}\right)} \cdot d \]

      8. sqr-pow 67.2%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]

   8. Simplified 67.2%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]
   9. Step-by-step derivation

      1. unpow-prod-down 85.9%

         \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]

   10. Applied egg-rr 85.9%

       \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1 \cdot 10^{+106}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{4 \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\ell}{M \cdot \left(h \cdot M\right)}\right)}\right)\\ \mathbf{elif}\;d \leq -7.4 \cdot 10^{-283}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(\frac{D}{\ell} \cdot \left(M \cdot \left(\frac{M}{d} \cdot \frac{h}{d}\right)\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq 8 \cdot 10^{-215}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d}\right)\\ \mathbf{elif}\;d \leq 2.75 \cdot 10^{+138}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(\frac{D}{\ell} \cdot \left(M \cdot \left(\frac{M}{d} \cdot \frac{h}{d}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 12: 55.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;M \leq 4.2 \cdot 10^{-199}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;M \leq 3.5 \cdot 10^{-143}:\\ \;\;\;\;t_0 \cdot \left(1 + \frac{-0.5}{\frac{\left(\ell \cdot 4\right) \cdot \left(d \cdot d\right)}{h \cdot \left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)}}\right)\\ \mathbf{elif}\;M \leq 7.4 \cdot 10^{-106}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(1 + \frac{-0.5}{4 \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\ell}{M \cdot \left(h \cdot M\right)}\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (* (/ d h) (/ d l))))
        (t_1 (* (sqrt (/ d h)) (sqrt (/ d l)))))
   (if (<= M 4.2e-199)
     t_1
     (if (<= M 3.5e-143)
       (*
        t_0
        (+ 1.0 (/ -0.5 (/ (* (* l 4.0) (* d d)) (* h (* (* D M) (* D M)))))))
       (if (<= M 7.4e-106)
         t_1
         (*
          t_0
          (+
           1.0
           (/ -0.5 (* 4.0 (* (* (/ d D) (/ d D)) (/ l (* M (* h M)))))))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(((d / h) * (d / l)));
	double t_1 = sqrt((d / h)) * sqrt((d / l));
	double tmp;
	if (M <= 4.2e-199) {
		tmp = t_1;
	} else if (M <= 3.5e-143) {
		tmp = t_0 * (1.0 + (-0.5 / (((l * 4.0) * (d * d)) / (h * ((D * M) * (D * M))))));
	} else if (M <= 7.4e-106) {
		tmp = t_1;
	} else {
		tmp = t_0 * (1.0 + (-0.5 / (4.0 * (((d / D) * (d / D)) * (l / (M * (h * M)))))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(((d / h) * (d / l)))
    t_1 = sqrt((d / h)) * sqrt((d / l))
    if (m <= 4.2d-199) then
        tmp = t_1
    else if (m <= 3.5d-143) then
        tmp = t_0 * (1.0d0 + ((-0.5d0) / (((l * 4.0d0) * (d * d)) / (h * ((d_1 * m) * (d_1 * m))))))
    else if (m <= 7.4d-106) then
        tmp = t_1
    else
        tmp = t_0 * (1.0d0 + ((-0.5d0) / (4.0d0 * (((d / d_1) * (d / d_1)) * (l / (m * (h * m)))))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt(((d / h) * (d / l)));
	double t_1 = Math.sqrt((d / h)) * Math.sqrt((d / l));
	double tmp;
	if (M <= 4.2e-199) {
		tmp = t_1;
	} else if (M <= 3.5e-143) {
		tmp = t_0 * (1.0 + (-0.5 / (((l * 4.0) * (d * d)) / (h * ((D * M) * (D * M))))));
	} else if (M <= 7.4e-106) {
		tmp = t_1;
	} else {
		tmp = t_0 * (1.0 + (-0.5 / (4.0 * (((d / D) * (d / D)) * (l / (M * (h * M)))))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.sqrt(((d / h) * (d / l)))
	t_1 = math.sqrt((d / h)) * math.sqrt((d / l))
	tmp = 0
	if M <= 4.2e-199:
		tmp = t_1
	elif M <= 3.5e-143:
		tmp = t_0 * (1.0 + (-0.5 / (((l * 4.0) * (d * d)) / (h * ((D * M) * (D * M))))))
	elif M <= 7.4e-106:
		tmp = t_1
	else:
		tmp = t_0 * (1.0 + (-0.5 / (4.0 * (((d / D) * (d / D)) * (l / (M * (h * M)))))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(Float64(d / h) * Float64(d / l)))
	t_1 = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)))
	tmp = 0.0
	if (M <= 4.2e-199)
		tmp = t_1;
	elseif (M <= 3.5e-143)
		tmp = Float64(t_0 * Float64(1.0 + Float64(-0.5 / Float64(Float64(Float64(l * 4.0) * Float64(d * d)) / Float64(h * Float64(Float64(D * M) * Float64(D * M)))))));
	elseif (M <= 7.4e-106)
		tmp = t_1;
	else
		tmp = Float64(t_0 * Float64(1.0 + Float64(-0.5 / Float64(4.0 * Float64(Float64(Float64(d / D) * Float64(d / D)) * Float64(l / Float64(M * Float64(h * M))))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(((d / h) * (d / l)));
	t_1 = sqrt((d / h)) * sqrt((d / l));
	tmp = 0.0;
	if (M <= 4.2e-199)
		tmp = t_1;
	elseif (M <= 3.5e-143)
		tmp = t_0 * (1.0 + (-0.5 / (((l * 4.0) * (d * d)) / (h * ((D * M) * (D * M))))));
	elseif (M <= 7.4e-106)
		tmp = t_1;
	else
		tmp = t_0 * (1.0 + (-0.5 / (4.0 * (((d / D) * (d / D)) * (l / (M * (h * M)))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, 4.2e-199], t$95$1, If[LessEqual[M, 3.5e-143], N[(t$95$0 * N[(1.0 + N[(-0.5 / N[(N[(N[(l * 4.0), $MachinePrecision] * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(h * N[(N[(D * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 7.4e-106], t$95$1, N[(t$95$0 * N[(1.0 + N[(-0.5 / N[(4.0 * N[(N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] * N[(l / N[(M * N[(h * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if M < 4.20000000000000004e-199 or 3.50000000000000005e-143 < M < 7.39999999999999959e-106

   1. Initial program 70.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 70.4%

         \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      2. metadata-eval 70.4%

         \[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. unpow1/2 70.4%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval 70.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. unpow1/2 70.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. sub-neg 70.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      7. +-commutative 70.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right) \]

      8. *-commutative 70.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) + 1\right)\right) \]

      9. distribute-rgt-neg-in 70.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\frac{h}{\ell} \cdot \left(-\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} + 1\right)\right) \]

