Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.9% → 82.0%

Time: 24.6s

Alternatives: 20

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: 66.9% 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: 82.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{-d}\\ \mathbf{if}\;d \leq -2.7 \cdot 10^{+29}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{t\_0}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{\frac{D\_m}{d}}{\frac{2}{M}}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{t\_0}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + -0.5 \cdot {\left(\left(D\_m \cdot \left(M \cdot \frac{0.5}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(h, -0.5 \cdot \frac{{\left(M \cdot \frac{\frac{D\_m}{d}}{2}\right)}^{2}}{\ell}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (sqrt (- d))))
   (if (<= d -2.7e+29)
     (*
      (sqrt (/ d l))
      (*
       (/ t_0 (sqrt (- h)))
       (+ 1.0 (* (/ h l) (* (pow (/ (/ D_m d) (/ 2.0 M)) 2.0) -0.5)))))
     (if (<= d -4e-310)
       (*
        (/ t_0 (sqrt (- l)))
        (*
         (sqrt (/ d h))
         (+
          1.0
          (* -0.5 (pow (* (* D_m (* M (/ 0.5 d))) (sqrt (/ h l))) 2.0)))))
       (*
        d
        (/
         (fma h (* -0.5 (/ (pow (* M (/ (/ D_m d) 2.0)) 2.0) l)) 1.0)
         (* (sqrt l) (sqrt h))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = sqrt(-d);
	double tmp;
	if (d <= -2.7e+29) {
		tmp = sqrt((d / l)) * ((t_0 / sqrt(-h)) * (1.0 + ((h / l) * (pow(((D_m / d) / (2.0 / M)), 2.0) * -0.5))));
	} else if (d <= -4e-310) {
		tmp = (t_0 / sqrt(-l)) * (sqrt((d / h)) * (1.0 + (-0.5 * pow(((D_m * (M * (0.5 / d))) * sqrt((h / l))), 2.0))));
	} else {
		tmp = d * (fma(h, (-0.5 * (pow((M * ((D_m / d) / 2.0)), 2.0) / l)), 1.0) / (sqrt(l) * sqrt(h)));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = sqrt(Float64(-d))
	tmp = 0.0
	if (d <= -2.7e+29)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(Float64(t_0 / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(D_m / d) / Float64(2.0 / M)) ^ 2.0) * -0.5)))));
	elseif (d <= -4e-310)
		tmp = Float64(Float64(t_0 / sqrt(Float64(-l))) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(-0.5 * (Float64(Float64(D_m * Float64(M * Float64(0.5 / d))) * sqrt(Float64(h / l))) ^ 2.0)))));
	else
		tmp = Float64(d * Float64(fma(h, Float64(-0.5 * Float64((Float64(M * Float64(Float64(D_m / d) / 2.0)) ^ 2.0) / l)), 1.0) / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[d, -2.7e+29], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m / d), $MachinePrecision] / N[(2.0 / M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4e-310], N[(N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[Power[N[(N[(D$95$m * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(h * N[(-0.5 * N[(N[Power[N[(M * N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.7e29

   1. Initial program 80.2%

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

   3. Simplified 78.7%

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

      1. associate-*r/ 80.3%

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

      2. *-un-lft-identity 80.3%

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

      3. times-frac 78.7%

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

      4. associate-/l/ 78.7%

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

      5. *-commutative 78.7%

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

      6. times-frac 80.3%

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

      7. *-un-lft-identity 80.3%

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

      8. *-commutative 80.3%

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

      9. frac-times 80.3%

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

      10. clear-num 80.2%

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

      11. un-div-inv 80.2%

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

   6. Applied egg-rr 80.2%

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

      1. frac-2neg 80.2%

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

      2. sqrt-div 90.3%

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

   8. Applied egg-rr 90.3%

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

   if -2.7e29 < d < -3.999999999999988e-310

   1. Initial program 66.1%

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

   3. Simplified 64.8%

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

      1. *-commutative 64.8%

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

      2. clear-num 64.8%

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

      3. un-div-inv 66.2%

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

      4. *-commutative 66.2%

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

      5. associate-*r/ 67.4%

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

      6. *-un-lft-identity 67.4%

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

      7. times-frac 66.2%

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

      8. associate-/l/ 66.2%

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

      9. *-commutative 66.2%

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

      10. times-frac 67.4%

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

      11. *-commutative 67.4%

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

      12. *-un-lft-identity 67.4%

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

      13. associate-/l* 66.1%

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

      14. *-un-lft-identity 66.1%

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

      15. times-frac 66.1%

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

      16. metadata-eval 66.1%

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

   6. Applied egg-rr 66.1%

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

      1. associate-/l* 66.1%

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

      2. associate-*r* 66.1%

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

      3. associate-/l* 67.4%

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

      4. *-commutative 67.4%

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

      5. associate-*r* 67.4%

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

      6. associate-/l* 67.4%

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

   8. Simplified 67.4%

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

      1. add-sqr-sqrt 67.4%

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

      2. pow2 67.4%

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

      3. div-inv 66.0%

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

      4. clear-num 66.0%

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

      5. sqrt-prod 66.0%

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

      6. sqrt-pow1 67.6%

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

      7. metadata-eval 67.6%

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

      8. pow1 67.6%

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

      9. associate-*l* 66.3%

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

   10. Applied egg-rr 66.3%

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

       1. frac-2neg 66.3%

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

       2. sqrt-div 76.3%

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

   12. Applied egg-rr 76.3%

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

   if -3.999999999999988e-310 < d

   1. Initial program 66.1%

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

   3. Simplified 66.1%

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

   6. Applied egg-rr 72.8%

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

      1. distribute-rgt1-in 81.4%

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

      2. +-commutative 81.4%

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

      3. associate-*r/ 82.4%

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

      4. *-commutative 82.4%

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

      5. associate-/l* 84.0%

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

   8. Simplified 87.5%

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

Alternative 2: 81.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{-d}\\ \mathbf{if}\;d \leq -6.8 \cdot 10^{+30}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{t\_0}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{\frac{D\_m}{d}}{\frac{2}{M}}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{t\_0}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + -0.5 \cdot \frac{{\left(\frac{0.5}{d} \cdot \left(D\_m \cdot M\right)\right)}^{2}}{\frac{\ell}{h}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(h, -0.5 \cdot \frac{{\left(M \cdot \frac{\frac{D\_m}{d}}{2}\right)}^{2}}{\ell}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (sqrt (- d))))
   (if (<= d -6.8e+30)
     (*
      (sqrt (/ d l))
      (*
       (/ t_0 (sqrt (- h)))
       (+ 1.0 (* (/ h l) (* (pow (/ (/ D_m d) (/ 2.0 M)) 2.0) -0.5)))))
     (if (<= d -4e-310)
       (*
        (/ t_0 (sqrt (- l)))
        (*
         (sqrt (/ d h))
         (+ 1.0 (* -0.5 (/ (pow (* (/ 0.5 d) (* D_m M)) 2.0) (/ l h))))))
       (*
        d
        (/
         (fma h (* -0.5 (/ (pow (* M (/ (/ D_m d) 2.0)) 2.0) l)) 1.0)
         (* (sqrt l) (sqrt h))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = sqrt(-d);
	double tmp;
	if (d <= -6.8e+30) {
		tmp = sqrt((d / l)) * ((t_0 / sqrt(-h)) * (1.0 + ((h / l) * (pow(((D_m / d) / (2.0 / M)), 2.0) * -0.5))));
	} else if (d <= -4e-310) {
		tmp = (t_0 / sqrt(-l)) * (sqrt((d / h)) * (1.0 + (-0.5 * (pow(((0.5 / d) * (D_m * M)), 2.0) / (l / h)))));
	} else {
		tmp = d * (fma(h, (-0.5 * (pow((M * ((D_m / d) / 2.0)), 2.0) / l)), 1.0) / (sqrt(l) * sqrt(h)));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = sqrt(Float64(-d))
	tmp = 0.0
	if (d <= -6.8e+30)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(Float64(t_0 / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(D_m / d) / Float64(2.0 / M)) ^ 2.0) * -0.5)))));
	elseif (d <= -4e-310)
		tmp = Float64(Float64(t_0 / sqrt(Float64(-l))) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(-0.5 * Float64((Float64(Float64(0.5 / d) * Float64(D_m * M)) ^ 2.0) / Float64(l / h))))));
	else
		tmp = Float64(d * Float64(fma(h, Float64(-0.5 * Float64((Float64(M * Float64(Float64(D_m / d) / 2.0)) ^ 2.0) / l)), 1.0) / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[d, -6.8e+30], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m / d), $MachinePrecision] / N[(2.0 / M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4e-310], N[(N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[Power[N[(N[(0.5 / d), $MachinePrecision] * N[(D$95$m * M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(h * N[(-0.5 * N[(N[Power[N[(M * N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -6.8000000000000005e30