      10. fma-def 70.4%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{h}{\ell}, -\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}, 1\right)}\right) \]

   3. Simplified 70.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2} \cdot -0.5, 1\right)\right)} \]

   4. Taylor expanded in h around 0 43.6%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]

   if 4.20000000000000004e-199 < M < 3.50000000000000005e-143

   1. Initial program 58.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 58.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 58.4%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 58.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 58.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 58.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 58.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 58.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 58.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 58.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 58.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 58.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 58.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 58.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 58.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 58.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 58.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 58.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. associate-*l/ 58.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. associate-*r/ 58.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      11. div-inv 58.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{1}{2}\right)}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      12. metadata-eval 58.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \left(\frac{D}{d} \cdot \color{blue}{0.5}\right)\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 58.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. pow1 58.7%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1}} \]

      2. *-commutative 58.7%

         \[\leadsto {\color{blue}{\left(\left(1 - \frac{\left({\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}}^{1} \]

      3. associate-/l* 58.4%

         \[\leadsto {\left(\left(1 - \color{blue}{\frac{{\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}^{1} \]

      4. *-commutative 58.4%

         \[\leadsto {\left(\left(1 - \frac{\color{blue}{0.5 \cdot {\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2}}}{\frac{\ell}{h}}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}^{1} \]

      5. associate-*l/ 58.4%

         \[\leadsto {\left(\left(1 - \frac{0.5 \cdot {\left(M \cdot \color{blue}{\frac{D \cdot 0.5}{d}}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}^{1} \]

      6. sqrt-unprod 58.2%

         \[\leadsto {\left(\left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}}\right)}^{1} \]

      7. *-commutative 58.2%

         \[\leadsto {\left(\left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}}\right)}^{1} \]

      8. frac-times 49.9%

         \[\leadsto {\left(\left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \sqrt{\color{blue}{\frac{d \cdot d}{\ell \cdot h}}}\right)}^{1} \]

      9. *-commutative 49.9%

         \[\leadsto {\left(\left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}}\right)}^{1} \]

   7. Applied egg-rr 49.9%

      \[\leadsto \color{blue}{{\left(\left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \sqrt{\frac{d \cdot d}{h \cdot \ell}}\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 49.9%

         \[\leadsto \color{blue}{\left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \sqrt{\frac{d \cdot d}{h \cdot \ell}}} \]

      2. *-commutative 49.9%

         \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right)} \]

      3. times-frac 58.2%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \]

      4. sub-neg 58.2%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right)\right)} \]

      5. associate-/l* 58.2%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \left(-\color{blue}{\frac{0.5}{\frac{\frac{\ell}{h}}{{\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}}}\right)\right) \]

      6. distribute-neg-frac 58.2%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \color{blue}{\frac{-0.5}{\frac{\frac{\ell}{h}}{{\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}}}\right) \]

      7. metadata-eval 58.2%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{\color{blue}{-0.5}}{\frac{\frac{\ell}{h}}{{\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}}\right) \]

   9. Simplified 58.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\frac{\frac{\ell}{h}}{{\left(D \cdot \frac{0.5}{\frac{d}{M}}\right)}^{2}}}\right)} \]

   10. Taylor expanded in l around 0 33.3%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\color{blue}{4 \cdot \frac{{d}^{2} \cdot \ell}{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}}}\right) \]
   11. Step-by-step derivation

       1. associate-*r/ 33.3%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\color{blue}{\frac{4 \cdot \left({d}^{2} \cdot \ell\right)}{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}}}\right) \]

       2. *-commutative 33.3%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\frac{4 \cdot \left({d}^{2} \cdot \ell\right)}{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}}\right) \]

       3. associate-*r* 33.3%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\frac{4 \cdot \left({d}^{2} \cdot \ell\right)}{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}}\right) \]

       4. *-commutative 33.3%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\frac{4 \cdot \color{blue}{\left(\ell \cdot {d}^{2}\right)}}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}\right) \]

       5. associate-*r* 33.3%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\frac{\color{blue}{\left(4 \cdot \ell\right) \cdot {d}^{2}}}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}\right) \]

       6. unpow2 33.3%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\frac{\left(4 \cdot \ell\right) \cdot \color{blue}{\left(d \cdot d\right)}}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}\right) \]

       7. unpow2 33.3%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\frac{\left(4 \cdot \ell\right) \cdot \left(d \cdot d\right)}{\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot h}}\right) \]

       8. unpow2 33.3%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\frac{\left(4 \cdot \ell\right) \cdot \left(d \cdot d\right)}{\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot h}}\right) \]

       9. unswap-sqr 49.9%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\frac{\left(4 \cdot \ell\right) \cdot \left(d \cdot d\right)}{\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot h}}\right) \]

   12. Simplified 49.9%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\color{blue}{\frac{\left(4 \cdot \ell\right) \cdot \left(d \cdot d\right)}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot h}}}\right) \]

   if 7.39999999999999959e-106 < M

   1. Initial program 70.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 70.3%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 70.3%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 70.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 70.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 70.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 70.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 70.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 70.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 70.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 70.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 70.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 70.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 70.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 73.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. associate-*l/ 73.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. associate-*r/ 73.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      11. div-inv 73.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{1}{2}\right)}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      12. metadata-eval 73.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \left(\frac{D}{d} \cdot \color{blue}{0.5}\right)\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 73.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. pow1 73.3%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1}} \]

      2. *-commutative 73.3%

         \[\leadsto {\color{blue}{\left(\left(1 - \frac{\left({\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}}^{1} \]

      3. associate-/l* 70.4%

         \[\leadsto {\left(\left(1 - \color{blue}{\frac{{\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}^{1} \]

      4. *-commutative 70.4%

         \[\leadsto {\left(\left(1 - \frac{\color{blue}{0.5 \cdot {\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2}}}{\frac{\ell}{h}}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}^{1} \]

      5. associate-*l/ 70.4%

         \[\leadsto {\left(\left(1 - \frac{0.5 \cdot {\left(M \cdot \color{blue}{\frac{D \cdot 0.5}{d}}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}^{1} \]

      6. sqrt-unprod 61.4%

         \[\leadsto {\left(\left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}}\right)}^{1} \]

      7. *-commutative 61.4%

         \[\leadsto {\left(\left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}}\right)}^{1} \]

      8. frac-times 43.1%

         \[\leadsto {\left(\left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \sqrt{\color{blue}{\frac{d \cdot d}{\ell \cdot h}}}\right)}^{1} \]

      9. *-commutative 43.1%

         \[\leadsto {\left(\left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}}\right)}^{1} \]

   7. Applied egg-rr 43.1%

      \[\leadsto \color{blue}{{\left(\left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \sqrt{\frac{d \cdot d}{h \cdot \ell}}\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 43.1%

         \[\leadsto \color{blue}{\left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \sqrt{\frac{d \cdot d}{h \cdot \ell}}} \]

      2. *-commutative 43.1%

         \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right)} \]

      3. times-frac 61.4%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \]

      4. sub-neg 61.4%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right)\right)} \]

      5. associate-/l* 61.4%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \left(-\color{blue}{\frac{0.5}{\frac{\frac{\ell}{h}}{{\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}}}\right)\right) \]

      6. distribute-neg-frac 61.4%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \color{blue}{\frac{-0.5}{\frac{\frac{\ell}{h}}{{\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}}}\right) \]

      7. metadata-eval 61.4%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{\color{blue}{-0.5}}{\frac{\frac{\ell}{h}}{{\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}}\right) \]

   9. Simplified 61.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\frac{\frac{\ell}{h}}{{\left(D \cdot \frac{0.5}{\frac{d}{M}}\right)}^{2}}}\right)} \]

   10. Taylor expanded in l around 0 42.5%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\color{blue}{4 \cdot \frac{{d}^{2} \cdot \ell}{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}}}\right) \]
   11. Step-by-step derivation

       1. associate-/r* 43.9%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{4 \cdot \color{blue}{\frac{\frac{{d}^{2} \cdot \ell}{{D}^{2}}}{h \cdot {M}^{2}}}}\right) \]

       2. *-commutative 43.9%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{4 \cdot \frac{\frac{{d}^{2} \cdot \ell}{{D}^{2}}}{\color{blue}{{M}^{2} \cdot h}}}\right) \]

       3. associate-/r* 42.5%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{4 \cdot \color{blue}{\frac{{d}^{2} \cdot \ell}{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}}}\right) \]

       4. times-frac 40.9%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{4 \cdot \color{blue}{\left(\frac{{d}^{2}}{{D}^{2}} \cdot \frac{\ell}{{M}^{2} \cdot h}\right)}}\right) \]

       5. unpow2 40.9%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{4 \cdot \left(\frac{\color{blue}{d \cdot d}}{{D}^{2}} \cdot \frac{\ell}{{M}^{2} \cdot h}\right)}\right) \]

       6. unpow2 40.9%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{4 \cdot \left(\frac{d \cdot d}{\color{blue}{D \cdot D}} \cdot \frac{\ell}{{M}^{2} \cdot h}\right)}\right) \]

       7. times-frac 50.2%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{4 \cdot \left(\color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D}\right)} \cdot \frac{\ell}{{M}^{2} \cdot h}\right)}\right) \]

       8. unpow2 50.2%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{4 \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\ell}{\color{blue}{\left(M \cdot M\right)} \cdot h}\right)}\right) \]

       9. associate-*l* 55.7%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{4 \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\ell}{\color{blue}{M \cdot \left(M \cdot h\right)}}\right)}\right) \]

   12. Simplified 55.7%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\color{blue}{4 \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\ell}{M \cdot \left(M \cdot h\right)}\right)}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4.2 \cdot 10^{-199}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;M \leq 3.5 \cdot 10^{-143}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\frac{\left(\ell \cdot 4\right) \cdot \left(d \cdot d\right)}{h \cdot \left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)}}\right)\\ \mathbf{elif}\;M \leq 7.4 \cdot 10^{-106}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{4 \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\ell}{M \cdot \left(h \cdot M\right)}\right)}\right)\\ \end{array} \]

Alternative 13: 56.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{if}\;d \leq -1 \cdot 10^{+98}:\\ \;\;\;\;t_0 \cdot \left(1 + \frac{-0.5}{4 \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\ell}{M \cdot \left(h \cdot M\right)}\right)}\right)\\ \mathbf{elif}\;d \leq 3.4 \cdot 10^{+138}:\\ \;\;\;\;t_0 \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(\frac{D}{\ell} \cdot \left(M \cdot \left(\frac{M}{d} \cdot \frac{h}{d}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (* (/ d h) (/ d l)))))
   (if (<= d -1e+98)
     (*
      t_0
      (+ 1.0 (/ -0.5 (* 4.0 (* (* (/ d D) (/ d D)) (/ l (* M (* h M))))))))
     (if (<= d 3.4e+138)
       (* t_0 (+ 1.0 (* -0.125 (* D (* (/ D l) (* M (* (/ M d) (/ h d))))))))
       (* d (* (pow h -0.5) (pow l -0.5)))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(((d / h) * (d / l)));
	double tmp;
	if (d <= -1e+98) {
		tmp = t_0 * (1.0 + (-0.5 / (4.0 * (((d / D) * (d / D)) * (l / (M * (h * M)))))));
	} else if (d <= 3.4e+138) {
		tmp = t_0 * (1.0 + (-0.125 * (D * ((D / l) * (M * ((M / d) * (h / d)))))));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((d / h) * (d / l)))
    if (d <= (-1d+98)) then
        tmp = t_0 * (1.0d0 + ((-0.5d0) / (4.0d0 * (((d / d_1) * (d / d_1)) * (l / (m * (h * m)))))))
    else if (d <= 3.4d+138) then
        tmp = t_0 * (1.0d0 + ((-0.125d0) * (d_1 * ((d_1 / l) * (m * ((m / d) * (h / d)))))))
    else
        tmp = d * ((h ** (-0.5d0)) * (l ** (-0.5d0)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt(((d / h) * (d / l)));
	double tmp;
	if (d <= -1e+98) {
		tmp = t_0 * (1.0 + (-0.5 / (4.0 * (((d / D) * (d / D)) * (l / (M * (h * M)))))));
	} else if (d <= 3.4e+138) {
		tmp = t_0 * (1.0 + (-0.125 * (D * ((D / l) * (M * ((M / d) * (h / d)))))));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.sqrt(((d / h) * (d / l)))
	tmp = 0
	if d <= -1e+98:
		tmp = t_0 * (1.0 + (-0.5 / (4.0 * (((d / D) * (d / D)) * (l / (M * (h * M)))))))
	elif d <= 3.4e+138:
		tmp = t_0 * (1.0 + (-0.125 * (D * ((D / l) * (M * ((M / d) * (h / d)))))))
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(Float64(d / h) * Float64(d / l)))
	tmp = 0.0
	if (d <= -1e+98)
		tmp = Float64(t_0 * Float64(1.0 + Float64(-0.5 / Float64(4.0 * Float64(Float64(Float64(d / D) * Float64(d / D)) * Float64(l / Float64(M * Float64(h * M))))))));
	elseif (d <= 3.4e+138)
		tmp = Float64(t_0 * Float64(1.0 + Float64(-0.125 * Float64(D * Float64(Float64(D / l) * Float64(M * Float64(Float64(M / d) * Float64(h / d))))))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(((d / h) * (d / l)));
	tmp = 0.0;
	if (d <= -1e+98)
		tmp = t_0 * (1.0 + (-0.5 / (4.0 * (((d / D) * (d / D)) * (l / (M * (h * M)))))));
	elseif (d <= 3.4e+138)
		tmp = t_0 * (1.0 + (-0.125 * (D * ((D / l) * (M * ((M / d) * (h / d)))))));
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -1e+98], N[(t$95$0 * N[(1.0 + N[(-0.5 / N[(4.0 * N[(N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] * N[(l / N[(M * N[(h * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.4e+138], N[(t$95$0 * N[(1.0 + N[(-0.125 * N[(D * N[(N[(D / l), $MachinePrecision] * N[(M * N[(N[(M / d), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -9.99999999999999998e97