   1. Initial program 79.3%

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

   3. Simplified 77.6%

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

      1. associate-*r/ 79.3%

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

      2. *-un-lft-identity 79.3%

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

      3. times-frac 77.6%

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

      4. associate-/l/ 77.6%

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

      5. *-commutative 77.6%

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

      6. times-frac 79.3%

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

      7. *-un-lft-identity 79.3%

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

      8. *-commutative 79.3%

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

      9. frac-times 79.3%

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

      10. clear-num 79.3%

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

      11. un-div-inv 79.3%

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

   6. Applied egg-rr 79.3%

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

      1. frac-2neg 79.3%

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

      2. sqrt-div 89.9%

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

   8. Applied egg-rr 89.9%

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

   if -6.8000000000000005e30 < d < -3.999999999999988e-310

   1. Initial program 67.4%

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

   3. Simplified 66.2%

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

      1. *-commutative 66.2%

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

      2. clear-num 66.1%

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

      3. un-div-inv 67.5%

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

      4. *-commutative 67.5%

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

      5. associate-*r/ 68.6%

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

      6. *-un-lft-identity 68.6%

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

      7. times-frac 67.5%

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

      8. associate-/l/ 67.5%

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

      9. *-commutative 67.5%

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

      10. times-frac 68.6%

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

      11. *-commutative 68.6%

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

      12. *-un-lft-identity 68.6%

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

      13. associate-/l* 67.4%

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

      14. *-un-lft-identity 67.4%

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

      15. times-frac 67.4%

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

      16. metadata-eval 67.4%

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

   6. Applied egg-rr 67.4%

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

      1. associate-/l* 67.4%

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

      2. associate-*r* 67.4%

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

      3. associate-/l* 68.6%

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

      4. *-commutative 68.6%

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

      5. associate-*r* 68.6%

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

      6. associate-/l* 68.7%

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

   8. Simplified 68.7%

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

      1. frac-2neg 67.5%

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

      2. sqrt-div 77.2%

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

   10. Applied egg-rr 78.3%

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

   if -3.999999999999988e-310 < d

   1. Initial program 66.1%

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

   3. Simplified 66.1%

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

   6. Applied egg-rr 72.8%

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

      1. distribute-rgt1-in 81.4%

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

      2. +-commutative 81.4%

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

      3. associate-*r/ 82.4%

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

      4. *-commutative 82.4%

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

      5. associate-/l* 84.0%

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

   8. Simplified 87.5%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.8 \cdot 10^{+30}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{\frac{D}{d}}{\frac{2}{M}}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + -0.5 \cdot \frac{{\left(\frac{0.5}{d} \cdot \left(D \cdot M\right)\right)}^{2}}{\frac{\ell}{h}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(h, -0.5 \cdot \frac{{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2}}{\ell}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -2.65 \cdot 10^{-233}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(D\_m \cdot \frac{\frac{M}{2}}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt{0}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(h, -0.5 \cdot \frac{{\left(M \cdot \frac{\frac{D\_m}{d}}{2}\right)}^{2}}{\ell}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= d -2.65e-233)
   (*
    (sqrt (/ d l))
    (*
     (/ (sqrt (- d)) (sqrt (- h)))
     (+ 1.0 (* (/ h l) (* -0.5 (pow (* D_m (/ (/ M 2.0) d)) 2.0))))))
   (if (<= d -4e-310)
     (/ d (sqrt 0.0))
     (*
      d
      (/
       (fma h (* -0.5 (/ (pow (* M (/ (/ D_m d) 2.0)) 2.0) l)) 1.0)
       (* (sqrt l) (sqrt h)))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (d <= -2.65e-233) {
		tmp = sqrt((d / l)) * ((sqrt(-d) / sqrt(-h)) * (1.0 + ((h / l) * (-0.5 * pow((D_m * ((M / 2.0) / d)), 2.0)))));
	} else if (d <= -4e-310) {
		tmp = d / sqrt(0.0);
	} else {
		tmp = d * (fma(h, (-0.5 * (pow((M * ((D_m / d) / 2.0)), 2.0) / l)), 1.0) / (sqrt(l) * sqrt(h)));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (d <= -2.65e-233)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(D_m * Float64(Float64(M / 2.0) / d)) ^ 2.0))))));
	elseif (d <= -4e-310)
		tmp = Float64(d / sqrt(0.0));
	else
		tmp = Float64(d * Float64(fma(h, Float64(-0.5 * Float64((Float64(M * Float64(Float64(D_m / d) / 2.0)) ^ 2.0) / l)), 1.0) / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[d, -2.65e-233], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(D$95$m * N[(N[(M / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4e-310], N[(d / N[Sqrt[0.0], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(h * N[(-0.5 * N[(N[Power[N[(M * N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.64999999999999986e-233

   1. Initial program 79.7%

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

   3. Simplified 78.1%

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

      1. frac-2neg 78.9%

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

      2. sqrt-div 84.4%

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

   6. Applied egg-rr 83.6%

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

   if -2.64999999999999986e-233 < d < -3.999999999999988e-310

   1. Initial program 27.5%

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

   3. Simplified 27.2%

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

   5. Taylor expanded in d around inf 24.8%

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

      1. *-commutative 24.8%

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

   7. Simplified 24.8%

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

      1. pow1 24.8%

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

      2. metadata-eval 24.8%

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

      3. sqrt-pow1 3.0%

         \[\leadsto \color{blue}{\sqrt{{d}^{2}}} \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      4. sqrt-prod 3.0%

         \[\leadsto \color{blue}{\sqrt{{d}^{2} \cdot \frac{1}{\ell \cdot h}}} \]

      5. div-inv 3.0%

         \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{\ell \cdot h}}} \]

      6. sqrt-div 3.0%

         \[\leadsto \color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{\ell \cdot h}}} \]

      7. sqrt-pow1 19.6%

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

      8. metadata-eval 19.6%

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

      9. pow1 19.6%

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

      10. *-commutative 19.6%

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

   9. Applied egg-rr 19.6%

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

       1. expm1-log1p-u 19.6%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(h \cdot \ell\right)\right)}}} \]