   1. Initial program 71.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 71.8%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 71.8%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 71.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 71.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 71.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 71.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 71.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 71.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 71.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 71.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 71.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 71.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 71.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 72.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 72.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 72.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 72.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. associate-*l/ 72.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. associate-*r/ 72.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      11. div-inv 72.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{1}{2}\right)}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      12. metadata-eval 72.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \left(\frac{D}{d} \cdot \color{blue}{0.5}\right)\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 72.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. pow1 72.3%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1}} \]

      2. *-commutative 72.3%

         \[\leadsto {\color{blue}{\left(\left(1 - \frac{\left({\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}}^{1} \]

      3. associate-/l* 71.8%

         \[\leadsto {\left(\left(1 - \color{blue}{\frac{{\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}^{1} \]

      4. *-commutative 71.8%

         \[\leadsto {\left(\left(1 - \frac{\color{blue}{0.5 \cdot {\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2}}}{\frac{\ell}{h}}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}^{1} \]

      5. associate-*l/ 71.8%

         \[\leadsto {\left(\left(1 - \frac{0.5 \cdot {\left(M \cdot \color{blue}{\frac{D \cdot 0.5}{d}}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}^{1} \]

      6. sqrt-unprod 57.2%

         \[\leadsto {\left(\left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}}\right)}^{1} \]

      7. *-commutative 57.2%

         \[\leadsto {\left(\left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}}\right)}^{1} \]

      8. frac-times 39.1%

         \[\leadsto {\left(\left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \sqrt{\color{blue}{\frac{d \cdot d}{\ell \cdot h}}}\right)}^{1} \]

      9. *-commutative 39.1%

         \[\leadsto {\left(\left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}}\right)}^{1} \]

   7. Applied egg-rr 39.1%

      \[\leadsto \color{blue}{{\left(\left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \sqrt{\frac{d \cdot d}{h \cdot \ell}}\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 39.1%

         \[\leadsto \color{blue}{\left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \sqrt{\frac{d \cdot d}{h \cdot \ell}}} \]

      2. *-commutative 39.1%

         \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right)} \]

      3. times-frac 57.2%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \]

      4. sub-neg 57.2%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right)\right)} \]

      5. associate-/l* 57.2%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \left(-\color{blue}{\frac{0.5}{\frac{\frac{\ell}{h}}{{\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}}}\right)\right) \]

      6. distribute-neg-frac 57.2%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \color{blue}{\frac{-0.5}{\frac{\frac{\ell}{h}}{{\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}}}\right) \]

      7. metadata-eval 57.2%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{\color{blue}{-0.5}}{\frac{\frac{\ell}{h}}{{\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}}\right) \]

   9. Simplified 57.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\frac{\frac{\ell}{h}}{{\left(D \cdot \frac{0.5}{\frac{d}{M}}\right)}^{2}}}\right)} \]

   10. Taylor expanded in l around 0 31.4%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\color{blue}{4 \cdot \frac{{d}^{2} \cdot \ell}{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}}}\right) \]
   11. Step-by-step derivation

       1. associate-/r* 35.8%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{4 \cdot \color{blue}{\frac{\frac{{d}^{2} \cdot \ell}{{D}^{2}}}{h \cdot {M}^{2}}}}\right) \]

       2. *-commutative 35.8%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{4 \cdot \frac{\frac{{d}^{2} \cdot \ell}{{D}^{2}}}{\color{blue}{{M}^{2} \cdot h}}}\right) \]

       3. associate-/r* 31.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{4 \cdot \color{blue}{\frac{{d}^{2} \cdot \ell}{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}}}\right) \]

       4. times-frac 33.1%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{4 \cdot \color{blue}{\left(\frac{{d}^{2}}{{D}^{2}} \cdot \frac{\ell}{{M}^{2} \cdot h}\right)}}\right) \]

       5. unpow2 33.1%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{4 \cdot \left(\frac{\color{blue}{d \cdot d}}{{D}^{2}} \cdot \frac{\ell}{{M}^{2} \cdot h}\right)}\right) \]

       6. unpow2 33.1%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{4 \cdot \left(\frac{d \cdot d}{\color{blue}{D \cdot D}} \cdot \frac{\ell}{{M}^{2} \cdot h}\right)}\right) \]

       7. times-frac 45.0%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{4 \cdot \left(\color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D}\right)} \cdot \frac{\ell}{{M}^{2} \cdot h}\right)}\right) \]

       8. unpow2 45.0%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{4 \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\ell}{\color{blue}{\left(M \cdot M\right)} \cdot h}\right)}\right) \]

       9. associate-*l* 47.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{4 \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\ell}{\color{blue}{M \cdot \left(M \cdot h\right)}}\right)}\right) \]

   12. Simplified 47.4%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\color{blue}{4 \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\ell}{M \cdot \left(M \cdot h\right)}\right)}}\right) \]

   if -9.99999999999999998e97 < d < 3.40000000000000011e138

   1. Initial program 68.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 68.1%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 68.1%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 68.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 68.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 68.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 68.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 68.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 68.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 68.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. pow1/2 68.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{0.5}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      2. sqr-pow 68.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{0.5}{2}\right)}\right)}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      3. metadata-eval 68.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.25}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{0.5}{2}\right)}\right)\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval 68.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.25}}\right)\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied egg-rr 68.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   6. Taylor expanded in M around 0 48.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   7. Step-by-step derivation

      1. *-commutative 48.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}} \cdot 0.125}\right) \]

      2. times-frac 49.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \cdot 0.125\right) \]

      3. unpow2 49.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right) \cdot 0.125\right) \]

      4. associate-/l* 52.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{\frac{\ell}{D}}} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right) \cdot 0.125\right) \]

      5. associate-/l* 51.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \left(\frac{D}{\frac{\ell}{D}} \cdot \color{blue}{\frac{{M}^{2}}{\frac{{d}^{2}}{h}}}\right) \cdot 0.125\right) \]

      6. unpow2 51.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{\color{blue}{M \cdot M}}{\frac{{d}^{2}}{h}}\right) \cdot 0.125\right) \]

      7. unpow2 51.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{M \cdot M}{\frac{\color{blue}{d \cdot d}}{h}}\right) \cdot 0.125\right) \]

      8. associate-*l/ 54.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{M \cdot M}{\color{blue}{\frac{d}{h} \cdot d}}\right) \cdot 0.125\right) \]

      9. *-commutative 54.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{M \cdot M}{\color{blue}{d \cdot \frac{d}{h}}}\right) \cdot 0.125\right) \]

      10. times-frac 64.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \left(\frac{D}{\frac{\ell}{D}} \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{M}{\frac{d}{h}}\right)}\right) \cdot 0.125\right) \]

   8. Simplified 64.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{M}{\frac{d}{h}}\right)\right) \cdot 0.125}\right) \]
   9. Step-by-step derivation

      1. pow1 64.0%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{M}{\frac{d}{h}}\right)\right) \cdot 0.125\right)\right)}^{1}} \]

   10. Applied egg-rr 43.4%

       \[\leadsto \color{blue}{{\left(\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{\ell} \cdot D\right) \cdot \left(\left(\frac{M}{d} \cdot \left(M \cdot \frac{h}{d}\right)\right) \cdot 0.125\right)\right)\right)}^{1}} \]
   11. Step-by-step derivation

       1. unpow1 43.4%

          \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{\ell} \cdot D\right) \cdot \left(\left(\frac{M}{d} \cdot \left(M \cdot \frac{h}{d}\right)\right) \cdot 0.125\right)\right)} \]

       2. times-frac 54.6%

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{D}{\ell} \cdot D\right) \cdot \left(\left(\frac{M}{d} \cdot \left(M \cdot \frac{h}{d}\right)\right) \cdot 0.125\right)\right) \]

       3. sub-neg 54.6%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{D}{\ell} \cdot D\right) \cdot \left(\left(\frac{M}{d} \cdot \left(M \cdot \frac{h}{d}\right)\right) \cdot 0.125\right)\right)\right)} \]

       4. associate-*r* 54.6%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \left(-\color{blue}{\left(\left(\frac{D}{\ell} \cdot D\right) \cdot \left(\frac{M}{d} \cdot \left(M \cdot \frac{h}{d}\right)\right)\right) \cdot 0.125}\right)\right) \]

       5. distribute-rgt-neg-in 54.6%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \color{blue}{\left(\left(\frac{D}{\ell} \cdot D\right) \cdot \left(\frac{M}{d} \cdot \left(M \cdot \frac{h}{d}\right)\right)\right) \cdot \left(-0.125\right)}\right) \]

   12. Simplified 55.2%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \left(D \cdot \left(\frac{D}{\ell} \cdot \left(M \cdot \left(\frac{h}{d} \cdot \frac{M}{d}\right)\right)\right)\right) \cdot -0.125\right)} \]

   if 3.40000000000000011e138 < d

   1. Initial program 75.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 75.7%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 75.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 75.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 75.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 75.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 75.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 75.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 75.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 75.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 67.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 67.2%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. *-commutative 67.2%

         \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

   6. Applied egg-rr 67.2%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
   7. Step-by-step derivation

      1. *-lft-identity 67.2%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. unpow-1 67.2%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      3. sqr-pow 67.2%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      4. rem-sqrt-square 67.2%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      5. metadata-eval 67.2%

         \[\leadsto \left|{\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right| \cdot d \]

      6. sqr-pow 66.9%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)}}\right| \cdot d \]

      7. fabs-sqr 66.9%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)}\right)} \cdot d \]

      8. sqr-pow 67.2%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]

   8. Simplified 67.2%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]
   9. Step-by-step derivation

      1. unpow-prod-down 85.9%

         \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]

   10. Applied egg-rr 85.9%

       \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1 \cdot 10^{+98}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{4 \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\ell}{M \cdot \left(h \cdot M\right)}\right)}\right)\\ \mathbf{elif}\;d \leq 3.4 \cdot 10^{+138}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(\frac{D}{\ell} \cdot \left(M \cdot \left(\frac{M}{d} \cdot \frac{h}{d}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 14: 55.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{if}\;d \leq -2.3 \cdot 10^{+90} \lor \neg \left(d \leq 1.38 \cdot 10^{+101}\right):\\ \;\;\;\;t_0 \cdot \left(1 + \frac{-0.5}{4 \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\ell}{M \cdot \left(h \cdot M\right)}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(\frac{D}{\ell} \cdot \left(M \cdot \left(\frac{M}{d} \cdot \frac{h}{d}\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (* (/ d h) (/ d l)))))
   (if (or (<= d -2.3e+90) (not (<= d 1.38e+101)))
     (*
      t_0
      (+ 1.0 (/ -0.5 (* 4.0 (* (* (/ d D) (/ d D)) (/ l (* M (* h M))))))))
     (* t_0 (+ 1.0 (* -0.125 (* D (* (/ D l) (* M (* (/ M d) (/ h d)))))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(((d / h) * (d / l)));
	double tmp;
	if ((d <= -2.3e+90) || !(d <= 1.38e+101)) {
		tmp = t_0 * (1.0 + (-0.5 / (4.0 * (((d / D) * (d / D)) * (l / (M * (h * M)))))));
	} else {
		tmp = t_0 * (1.0 + (-0.125 * (D * ((D / l) * (M * ((M / d) * (h / d)))))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((d / h) * (d / l)))
    if ((d <= (-2.3d+90)) .or. (.not. (d <= 1.38d+101))) then
        tmp = t_0 * (1.0d0 + ((-0.5d0) / (4.0d0 * (((d / d_1) * (d / d_1)) * (l / (m * (h * m)))))))
    else
        tmp = t_0 * (1.0d0 + ((-0.125d0) * (d_1 * ((d_1 / l) * (m * ((m / d) * (h / d)))))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt(((d / h) * (d / l)));
	double tmp;
	if ((d <= -2.3e+90) || !(d <= 1.38e+101)) {
		tmp = t_0 * (1.0 + (-0.5 / (4.0 * (((d / D) * (d / D)) * (l / (M * (h * M)))))));
	} else {
		tmp = t_0 * (1.0 + (-0.125 * (D * ((D / l) * (M * ((M / d) * (h / d)))))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.sqrt(((d / h) * (d / l)))
	tmp = 0
	if (d <= -2.3e+90) or not (d <= 1.38e+101):
		tmp = t_0 * (1.0 + (-0.5 / (4.0 * (((d / D) * (d / D)) * (l / (M * (h * M)))))))
	else:
		tmp = t_0 * (1.0 + (-0.125 * (D * ((D / l) * (M * ((M / d) * (h / d)))))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(Float64(d / h) * Float64(d / l)))
	tmp = 0.0
	if ((d <= -2.3e+90) || !(d <= 1.38e+101))
		tmp = Float64(t_0 * Float64(1.0 + Float64(-0.5 / Float64(4.0 * Float64(Float64(Float64(d / D) * Float64(d / D)) * Float64(l / Float64(M * Float64(h * M))))))));
	else
		tmp = Float64(t_0 * Float64(1.0 + Float64(-0.125 * Float64(D * Float64(Float64(D / l) * Float64(M * Float64(Float64(M / d) * Float64(h / d))))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(((d / h) * (d / l)));
	tmp = 0.0;
	if ((d <= -2.3e+90) || ~((d <= 1.38e+101)))
		tmp = t_0 * (1.0 + (-0.5 / (4.0 * (((d / D) * (d / D)) * (l / (M * (h * M)))))));
	else
		tmp = t_0 * (1.0 + (-0.125 * (D * ((D / l) * (M * ((M / d) * (h / d)))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[d, -2.3e+90], N[Not[LessEqual[d, 1.38e+101]], $MachinePrecision]], N[(t$95$0 * N[(1.0 + N[(-0.5 / N[(4.0 * N[(N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] * N[(l / N[(M * N[(h * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 + N[(-0.125 * N[(D * N[(N[(D / l), $MachinePrecision] * N[(M * N[(N[(M / d), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -2.3e90 or 1.38e101 < d