       2. expm1-undefine 40.0%

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

       3. *-commutative 40.0%

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

   11. Applied egg-rr 40.0%

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

       1. log1p-undefine 40.0%

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

       2. rem-exp-log 40.0%

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

       3. associate-+r- 40.0%

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

       4. *-commutative 40.0%

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

       5. fma-neg 40.0%

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

       6. metadata-eval 40.0%

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

   13. Simplified 40.0%

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

   14. Taylor expanded in h around 0 58.9%

       \[\leadsto \frac{d}{\sqrt{1 + \color{blue}{-1}}} \]

   if -3.999999999999988e-310 < d

   1. Initial program 66.1%

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

   3. Simplified 66.1%

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

   6. Applied egg-rr 72.8%

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

      1. distribute-rgt1-in 81.4%

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

      2. +-commutative 81.4%

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

      3. associate-*r/ 82.4%

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

      4. *-commutative 82.4%

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

      5. associate-/l* 84.0%

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

   8. Simplified 87.5%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.65 \cdot 10^{-233}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt{0}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(h, -0.5 \cdot \frac{{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2}}{\ell}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \frac{{\left(M \cdot \frac{\frac{D\_m}{d}}{2}\right)}^{2}}{\ell}\\ \mathbf{if}\;d \leq -5.8 \cdot 10^{-231}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(h \cdot t\_0\right)\right)\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt{0}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(h, -0.5 \cdot t\_0, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (/ (pow (* M (/ (/ D_m d) 2.0)) 2.0) l)))
   (if (<= d -5.8e-231)
     (* (sqrt (/ d l)) (* (sqrt (/ d h)) (+ 1.0 (* -0.5 (* h t_0)))))
     (if (<= d -4e-310)
       (/ d (sqrt 0.0))
       (* d (/ (fma h (* -0.5 t_0) 1.0) (* (sqrt l) (sqrt h))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = pow((M * ((D_m / d) / 2.0)), 2.0) / l;
	double tmp;
	if (d <= -5.8e-231) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + (-0.5 * (h * t_0))));
	} else if (d <= -4e-310) {
		tmp = d / sqrt(0.0);
	} else {
		tmp = d * (fma(h, (-0.5 * t_0), 1.0) / (sqrt(l) * sqrt(h)));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64((Float64(M * Float64(Float64(D_m / d) / 2.0)) ^ 2.0) / l)
	tmp = 0.0
	if (d <= -5.8e-231)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(-0.5 * Float64(h * t_0)))));
	elseif (d <= -4e-310)
		tmp = Float64(d / sqrt(0.0));
	else
		tmp = Float64(d * Float64(fma(h, Float64(-0.5 * t_0), 1.0) / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[Power[N[(M * N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[d, -5.8e-231], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(h * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4e-310], N[(d / N[Sqrt[0.0], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(h * N[(-0.5 * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.8000000000000001e-231

   1. Initial program 79.7%

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

   3. Simplified 78.1%

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

      1. *-commutative 78.1%

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

      2. clear-num 78.1%

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

      3. un-div-inv 79.0%

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

      4. *-commutative 79.0%

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

      5. associate-*r/ 80.6%

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

      6. *-un-lft-identity 80.6%

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

      7. times-frac 79.0%

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

      8. associate-/l/ 79.0%

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

      9. *-commutative 79.0%

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

      10. times-frac 80.6%

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

      11. *-commutative 80.6%

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

      12. *-un-lft-identity 80.6%

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

      13. associate-/l* 79.8%

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

      14. *-un-lft-identity 79.8%

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

      15. times-frac 79.8%

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

      16. metadata-eval 79.8%

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

   6. Applied egg-rr 79.8%

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

      1. associate-/r/ 83.2%

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

      2. associate-*l/ 81.5%

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

      3. associate-/l* 79.0%

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

      4. associate-*l* 79.0%

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

      5. *-commutative 79.0%

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

      6. associate-*l/ 81.5%

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

      7. associate-/l* 83.2%

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

      8. associate-*r* 83.2%

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

      9. *-commutative 83.2%

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

      10. metadata-eval 83.2%

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

      11. associate-/l* 83.2%

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

      12. *-rgt-identity 83.2%

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

      13. associate-*r/ 83.2%

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

      14. *-commutative 83.2%

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

      15. associate-/l* 83.2%

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

   8. Simplified 83.2%

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

   if -5.8000000000000001e-231 < d < -3.999999999999988e-310

   1. Initial program 27.5%

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

   3. Simplified 27.2%

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

   5. Taylor expanded in d around inf 24.8%

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

      1. *-commutative 24.8%

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

   7. Simplified 24.8%

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

      1. pow1 24.8%

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

      2. metadata-eval 24.8%

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

      3. sqrt-pow1 3.0%

         \[\leadsto \color{blue}{\sqrt{{d}^{2}}} \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      4. sqrt-prod 3.0%

         \[\leadsto \color{blue}{\sqrt{{d}^{2} \cdot \frac{1}{\ell \cdot h}}} \]

      5. div-inv 3.0%

         \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{\ell \cdot h}}} \]

      6. sqrt-div 3.0%

         \[\leadsto \color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{\ell \cdot h}}} \]

      7. sqrt-pow1 19.6%

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

      8. metadata-eval 19.6%

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

      9. pow1 19.6%

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

      10. *-commutative 19.6%

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

   9. Applied egg-rr 19.6%

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

       1. expm1-log1p-u 19.6%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(h \cdot \ell\right)\right)}}} \]

       2. expm1-undefine 40.0%

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

       3. *-commutative 40.0%

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

   11. Applied egg-rr 40.0%

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

       1. log1p-undefine 40.0%

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

       2. rem-exp-log 40.0%

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

       3. associate-+r- 40.0%

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

       4. *-commutative 40.0%

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

       5. fma-neg 40.0%

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

       6. metadata-eval 40.0%

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

   13. Simplified 40.0%

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

   14. Taylor expanded in h around 0 58.9%

       \[\leadsto \frac{d}{\sqrt{1 + \color{blue}{-1}}} \]

   if -3.999999999999988e-310 < d

   1. Initial program 66.1%

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

   3. Simplified 66.1%

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

   6. Applied egg-rr 72.8%

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

      1. distribute-rgt1-in 81.4%

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

      2. +-commutative 81.4%

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

      3. associate-*r/ 82.4%

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

      4. *-commutative 82.4%

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

      5. associate-/l* 84.0%

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

   8. Simplified 87.5%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.8 \cdot 10^{-231}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt{0}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(h, -0.5 \cdot \frac{{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2}}{\ell}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -42000:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{-272}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \mathsf{fma}\left(h \cdot \frac{{\left(\frac{0.5}{d} \cdot \left(D\_m \cdot M\right)\right)}^{2}}{\ell}, -0.5, 1\right)\\ \mathbf{elif}\;\ell \leq 8.2 \cdot 10^{+192}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(0.5 \cdot \left(D\_m \cdot \frac{M}{d}\right)\right)}^{2}, 1\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= l -42000.0)
   (*
    (* d (sqrt (/ 1.0 (* l h))))
    (+ -1.0 (* 0.5 (* (/ h l) (pow (* (/ D_m d) (/ M 2.0)) 2.0)))))
   (if (<= l 8e-272)
     (*
      (sqrt (* (/ d l) (/ d h)))
      (fma (* h (/ (pow (* (/ 0.5 d) (* D_m M)) 2.0) l)) -0.5 1.0))
     (if (<= l 8.2e+192)
       (*
        d
        (/
         (fma (/ h l) (* -0.5 (pow (* 0.5 (* D_m (/ M d))) 2.0)) 1.0)
         (sqrt (* l h))))
       (* d (/ (sqrt (/ 1.0 l)) (sqrt h)))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (l <= -42000.0) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * ((h / l) * pow(((D_m / d) * (M / 2.0)), 2.0))));
	} else if (l <= 8e-272) {
		tmp = sqrt(((d / l) * (d / h))) * fma((h * (pow(((0.5 / d) * (D_m * M)), 2.0) / l)), -0.5, 1.0);
	} else if (l <= 8.2e+192) {
		tmp = d * (fma((h / l), (-0.5 * pow((0.5 * (D_m * (M / d))), 2.0)), 1.0) / sqrt((l * h)));
	} else {
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (l <= -42000.0)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(-1.0 + Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(M / 2.0)) ^ 2.0)))));
	elseif (l <= 8e-272)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * fma(Float64(h * Float64((Float64(Float64(0.5 / d) * Float64(D_m * M)) ^ 2.0) / l)), -0.5, 1.0));
	elseif (l <= 8.2e+192)
		tmp = Float64(d * Float64(fma(Float64(h / l), Float64(-0.5 * (Float64(0.5 * Float64(D_m * Float64(M / d))) ^ 2.0)), 1.0) / sqrt(Float64(l * h))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / l)) / sqrt(h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[l, -42000.0], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 8e-272], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(h * N[(N[Power[N[(N[(0.5 / d), $MachinePrecision] * N[(D$95$m * M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 8.2e+192], N[(d * N[(N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(0.5 * N[(D$95$m * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -42000

   1. Initial program 61.4%

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

   3. Simplified 59.9%

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

      1. sqrt-unprod 43.0%

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

      2. pow1/2 43.0%

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

      3. frac-times 32.3%

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

      4. pow2 32.3%

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

   6. Applied egg-rr 32.3%

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

      1. unpow1/2 32.3%

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

      2. *-commutative 32.3%

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

   8. Simplified 32.3%

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

   9. Taylor expanded in d around -inf 64.1%

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

   if -42000 < l < 7.99999999999999944e-272

   1. Initial program 82.3%

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

   3. Simplified 79.8%

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

      1. *-commutative 79.8%

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

      2. clear-num 79.8%

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

      3. un-div-inv 81.1%

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

      4. *-commutative 81.1%

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

      5. associate-*r/ 83.6%

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

      6. *-un-lft-identity 83.6%

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

      7. times-frac 81.1%

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

      8. associate-/l/ 81.1%

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

      9. *-commutative 81.1%

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

      10. times-frac 83.6%

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

      11. *-commutative 83.6%

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

      12. *-un-lft-identity 83.6%

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

      13. associate-/l* 83.7%

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

      14. *-un-lft-identity 83.7%

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

      15. times-frac 83.7%

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

      16. metadata-eval 83.7%

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

   6. Applied egg-rr 83.7%

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

      1. associate-/l* 83.7%

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

      2. associate-*r* 83.7%

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

      3. associate-/l* 83.6%

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

      4. *-commutative 83.6%

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

      5. associate-*r* 83.6%

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

      6. associate-/l* 83.6%

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

   8. Simplified 83.6%

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

      1. pow1 83.6%

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

      2. associate-*r* 83.6%

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

      3. pow1/2 83.6%

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

      4. pow1/2 83.6%

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

      5. pow-prod-down 78.8%

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

      6. +-commutative 78.8%

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

      7. *-commutative 78.8%

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

      8. fma-define 78.8%

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

      9. div-inv 77.5%

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

      10. associate-*l* 75.0%

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

      11. clear-num 75.0%

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

   10. Applied egg-rr 75.0%

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

       1. unpow1 75.0%

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

       2. unpow1/2 75.0%

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

       3. associate-*r/ 81.3%

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

       4. *-commutative 81.3%

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

       5. associate-*r/ 81.3%

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

       6. associate-*r* 83.9%

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

   12. Simplified 83.9%

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

   if 7.99999999999999944e-272 < l < 8.20000000000000006e192

   1. Initial program 71.3%

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

   3. Simplified 71.3%

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

      1. sqrt-unprod 59.3%

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

      2. pow1/2 59.3%

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

      3. frac-times 51.6%

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

      4. pow2 51.6%

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

   6. Applied egg-rr 51.6%

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

      1. unpow1/2 51.6%

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

      2. *-commutative 51.6%

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

   8. Simplified 51.6%

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

      1. pow1 51.6%

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

   10. Applied egg-rr 76.3%

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

       1. unpow1 76.3%

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

       2. associate-*l/ 75.5%

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

       3. associate-/l* 77.4%

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

       4. +-commutative 77.4%

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

       5. associate-*r* 77.4%

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

       6. *-commutative 77.4%

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

       7. fma-define 77.4%

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

       8. *-commutative 77.4%

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

       9. associate-*r* 77.4%

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

       10. associate-*l/ 77.4%

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

       11. *-commutative 77.4%

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

       12. associate-/l* 76.4%

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

   12. Simplified 76.4%

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

   if 8.20000000000000006e192 < l

   1. Initial program 38.1%

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

   3. Simplified 38.1%

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

   5. Taylor expanded in d around inf 37.1%

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

      1. *-commutative 37.1%

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

   7. Simplified 37.1%

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

      1. associate-/r* 41.6%

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

      2. sqrt-div 65.4%

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

   9. Applied egg-rr 65.4%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -42000:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{-272}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \mathsf{fma}\left(h \cdot \frac{{\left(\frac{0.5}{d} \cdot \left(D \cdot M\right)\right)}^{2}}{\ell}, -0.5, 1\right)\\ \mathbf{elif}\;\ell \leq 8.2 \cdot 10^{+192}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right)}^{2}, 1\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1.15 \cdot 10^{-227}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(M \cdot \frac{\frac{D\_m}{d}}{2}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt{0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= d -1.15e-227)
   (*
    (sqrt (/ d l))
    (*
     (sqrt (/ d h))
     (+ 1.0 (* -0.5 (* h (/ (pow (* M (/ (/ D_m d) 2.0)) 2.0) l))))))
   (if (<= d -4e-310)
     (/ d (sqrt 0.0))
     (*
      (/ d (* (sqrt l) (sqrt h)))
      (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D_m d) (/ M 2.0)) 2.0))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (d <= -1.15e-227) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + (-0.5 * (h * (pow((M * ((D_m / d) / 2.0)), 2.0) / l)))));
	} else if (d <= -4e-310) {
		tmp = d / sqrt(0.0);
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - (0.5 * ((h / l) * pow(((D_m / d) * (M / 2.0)), 2.0))));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: tmp
    if (d <= (-1.15d-227)) then
        tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0d0 + ((-0.5d0) * (h * (((m * ((d_m / d) / 2.0d0)) ** 2.0d0) / l)))))
    else if (d <= (-4d-310)) then
        tmp = d / sqrt(0.0d0)
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 - (0.5d0 * ((h / l) * (((d_m / d) * (m / 2.0d0)) ** 2.0d0))))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (d <= -1.15e-227) {
		tmp = Math.sqrt((d / l)) * (Math.sqrt((d / h)) * (1.0 + (-0.5 * (h * (Math.pow((M * ((D_m / d) / 2.0)), 2.0) / l)))));
	} else if (d <= -4e-310) {
		tmp = d / Math.sqrt(0.0);
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 - (0.5 * ((h / l) * Math.pow(((D_m / d) * (M / 2.0)), 2.0))));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if d <= -1.15e-227:
		tmp = math.sqrt((d / l)) * (math.sqrt((d / h)) * (1.0 + (-0.5 * (h * (math.pow((M * ((D_m / d) / 2.0)), 2.0) / l)))))
	elif d <= -4e-310:
		tmp = d / math.sqrt(0.0)
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 - (0.5 * ((h / l) * math.pow(((D_m / d) * (M / 2.0)), 2.0))))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (d <= -1.15e-227)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(-0.5 * Float64(h * Float64((Float64(M * Float64(Float64(D_m / d) / 2.0)) ^ 2.0) / l))))));
	elseif (d <= -4e-310)
		tmp = Float64(d / sqrt(0.0));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(M / 2.0)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (d <= -1.15e-227)
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + (-0.5 * (h * (((M * ((D_m / d) / 2.0)) ^ 2.0) / l)))));
	elseif (d <= -4e-310)
		tmp = d / sqrt(0.0);
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - (0.5 * ((h / l) * (((D_m / d) * (M / 2.0)) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[d, -1.15e-227], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(h * N[(N[Power[N[(M * N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4e-310], N[(d / N[Sqrt[0.0], $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.15000000000000006e-227