   1. Initial program 74.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 74.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 74.4%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 74.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 74.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 74.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 74.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 74.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 74.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 74.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 74.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 74.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 74.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 74.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 77.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 77.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 77.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 77.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. associate-*l/ 77.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. associate-*r/ 77.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      11. div-inv 77.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{1}{2}\right)}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      12. metadata-eval 77.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \left(\frac{D}{d} \cdot \color{blue}{0.5}\right)\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 77.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. pow1 77.1%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\right)}^{1}} \]

      2. *-commutative 77.1%

         \[\leadsto {\color{blue}{\left(\left(1 - \frac{\left({\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}}^{1} \]

      3. associate-/l* 74.4%

         \[\leadsto {\left(\left(1 - \color{blue}{\frac{{\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}^{1} \]

      4. *-commutative 74.4%

         \[\leadsto {\left(\left(1 - \frac{\color{blue}{0.5 \cdot {\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2}}}{\frac{\ell}{h}}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}^{1} \]

      5. associate-*l/ 74.4%

         \[\leadsto {\left(\left(1 - \frac{0.5 \cdot {\left(M \cdot \color{blue}{\frac{D \cdot 0.5}{d}}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}^{1} \]

      6. sqrt-unprod 63.6%

         \[\leadsto {\left(\left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}}\right)}^{1} \]

      7. *-commutative 63.6%

         \[\leadsto {\left(\left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}}\right)}^{1} \]

      8. frac-times 41.9%

         \[\leadsto {\left(\left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \sqrt{\color{blue}{\frac{d \cdot d}{\ell \cdot h}}}\right)}^{1} \]

      9. *-commutative 41.9%

         \[\leadsto {\left(\left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}}\right)}^{1} \]

   7. Applied egg-rr 41.9%

      \[\leadsto \color{blue}{{\left(\left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \sqrt{\frac{d \cdot d}{h \cdot \ell}}\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 41.9%

         \[\leadsto \color{blue}{\left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \sqrt{\frac{d \cdot d}{h \cdot \ell}}} \]

      2. *-commutative 41.9%

         \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right)} \]

      3. times-frac 63.6%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right) \]

      4. sub-neg 63.6%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\frac{0.5 \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right)\right)} \]

      5. associate-/l* 63.7%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \left(-\color{blue}{\frac{0.5}{\frac{\frac{\ell}{h}}{{\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}}}\right)\right) \]

      6. distribute-neg-frac 63.7%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \color{blue}{\frac{-0.5}{\frac{\frac{\ell}{h}}{{\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}}}\right) \]

      7. metadata-eval 63.7%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{\color{blue}{-0.5}}{\frac{\frac{\ell}{h}}{{\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}}\right) \]

   9. Simplified 63.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\frac{\frac{\ell}{h}}{{\left(D \cdot \frac{0.5}{\frac{d}{M}}\right)}^{2}}}\right)} \]

   10. Taylor expanded in l around 0 41.1%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\color{blue}{4 \cdot \frac{{d}^{2} \cdot \ell}{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}}}\right) \]
   11. Step-by-step derivation

       1. associate-/r* 43.3%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{4 \cdot \color{blue}{\frac{\frac{{d}^{2} \cdot \ell}{{D}^{2}}}{h \cdot {M}^{2}}}}\right) \]

       2. *-commutative 43.3%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{4 \cdot \frac{\frac{{d}^{2} \cdot \ell}{{D}^{2}}}{\color{blue}{{M}^{2} \cdot h}}}\right) \]

       3. associate-/r* 41.1%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{4 \cdot \color{blue}{\frac{{d}^{2} \cdot \ell}{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}}}\right) \]

       4. times-frac 39.6%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{4 \cdot \color{blue}{\left(\frac{{d}^{2}}{{D}^{2}} \cdot \frac{\ell}{{M}^{2} \cdot h}\right)}}\right) \]

       5. unpow2 39.6%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{4 \cdot \left(\frac{\color{blue}{d \cdot d}}{{D}^{2}} \cdot \frac{\ell}{{M}^{2} \cdot h}\right)}\right) \]

       6. unpow2 39.6%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{4 \cdot \left(\frac{d \cdot d}{\color{blue}{D \cdot D}} \cdot \frac{\ell}{{M}^{2} \cdot h}\right)}\right) \]

       7. times-frac 53.9%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{4 \cdot \left(\color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D}\right)} \cdot \frac{\ell}{{M}^{2} \cdot h}\right)}\right) \]

       8. unpow2 53.9%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{4 \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\ell}{\color{blue}{\left(M \cdot M\right)} \cdot h}\right)}\right) \]

       9. associate-*l* 56.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{4 \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\ell}{\color{blue}{M \cdot \left(M \cdot h\right)}}\right)}\right) \]

   12. Simplified 56.4%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{\color{blue}{4 \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\ell}{M \cdot \left(M \cdot h\right)}\right)}}\right) \]

   if -2.3e90 < d < 1.38e101

   1. Initial program 67.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 67.5%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 67.5%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 67.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 67.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 67.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 67.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 67.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 67.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 67.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. pow1/2 67.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{0.5}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      2. sqr-pow 67.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{0.5}{2}\right)}\right)}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      3. metadata-eval 67.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.25}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{0.5}{2}\right)}\right)\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval 67.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.25}}\right)\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied egg-rr 67.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   6. Taylor expanded in M around 0 48.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   7. Step-by-step derivation

      1. *-commutative 48.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}} \cdot 0.125}\right) \]

      2. times-frac 49.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \cdot 0.125\right) \]

      3. unpow2 49.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right) \cdot 0.125\right) \]

      4. associate-/l* 52.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{\frac{\ell}{D}}} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right) \cdot 0.125\right) \]

      5. associate-/l* 51.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \left(\frac{D}{\frac{\ell}{D}} \cdot \color{blue}{\frac{{M}^{2}}{\frac{{d}^{2}}{h}}}\right) \cdot 0.125\right) \]

      6. unpow2 51.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{\color{blue}{M \cdot M}}{\frac{{d}^{2}}{h}}\right) \cdot 0.125\right) \]

      7. unpow2 51.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{M \cdot M}{\frac{\color{blue}{d \cdot d}}{h}}\right) \cdot 0.125\right) \]

      8. associate-*l/ 55.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{M \cdot M}{\color{blue}{\frac{d}{h} \cdot d}}\right) \cdot 0.125\right) \]

      9. *-commutative 55.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{M \cdot M}{\color{blue}{d \cdot \frac{d}{h}}}\right) \cdot 0.125\right) \]

      10. times-frac 63.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \left(\frac{D}{\frac{\ell}{D}} \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{M}{\frac{d}{h}}\right)}\right) \cdot 0.125\right) \]

   8. Simplified 63.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{M}{\frac{d}{h}}\right)\right) \cdot 0.125}\right) \]
   9. Step-by-step derivation

      1. pow1 63.8%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{M}{\frac{d}{h}}\right)\right) \cdot 0.125\right)\right)}^{1}} \]

   10. Applied egg-rr 42.5%

       \[\leadsto \color{blue}{{\left(\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{\ell} \cdot D\right) \cdot \left(\left(\frac{M}{d} \cdot \left(M \cdot \frac{h}{d}\right)\right) \cdot 0.125\right)\right)\right)}^{1}} \]
   11. Step-by-step derivation

       1. unpow1 42.5%

          \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{\ell} \cdot D\right) \cdot \left(\left(\frac{M}{d} \cdot \left(M \cdot \frac{h}{d}\right)\right) \cdot 0.125\right)\right)} \]

       2. times-frac 54.1%

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{D}{\ell} \cdot D\right) \cdot \left(\left(\frac{M}{d} \cdot \left(M \cdot \frac{h}{d}\right)\right) \cdot 0.125\right)\right) \]

       3. sub-neg 54.1%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{D}{\ell} \cdot D\right) \cdot \left(\left(\frac{M}{d} \cdot \left(M \cdot \frac{h}{d}\right)\right) \cdot 0.125\right)\right)\right)} \]

       4. associate-*r* 54.1%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \left(-\color{blue}{\left(\left(\frac{D}{\ell} \cdot D\right) \cdot \left(\frac{M}{d} \cdot \left(M \cdot \frac{h}{d}\right)\right)\right) \cdot 0.125}\right)\right) \]

       5. distribute-rgt-neg-in 54.1%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \color{blue}{\left(\left(\frac{D}{\ell} \cdot D\right) \cdot \left(\frac{M}{d} \cdot \left(M \cdot \frac{h}{d}\right)\right)\right) \cdot \left(-0.125\right)}\right) \]

   12. Simplified 54.8%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \left(D \cdot \left(\frac{D}{\ell} \cdot \left(M \cdot \left(\frac{h}{d} \cdot \frac{M}{d}\right)\right)\right)\right) \cdot -0.125\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.3 \cdot 10^{+90} \lor \neg \left(d \leq 1.38 \cdot 10^{+101}\right):\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5}{4 \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\ell}{M \cdot \left(h \cdot M\right)}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(\frac{D}{\ell} \cdot \left(M \cdot \left(\frac{M}{d} \cdot \frac{h}{d}\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 15: 50.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(\frac{D}{\ell} \cdot \left(M \cdot \left(\frac{M}{d} \cdot \frac{h}{d}\right)\right)\right)\right)\right) \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (*
  (sqrt (* (/ d h) (/ d l)))
  (+ 1.0 (* -0.125 (* D (* (/ D l) (* M (* (/ M d) (/ h d)))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	return sqrt(((d / h) * (d / l))) * (1.0 + (-0.125 * (D * ((D / l) * (M * ((M / d) * (h / d)))))));
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = sqrt(((d / h) * (d / l))) * (1.0d0 + ((-0.125d0) * (d_1 * ((d_1 / l) * (m * ((m / d) * (h / d)))))))
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	return Math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.125 * (D * ((D / l) * (M * ((M / d) * (h / d)))))));
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	return math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.125 * (D * ((D / l) * (M * ((M / d) * (h / d)))))))

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	return Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 + Float64(-0.125 * Float64(D * Float64(Float64(D / l) * Float64(M * Float64(Float64(M / d) * Float64(h / d))))))))
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp = code(d, h, l, M, D)
	tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.125 * (D * ((D / l) * (M * ((M / d) * (h / d)))))));
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.125 * N[(D * N[(N[(D / l), $MachinePrecision] * N[(M * N[(N[(M / d), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. Step-by-step derivation

   1. metadata-eval 69.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. unpow1/2 69.8%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. metadata-eval 69.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. unpow1/2 69.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. *-commutative 69.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

   6. associate-*l* 69.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

   7. times-frac 70.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. metadata-eval 70.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

3. Simplified 70.0%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
4. Step-by-step derivation

   1. pow1/2 70.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{0.5}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   2. sqr-pow 69.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{0.5}{2}\right)}\right)}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   3. metadata-eval 69.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.25}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{0.5}{2}\right)}\right)\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   4. metadata-eval 69.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.25}}\right)\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

5. Applied egg-rr 69.9%

   \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

6. Taylor expanded in M around 0 48.0%

   \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
7. Step-by-step derivation

   1. *-commutative 48.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}} \cdot 0.125}\right) \]

   2. times-frac 48.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \cdot 0.125\right) \]

   3. unpow2 48.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right) \cdot 0.125\right) \]

   4. associate-/l* 50.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \left(\color{blue}{\frac{D}{\frac{\ell}{D}}} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right) \cdot 0.125\right) \]

   5. associate-/l* 51.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \left(\frac{D}{\frac{\ell}{D}} \cdot \color{blue}{\frac{{M}^{2}}{\frac{{d}^{2}}{h}}}\right) \cdot 0.125\right) \]

   6. unpow2 51.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{\color{blue}{M \cdot M}}{\frac{{d}^{2}}{h}}\right) \cdot 0.125\right) \]

   7. unpow2 51.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{M \cdot M}{\frac{\color{blue}{d \cdot d}}{h}}\right) \cdot 0.125\right) \]

   8. associate-*l/ 55.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{M \cdot M}{\color{blue}{\frac{d}{h} \cdot d}}\right) \cdot 0.125\right) \]

   9. *-commutative 55.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{M \cdot M}{\color{blue}{d \cdot \frac{d}{h}}}\right) \cdot 0.125\right) \]