   1. Initial program 79.7%

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

   3. Simplified 78.1%

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

      1. *-commutative 78.1%

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

      2. clear-num 78.1%

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

      3. un-div-inv 79.0%

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

      4. *-commutative 79.0%

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

      5. associate-*r/ 80.6%

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

      6. *-un-lft-identity 80.6%

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

      7. times-frac 79.0%

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

      8. associate-/l/ 79.0%

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

      9. *-commutative 79.0%

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

      10. times-frac 80.6%

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

      11. *-commutative 80.6%

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

      12. *-un-lft-identity 80.6%

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

      13. associate-/l* 79.8%

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

      14. *-un-lft-identity 79.8%

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

      15. times-frac 79.8%

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

      16. metadata-eval 79.8%

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

   6. Applied egg-rr 79.8%

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

      1. associate-/r/ 83.2%

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

      2. associate-*l/ 81.5%

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

      3. associate-/l* 79.0%

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

      4. associate-*l* 79.0%

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

      5. *-commutative 79.0%

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

      6. associate-*l/ 81.5%

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

      7. associate-/l* 83.2%

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

      8. associate-*r* 83.2%

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

      9. *-commutative 83.2%

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

      10. metadata-eval 83.2%

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

      11. associate-/l* 83.2%

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

      12. *-rgt-identity 83.2%

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

      13. associate-*r/ 83.2%

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

      14. *-commutative 83.2%

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

      15. associate-/l* 83.2%

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

   8. Simplified 83.2%

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

   if -1.15000000000000006e-227 < d < -3.999999999999988e-310

   1. Initial program 27.5%

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

   3. Simplified 27.2%

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

   5. Taylor expanded in d around inf 24.8%

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

      1. *-commutative 24.8%

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

   7. Simplified 24.8%

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

      1. pow1 24.8%

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

      2. metadata-eval 24.8%

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

      3. sqrt-pow1 3.0%

         \[\leadsto \color{blue}{\sqrt{{d}^{2}}} \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      4. sqrt-prod 3.0%

         \[\leadsto \color{blue}{\sqrt{{d}^{2} \cdot \frac{1}{\ell \cdot h}}} \]

      5. div-inv 3.0%

         \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{\ell \cdot h}}} \]

      6. sqrt-div 3.0%

         \[\leadsto \color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{\ell \cdot h}}} \]

      7. sqrt-pow1 19.6%

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

      8. metadata-eval 19.6%

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

      9. pow1 19.6%

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

      10. *-commutative 19.6%

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

   9. Applied egg-rr 19.6%

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

       1. expm1-log1p-u 19.6%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(h \cdot \ell\right)\right)}}} \]

       2. expm1-undefine 40.0%

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

       3. *-commutative 40.0%

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

   11. Applied egg-rr 40.0%

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

       1. log1p-undefine 40.0%

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

       2. rem-exp-log 40.0%

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

       3. associate-+r- 40.0%

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

       4. *-commutative 40.0%

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

       5. fma-neg 40.0%

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

       6. metadata-eval 40.0%

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

   13. Simplified 40.0%

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

   14. Taylor expanded in h around 0 58.9%

       \[\leadsto \frac{d}{\sqrt{1 + \color{blue}{-1}}} \]

   if -3.999999999999988e-310 < d

   1. Initial program 66.1%

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

   3. Simplified 66.1%

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

      1. *-commutative 66.1%

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

      2. sqrt-div 72.2%

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

      3. sqrt-div 81.2%

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

      4. frac-times 81.2%

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

      5. add-sqr-sqrt 81.4%

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

   6. Applied egg-rr 81.4%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.15 \cdot 10^{-227}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt{0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \frac{M}{2}\right)}^{2}\right)\\ \mathbf{if}\;\ell \leq -42000:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + t\_0\right)\\ \mathbf{elif}\;\ell \leq 4.8 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \mathsf{fma}\left(h \cdot \frac{{\left(\frac{0.5}{d} \cdot \left(D\_m \cdot M\right)\right)}^{2}}{\ell}, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - t\_0\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (* (/ h l) (pow (* (/ D_m d) (/ M 2.0)) 2.0)))))
   (if (<= l -42000.0)
     (* (* d (sqrt (/ 1.0 (* l h)))) (+ -1.0 t_0))
     (if (<= l 4.8e-305)
       (*
        (sqrt (* (/ d l) (/ d h)))
        (fma (* h (/ (pow (* (/ 0.5 d) (* D_m M)) 2.0) l)) -0.5 1.0))
       (* (/ d (* (sqrt l) (sqrt h))) (- 1.0 t_0))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = 0.5 * ((h / l) * pow(((D_m / d) * (M / 2.0)), 2.0));
	double tmp;
	if (l <= -42000.0) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + t_0);
	} else if (l <= 4.8e-305) {
		tmp = sqrt(((d / l) * (d / h))) * fma((h * (pow(((0.5 / d) * (D_m * M)), 2.0) / l)), -0.5, 1.0);
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - t_0);
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(M / 2.0)) ^ 2.0)))
	tmp = 0.0
	if (l <= -42000.0)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(-1.0 + t_0));
	elseif (l <= 4.8e-305)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * fma(Float64(h * Float64((Float64(Float64(0.5 / d) * Float64(D_m * M)) ^ 2.0) / l)), -0.5, 1.0));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 - t_0));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -42000.0], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.8e-305], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(h * N[(N[Power[N[(N[(0.5 / d), $MachinePrecision] * N[(D$95$m * M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -42000

   1. Initial program 61.4%

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

   3. Simplified 59.9%

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

      1. sqrt-unprod 43.0%

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

      2. pow1/2 43.0%

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

      3. frac-times 32.3%

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

      4. pow2 32.3%

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

   6. Applied egg-rr 32.3%

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

      1. unpow1/2 32.3%

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

      2. *-commutative 32.3%

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

   8. Simplified 32.3%

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

   9. Taylor expanded in d around -inf 64.1%

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

   if -42000 < l < 4.80000000000000039e-305

   1. Initial program 81.9%

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

   3. Simplified 80.6%

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

      1. *-commutative 80.6%

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

      2. clear-num 80.6%

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

      3. un-div-inv 81.9%

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

      4. *-commutative 81.9%

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

      5. associate-*r/ 83.2%

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

      6. *-un-lft-identity 83.2%

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

      7. times-frac 81.9%

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

      8. associate-/l/ 81.9%

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

      9. *-commutative 81.9%

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

      10. times-frac 83.2%

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

      11. *-commutative 83.2%

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

      12. *-un-lft-identity 83.2%

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

      13. associate-/l* 83.2%

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

      14. *-un-lft-identity 83.2%

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

      15. times-frac 83.2%

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

      16. metadata-eval 83.2%

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

   6. Applied egg-rr 83.2%

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

      1. associate-/l* 83.2%

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

      2. associate-*r* 83.2%

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

      3. associate-/l* 83.2%

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

      4. *-commutative 83.2%

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

      5. associate-*r* 83.2%

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

      6. associate-/l* 83.2%

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

   8. Simplified 83.2%

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

      1. pow1 83.2%

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

      2. associate-*r* 83.2%

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

      3. pow1/2 83.2%

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

      4. pow1/2 83.2%

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

      5. pow-prod-down 79.5%

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

      6. +-commutative 79.5%

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

      7. *-commutative 79.5%

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

      8. fma-define 79.5%

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

      9. div-inv 78.2%

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

      10. associate-*l* 76.9%

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

      11. clear-num 76.9%

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

   10. Applied egg-rr 76.9%

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

       1. unpow1 76.9%

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

       2. unpow1/2 76.9%

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

       3. associate-*r/ 83.4%

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

       4. *-commutative 83.4%

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

       5. associate-*r/ 83.4%

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

       6. associate-*r* 84.8%

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

   12. Simplified 84.8%

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

   if 4.80000000000000039e-305 < l

   1. Initial program 66.1%

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

   3. Simplified 66.1%

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

      1. *-commutative 66.1%

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

      2. sqrt-div 72.2%

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

      3. sqrt-div 81.2%

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

      4. frac-times 81.2%

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

      5. add-sqr-sqrt 81.4%

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

   6. Applied egg-rr 81.4%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -42000:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 4.8 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \mathsf{fma}\left(h \cdot \frac{{\left(\frac{0.5}{d} \cdot \left(D \cdot M\right)\right)}^{2}}{\ell}, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := {\left(\frac{0.5}{d} \cdot \left(D\_m \cdot M\right)\right)}^{2}\\ \mathbf{if}\;\ell \leq -42000:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 10^{-307}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \mathsf{fma}\left(h \cdot \frac{t\_0}{\ell}, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + t\_0 \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right) \cdot \frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (pow (* (/ 0.5 d) (* D_m M)) 2.0)))
   (if (<= l -42000.0)
     (*
      (* d (sqrt (/ 1.0 (* l h))))
      (+ -1.0 (* 0.5 (* (/ h l) (pow (* (/ D_m d) (/ M 2.0)) 2.0)))))
     (if (<= l 1e-307)
       (* (sqrt (* (/ d l) (/ d h))) (fma (* h (/ t_0 l)) -0.5 1.0))
       (* (+ 1.0 (* t_0 (* (/ h l) -0.5))) (/ (/ d (sqrt h)) (sqrt l)))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = pow(((0.5 / d) * (D_m * M)), 2.0);
	double tmp;
	if (l <= -42000.0) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * ((h / l) * pow(((D_m / d) * (M / 2.0)), 2.0))));
	} else if (l <= 1e-307) {
		tmp = sqrt(((d / l) * (d / h))) * fma((h * (t_0 / l)), -0.5, 1.0);
	} else {
		tmp = (1.0 + (t_0 * ((h / l) * -0.5))) * ((d / sqrt(h)) / sqrt(l));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64(0.5 / d) * Float64(D_m * M)) ^ 2.0
	tmp = 0.0
	if (l <= -42000.0)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(-1.0 + Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(M / 2.0)) ^ 2.0)))));
	elseif (l <= 1e-307)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * fma(Float64(h * Float64(t_0 / l)), -0.5, 1.0));
	else
		tmp = Float64(Float64(1.0 + Float64(t_0 * Float64(Float64(h / l) * -0.5))) * Float64(Float64(d / sqrt(h)) / sqrt(l)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Power[N[(N[(0.5 / d), $MachinePrecision] * N[(D$95$m * M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[l, -42000.0], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1e-307], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(h * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(t$95$0 * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -42000