   10. times-frac 64.3%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \left(\frac{D}{\frac{\ell}{D}} \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{M}{\frac{d}{h}}\right)}\right) \cdot 0.125\right) \]

8. Simplified 64.3%

   \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{M}{\frac{d}{h}}\right)\right) \cdot 0.125}\right) \]
9. Step-by-step derivation

   1. pow1 64.3%

      \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\right) \cdot \left(1 - \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{M}{\frac{d}{h}}\right)\right) \cdot 0.125\right)\right)}^{1}} \]

10. Applied egg-rr 40.4%

    \[\leadsto \color{blue}{{\left(\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{\ell} \cdot D\right) \cdot \left(\left(\frac{M}{d} \cdot \left(M \cdot \frac{h}{d}\right)\right) \cdot 0.125\right)\right)\right)}^{1}} \]
11. Step-by-step derivation

    1. unpow1 40.4%

       \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(1 - \left(\frac{D}{\ell} \cdot D\right) \cdot \left(\left(\frac{M}{d} \cdot \left(M \cdot \frac{h}{d}\right)\right) \cdot 0.125\right)\right)} \]

    2. times-frac 54.6%

       \[\leadsto \sqrt{\color{blue}{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{D}{\ell} \cdot D\right) \cdot \left(\left(\frac{M}{d} \cdot \left(M \cdot \frac{h}{d}\right)\right) \cdot 0.125\right)\right) \]

    3. sub-neg 54.6%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{D}{\ell} \cdot D\right) \cdot \left(\left(\frac{M}{d} \cdot \left(M \cdot \frac{h}{d}\right)\right) \cdot 0.125\right)\right)\right)} \]

    4. associate-*r* 54.6%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \left(-\color{blue}{\left(\left(\frac{D}{\ell} \cdot D\right) \cdot \left(\frac{M}{d} \cdot \left(M \cdot \frac{h}{d}\right)\right)\right) \cdot 0.125}\right)\right) \]

    5. distribute-rgt-neg-in 54.6%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \color{blue}{\left(\left(\frac{D}{\ell} \cdot D\right) \cdot \left(\frac{M}{d} \cdot \left(M \cdot \frac{h}{d}\right)\right)\right) \cdot \left(-0.125\right)}\right) \]

12. Simplified 55.9%

    \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \left(D \cdot \left(\frac{D}{\ell} \cdot \left(M \cdot \left(\frac{h}{d} \cdot \frac{M}{d}\right)\right)\right)\right) \cdot -0.125\right)} \]

13. Final simplification 55.9%

    \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(\frac{D}{\ell} \cdot \left(M \cdot \left(\frac{M}{d} \cdot \frac{h}{d}\right)\right)\right)\right)\right) \]

Alternative 16: 26.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D) :precision binary64 (* d (sqrt (/ (/ 1.0 h) l))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	return d * sqrt(((1.0 / h) / l));
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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(((1.0d0 / h) / l))
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	return d * Math.sqrt(((1.0 / h) / l));
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	return d * math.sqrt(((1.0 / h) / l))

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	return Float64(d * sqrt(Float64(Float64(1.0 / h) / l)))
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp = code(d, h, l, M, D)
	tmp = d * sqrt(((1.0 / h) / l));
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

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. Step-by-step derivation

   1. metadata-eval 69.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. unpow1/2 69.8%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. metadata-eval 69.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. unpow1/2 69.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. *-commutative 69.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

   6. associate-*l* 69.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

   7. times-frac 70.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. metadata-eval 70.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

3. Simplified 70.0%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
4. Step-by-step derivation

   1. associate-*r* 70.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

   2. frac-times 69.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

   3. *-commutative 69.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

   4. metadata-eval 69.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. associate-*r/ 72.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

   6. metadata-eval 72.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

   7. *-commutative 72.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

   8. frac-times 72.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   9. associate-*l/ 72.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   10. associate-*r/ 72.8%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   11. div-inv 72.8%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{1}{2}\right)}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   12. metadata-eval 72.8%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \left(\frac{D}{d} \cdot \color{blue}{0.5}\right)\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

5. Applied egg-rr 72.8%

   \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
6. Step-by-step derivation

   1. add-cube-cbrt 72.2%

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt[3]{\sqrt{\frac{d}{h}}} \cdot \sqrt[3]{\sqrt{\frac{d}{h}}}\right) \cdot \sqrt[3]{\sqrt{\frac{d}{h}}}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

7. Applied egg-rr 72.2%

   \[\leadsto \left(\color{blue}{\left(\left(\sqrt[3]{\sqrt{\frac{d}{h}}} \cdot \sqrt[3]{\sqrt{\frac{d}{h}}}\right) \cdot \sqrt[3]{\sqrt{\frac{d}{h}}}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

8. Taylor expanded in d around inf 27.1%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
9. Step-by-step derivation

   1. *-commutative 27.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   2. *-commutative 27.1%

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   3. associate-/r* 27.1%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

10. Simplified 27.1%

    \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]

11. Final simplification 27.1%

    \[\leadsto d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}} \]

Alternative 17: 26.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ d \cdot {\left(h \cdot \ell\right)}^{-0.5} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D) :precision binary64 (* d (pow (* h l) -0.5)))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	return d * pow((h * l), -0.5);
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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 * l) ** (-0.5d0))
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	return d * Math.pow((h * l), -0.5);
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	return d * math.pow((h * l), -0.5)

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	return Float64(d * (Float64(h * l) ^ -0.5))
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp = code(d, h, l, M, D)
	tmp = d * ((h * l) ^ -0.5);
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

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. Step-by-step derivation

   1. metadata-eval 69.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. unpow1/2 69.8%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. metadata-eval 69.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. unpow1/2 69.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. *-commutative 69.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

   6. associate-*l* 69.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

   7. times-frac 70.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. metadata-eval 70.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

3. Simplified 70.0%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

4. Taylor expanded in d around inf 27.1%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
5. Step-by-step derivation

   1. *-un-lft-identity 27.1%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

   2. *-commutative 27.1%

      \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

6. Applied egg-rr 27.1%

   \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
7. Step-by-step derivation

   1. *-lft-identity 27.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

   2. unpow-1 27.1%

      \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

   3. sqr-pow 27.1%

      \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

   4. rem-sqrt-square 27.0%

      \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

   5. metadata-eval 27.0%

      \[\leadsto \left|{\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right| \cdot d \]

   6. sqr-pow 27.0%

      \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)}}\right| \cdot d \]

   7. fabs-sqr 27.0%

      \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)}\right)} \cdot d \]

   8. sqr-pow 27.0%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]

8. Simplified 27.0%

   \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]

9. Final simplification 27.0%

   \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{-0.5} \]

Reproduce

herbie shell --seed 2023250 
(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)))))