   1. Initial program 61.4%

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

   3. Simplified 59.9%

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

      1. sqrt-unprod 43.0%

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

      2. pow1/2 43.0%

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

      3. frac-times 32.3%

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

      4. pow2 32.3%

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

   6. Applied egg-rr 32.3%

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

      1. unpow1/2 32.3%

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

      2. *-commutative 32.3%

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

   8. Simplified 32.3%

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

   9. Taylor expanded in d around -inf 64.1%

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

   if -42000 < l < 9.99999999999999909e-308

   1. Initial program 81.9%

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

   3. Simplified 80.6%

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

      1. *-commutative 80.6%

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

      2. clear-num 80.6%

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

      3. un-div-inv 81.9%

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

      4. *-commutative 81.9%

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

      5. associate-*r/ 83.2%

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

      6. *-un-lft-identity 83.2%

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

      7. times-frac 81.9%

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

      8. associate-/l/ 81.9%

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

      9. *-commutative 81.9%

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

      10. times-frac 83.2%

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

      11. *-commutative 83.2%

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

      12. *-un-lft-identity 83.2%

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

      13. associate-/l* 83.2%

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

      14. *-un-lft-identity 83.2%

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

      15. times-frac 83.2%

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

      16. metadata-eval 83.2%

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

   6. Applied egg-rr 83.2%

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

      1. associate-/l* 83.2%

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

      2. associate-*r* 83.2%

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

      3. associate-/l* 83.2%

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

      4. *-commutative 83.2%

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

      5. associate-*r* 83.2%

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

      6. associate-/l* 83.2%

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

   8. Simplified 83.2%

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

      1. pow1 83.2%

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

      2. associate-*r* 83.2%

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

      3. pow1/2 83.2%

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

      4. pow1/2 83.2%

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

      5. pow-prod-down 79.5%

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

      6. +-commutative 79.5%

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

      7. *-commutative 79.5%

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

      8. fma-define 79.5%

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

      9. div-inv 78.2%

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

      10. associate-*l* 76.9%

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

      11. clear-num 76.9%

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

   10. Applied egg-rr 76.9%

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

       1. unpow1 76.9%

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

       2. unpow1/2 76.9%

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

       3. associate-*r/ 83.4%

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

       4. *-commutative 83.4%

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

       5. associate-*r/ 83.4%

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

       6. associate-*r* 84.8%

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

   12. Simplified 84.8%

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

   if 9.99999999999999909e-308 < l

   1. Initial program 66.1%

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

   3. Simplified 66.1%

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

   6. Applied egg-rr 72.8%

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

      1. distribute-rgt1-in 81.4%

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

      2. +-commutative 81.4%

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

      3. associate-*r* 81.4%

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

      4. associate-*r* 81.4%

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

      5. associate-/l* 81.4%

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

      6. *-commutative 81.4%

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

      7. associate-*r* 81.4%

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

      8. associate-/l* 81.4%

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

      9. associate-/l/ 77.9%

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

   8. Simplified 77.9%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -42000:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 10^{-307}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \mathsf{fma}\left(h \cdot \frac{{\left(\frac{0.5}{d} \cdot \left(D \cdot M\right)\right)}^{2}}{\ell}, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + {\left(\frac{0.5}{d} \cdot \left(D \cdot M\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right) \cdot \frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := {\left(\ell \cdot h\right)}^{-0.5}\\ t_1 := d \cdot \left(-t\_0\right)\\ \mathbf{if}\;\ell \leq -7 \cdot 10^{+14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq -1.22 \cdot 10^{-275}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+174}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot \left(d \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (pow (* l h) -0.5)) (t_1 (* d (- t_0))))
   (if (<= l -7e+14)
     t_1
     (if (<= l -1.22e-275)
       (*
        (sqrt (* (/ d l) (/ d h)))
        (+ 1.0 (* -0.5 (* (/ h l) (pow (* (/ D_m d) (* M 0.5)) 2.0)))))
       (if (<= l -1e-310)
         t_1
         (if (<= l 5e+174)
           (*
            (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D_m d) (/ M 2.0)) 2.0))))
            (* d t_0))
           (* d (/ (sqrt (/ 1.0 l)) (sqrt h)))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = pow((l * h), -0.5);
	double t_1 = d * -t_0;
	double tmp;
	if (l <= -7e+14) {
		tmp = t_1;
	} else if (l <= -1.22e-275) {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * ((h / l) * pow(((D_m / d) * (M * 0.5)), 2.0))));
	} else if (l <= -1e-310) {
		tmp = t_1;
	} else if (l <= 5e+174) {
		tmp = (1.0 - (0.5 * ((h / l) * pow(((D_m / d) * (M / 2.0)), 2.0)))) * (d * t_0);
	} else {
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (l * h) ** (-0.5d0)
    t_1 = d * -t_0
    if (l <= (-7d+14)) then
        tmp = t_1
    else if (l <= (-1.22d-275)) then
        tmp = sqrt(((d / l) * (d / h))) * (1.0d0 + ((-0.5d0) * ((h / l) * (((d_m / d) * (m * 0.5d0)) ** 2.0d0))))
    else if (l <= (-1d-310)) then
        tmp = t_1
    else if (l <= 5d+174) then
        tmp = (1.0d0 - (0.5d0 * ((h / l) * (((d_m / d) * (m / 2.0d0)) ** 2.0d0)))) * (d * t_0)
    else
        tmp = d * (sqrt((1.0d0 / l)) / sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = Math.pow((l * h), -0.5);
	double t_1 = d * -t_0;
	double tmp;
	if (l <= -7e+14) {
		tmp = t_1;
	} else if (l <= -1.22e-275) {
		tmp = Math.sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * ((h / l) * Math.pow(((D_m / d) * (M * 0.5)), 2.0))));
	} else if (l <= -1e-310) {
		tmp = t_1;
	} else if (l <= 5e+174) {
		tmp = (1.0 - (0.5 * ((h / l) * Math.pow(((D_m / d) * (M / 2.0)), 2.0)))) * (d * t_0);
	} else {
		tmp = d * (Math.sqrt((1.0 / l)) / Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = math.pow((l * h), -0.5)
	t_1 = d * -t_0
	tmp = 0
	if l <= -7e+14:
		tmp = t_1
	elif l <= -1.22e-275:
		tmp = math.sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * ((h / l) * math.pow(((D_m / d) * (M * 0.5)), 2.0))))
	elif l <= -1e-310:
		tmp = t_1
	elif l <= 5e+174:
		tmp = (1.0 - (0.5 * ((h / l) * math.pow(((D_m / d) * (M / 2.0)), 2.0)))) * (d * t_0)
	else:
		tmp = d * (math.sqrt((1.0 / l)) / math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(l * h) ^ -0.5
	t_1 = Float64(d * Float64(-t_0))
	tmp = 0.0
	if (l <= -7e+14)
		tmp = t_1;
	elseif (l <= -1.22e-275)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(M * 0.5)) ^ 2.0)))));
	elseif (l <= -1e-310)
		tmp = t_1;
	elseif (l <= 5e+174)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(M / 2.0)) ^ 2.0)))) * Float64(d * t_0));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / l)) / sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = (l * h) ^ -0.5;
	t_1 = d * -t_0;
	tmp = 0.0;
	if (l <= -7e+14)
		tmp = t_1;
	elseif (l <= -1.22e-275)
		tmp = sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * ((h / l) * (((D_m / d) * (M * 0.5)) ^ 2.0))));
	elseif (l <= -1e-310)
		tmp = t_1;
	elseif (l <= 5e+174)
		tmp = (1.0 - (0.5 * ((h / l) * (((D_m / d) * (M / 2.0)) ^ 2.0)))) * (d * t_0);
	else
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]}, Block[{t$95$1 = N[(d * (-t$95$0)), $MachinePrecision]}, If[LessEqual[l, -7e+14], t$95$1, If[LessEqual[l, -1.22e-275], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(M * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1e-310], t$95$1, If[LessEqual[l, 5e+174], N[(N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d * t$95$0), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -7e14 or -1.21999999999999995e-275 < l < -9.999999999999969e-311

   1. Initial program 55.5%

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

   3. Simplified 54.1%

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

   5. Taylor expanded in d around inf 9.1%

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

      1. *-commutative 9.1%

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

   7. Simplified 9.1%

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

   8. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. unpow-1 0.0%

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

      3. metadata-eval 0.0%

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

      4. pow-sqr 0.0%

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

      5. rem-sqrt-square 0.0%

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

      6. rem-square-sqrt 0.0%

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

      7. fabs-sqr 0.0%

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

      8. rem-square-sqrt 0.0%

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

      9. *-commutative 0.0%

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

      10. unpow2 0.0%

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

      11. rem-square-sqrt 58.3%

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

      12. neg-mul-1 58.3%

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

   10. Simplified 58.3%

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

   if -7e14 < l < -1.21999999999999995e-275

   1. Initial program 89.9%

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

   3. Simplified 90.0%

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

      1. add-sqr-sqrt 89.9%

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

      2. pow2 89.9%

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

      3. sqrt-prod 89.9%

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

      4. sqrt-pow1 92.8%

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

      5. metadata-eval 92.8%

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

      6. frac-times 92.8%

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

      7. pow1 92.8%

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

      8. associate-/l* 92.8%

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

      9. *-un-lft-identity 92.8%

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

      10. times-frac 92.8%

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

      11. metadata-eval 92.8%

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

   6. Applied egg-rr 92.8%

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

      1. pow1 92.8%

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

   8. Applied egg-rr 87.2%

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

      1. unpow1 87.2%

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

      2. *-commutative 87.2%

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

      3. *-commutative 87.2%

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

      4. *-commutative 87.2%

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

      5. *-commutative 87.2%

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

      6. *-commutative 87.2%

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

   10. Simplified 87.2%

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

   if -9.999999999999969e-311 < l < 4.9999999999999997e174

   1. Initial program 73.4%

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

   3. Simplified 73.4%

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

      1. sqrt-unprod 60.1%

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

      2. pow1/2 60.1%

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

      3. frac-times 53.2%

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

      4. pow2 53.2%

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

   6. Applied egg-rr 53.2%

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

      1. unpow1/2 53.2%

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

      2. *-commutative 53.2%

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

   8. Simplified 53.2%

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

      1. div-inv 53.1%

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

      2. sqrt-prod 61.1%

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

      3. sqrt-pow1 76.9%

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

      4. metadata-eval 76.9%

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

      5. pow1 76.9%

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

      6. *-commutative 76.9%

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

      7. inv-pow 76.9%

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

      8. sqrt-pow1 77.7%

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

      9. *-commutative 77.7%

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

      10. metadata-eval 77.7%

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

   10. Applied egg-rr 77.7%

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

   if 4.9999999999999997e174 < l

   1. Initial program 39.2%

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

   3. Simplified 39.2%

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

   5. Taylor expanded in d around inf 38.3%

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

      1. *-commutative 38.3%

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

   7. Simplified 38.3%

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

      1. associate-/r* 41.8%

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

      2. sqrt-div 60.7%

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

   9. Applied egg-rr 60.7%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7 \cdot 10^{+14}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;\ell \leq -1.22 \cdot 10^{-275}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+174}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := 1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \frac{M}{2}\right)}^{2}\right)\\ t_1 := {\left(\ell \cdot h\right)}^{-0.5}\\ t_2 := d \cdot \left(-t\_1\right)\\ \mathbf{if}\;\ell \leq -2.7 \cdot 10^{-18}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\ell \leq -7 \cdot 10^{-276}:\\ \;\;\;\;t\_0 \cdot \sqrt{\frac{d \cdot d}{\ell \cdot h}}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+175}:\\ \;\;\;\;t\_0 \cdot \left(d \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D_m d) (/ M 2.0)) 2.0)))))
        (t_1 (pow (* l h) -0.5))
        (t_2 (* d (- t_1))))
   (if (<= l -2.7e-18)
     t_2
     (if (<= l -7e-276)
       (* t_0 (sqrt (/ (* d d) (* l h))))
       (if (<= l -1e-310)
         t_2
         (if (<= l 1.8e+175)
           (* t_0 (* d t_1))
           (* d (/ (sqrt (/ 1.0 l)) (sqrt h)))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = 1.0 - (0.5 * ((h / l) * pow(((D_m / d) * (M / 2.0)), 2.0)));
	double t_1 = pow((l * h), -0.5);
	double t_2 = d * -t_1;
	double tmp;
	if (l <= -2.7e-18) {
		tmp = t_2;
	} else if (l <= -7e-276) {
		tmp = t_0 * sqrt(((d * d) / (l * h)));
	} else if (l <= -1e-310) {
		tmp = t_2;
	} else if (l <= 1.8e+175) {
		tmp = t_0 * (d * t_1);
	} else {
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 - (0.5d0 * ((h / l) * (((d_m / d) * (m / 2.0d0)) ** 2.0d0)))
    t_1 = (l * h) ** (-0.5d0)
    t_2 = d * -t_1
    if (l <= (-2.7d-18)) then
        tmp = t_2
    else if (l <= (-7d-276)) then
        tmp = t_0 * sqrt(((d * d) / (l * h)))
    else if (l <= (-1d-310)) then
        tmp = t_2
    else if (l <= 1.8d+175) then
        tmp = t_0 * (d * t_1)
    else
        tmp = d * (sqrt((1.0d0 / l)) / sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = 1.0 - (0.5 * ((h / l) * Math.pow(((D_m / d) * (M / 2.0)), 2.0)));
	double t_1 = Math.pow((l * h), -0.5);
	double t_2 = d * -t_1;
	double tmp;
	if (l <= -2.7e-18) {
		tmp = t_2;
	} else if (l <= -7e-276) {
		tmp = t_0 * Math.sqrt(((d * d) / (l * h)));
	} else if (l <= -1e-310) {
		tmp = t_2;
	} else if (l <= 1.8e+175) {
		tmp = t_0 * (d * t_1);
	} else {
		tmp = d * (Math.sqrt((1.0 / l)) / Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = 1.0 - (0.5 * ((h / l) * math.pow(((D_m / d) * (M / 2.0)), 2.0)))
	t_1 = math.pow((l * h), -0.5)
	t_2 = d * -t_1
	tmp = 0
	if l <= -2.7e-18:
		tmp = t_2
	elif l <= -7e-276:
		tmp = t_0 * math.sqrt(((d * d) / (l * h)))
	elif l <= -1e-310:
		tmp = t_2
	elif l <= 1.8e+175:
		tmp = t_0 * (d * t_1)
	else:
		tmp = d * (math.sqrt((1.0 / l)) / math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(M / 2.0)) ^ 2.0))))
	t_1 = Float64(l * h) ^ -0.5
	t_2 = Float64(d * Float64(-t_1))
	tmp = 0.0
	if (l <= -2.7e-18)
		tmp = t_2;
	elseif (l <= -7e-276)
		tmp = Float64(t_0 * sqrt(Float64(Float64(d * d) / Float64(l * h))));
	elseif (l <= -1e-310)
		tmp = t_2;
	elseif (l <= 1.8e+175)
		tmp = Float64(t_0 * Float64(d * t_1));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / l)) / sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = 1.0 - (0.5 * ((h / l) * (((D_m / d) * (M / 2.0)) ^ 2.0)));
	t_1 = (l * h) ^ -0.5;
	t_2 = d * -t_1;
	tmp = 0.0;
	if (l <= -2.7e-18)
		tmp = t_2;
	elseif (l <= -7e-276)
		tmp = t_0 * sqrt(((d * d) / (l * h)));
	elseif (l <= -1e-310)
		tmp = t_2;
	elseif (l <= 1.8e+175)
		tmp = t_0 * (d * t_1);
	else
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]}, Block[{t$95$2 = N[(d * (-t$95$1)), $MachinePrecision]}, If[LessEqual[l, -2.7e-18], t$95$2, If[LessEqual[l, -7e-276], N[(t$95$0 * N[Sqrt[N[(N[(d * d), $MachinePrecision] / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1e-310], t$95$2, If[LessEqual[l, 1.8e+175], N[(t$95$0 * N[(d * t$95$1), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.69999999999999989e-18 or -6.99999999999999986e-276 < l < -9.999999999999969e-311

   1. Initial program 58.8%

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

   3. Simplified 57.5%

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

   5. Taylor expanded in d around inf 9.7%

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

      1. *-commutative 9.7%

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

   7. Simplified 9.7%

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

   8. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. unpow-1 0.0%

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

      3. metadata-eval 0.0%

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

      4. pow-sqr 0.0%

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

      5. rem-sqrt-square 0.0%

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

      6. rem-square-sqrt 0.0%

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

      7. fabs-sqr 0.0%

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

      8. rem-square-sqrt 0.0%

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

      9. *-commutative 0.0%

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

      10. unpow2 0.0%

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

      11. rem-square-sqrt 60.0%

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

      12. neg-mul-1 60.0%

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

   10. Simplified 60.0%

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

   if -2.69999999999999989e-18 < l < -6.99999999999999986e-276

   1. Initial program 90.3%

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

   3. Simplified 90.4%

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

      1. sqrt-unprod 88.8%

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

      2. pow1/2 88.8%

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

      3. frac-times 65.7%

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

      4. pow2 65.7%

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

   6. Applied egg-rr 65.7%

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

      1. unpow1/2 65.7%

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

      2. *-commutative 65.7%

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

   8. Simplified 65.7%

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

      1. unpow2 65.7%

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

   10. Applied egg-rr 65.7%

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

   if -9.999999999999969e-311 < l < 1.80000000000000017e175

   1. Initial program 73.4%

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

   3. Simplified 73.4%

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

      1. sqrt-unprod 60.1%

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

      2. pow1/2 60.1%

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

      3. frac-times 53.2%

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

      4. pow2 53.2%

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

   6. Applied egg-rr 53.2%

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

      1. unpow1/2 53.2%

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

      2. *-commutative 53.2%

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

   8. Simplified 53.2%

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

      1. div-inv 53.1%

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

      2. sqrt-prod 61.1%

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

      3. sqrt-pow1 76.9%

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

      4. metadata-eval 76.9%

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

      5. pow1 76.9%

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

      6. *-commutative 76.9%

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

      7. inv-pow 76.9%

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

      8. sqrt-pow1 77.7%

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

      9. *-commutative 77.7%

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

      10. metadata-eval 77.7%

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

   10. Applied egg-rr 77.7%

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

   if 1.80000000000000017e175 < l

   1. Initial program 39.2%

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

   3. Simplified 39.2%

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

   5. Taylor expanded in d around inf 38.3%

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

      1. *-commutative 38.3%

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

   7. Simplified 38.3%

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

      1. associate-/r* 41.8%

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

      2. sqrt-div 60.7%

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

   9. Applied egg-rr 60.7%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.7 \cdot 10^{-18}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;\ell \leq -7 \cdot 10^{-276}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d \cdot d}{\ell \cdot h}}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+175}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := {\left(\ell \cdot h\right)}^{-0.5}\\ t_1 := d \cdot \left(-t\_0\right)\\ \mathbf{if}\;\ell \leq -2.8 \cdot 10^{-81}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq -6.3 \cdot 10^{-276}:\\ \;\;\;\;\frac{d}{\sqrt{0}}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{+175}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot \left(d \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (pow (* l h) -0.5)) (t_1 (* d (- t_0))))
   (if (<= l -2.8e-81)
     t_1
     (if (<= l -6.3e-276)
       (/ d (sqrt 0.0))
       (if (<= l -1e-310)
         t_1
         (if (<= l 1.5e+175)
           (*
            (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D_m d) (/ M 2.0)) 2.0))))
            (* d t_0))
           (* d (/ (sqrt (/ 1.0 l)) (sqrt h)))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = pow((l * h), -0.5);
	double t_1 = d * -t_0;
	double tmp;
	if (l <= -2.8e-81) {
		tmp = t_1;
	} else if (l <= -6.3e-276) {
		tmp = d / sqrt(0.0);
	} else if (l <= -1e-310) {
		tmp = t_1;
	} else if (l <= 1.5e+175) {
		tmp = (1.0 - (0.5 * ((h / l) * pow(((D_m / d) * (M / 2.0)), 2.0)))) * (d * t_0);
	} else {
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (l * h) ** (-0.5d0)
    t_1 = d * -t_0
    if (l <= (-2.8d-81)) then
        tmp = t_1
    else if (l <= (-6.3d-276)) then
        tmp = d / sqrt(0.0d0)
    else if (l <= (-1d-310)) then
        tmp = t_1
    else if (l <= 1.5d+175) then
        tmp = (1.0d0 - (0.5d0 * ((h / l) * (((d_m / d) * (m / 2.0d0)) ** 2.0d0)))) * (d * t_0)
    else
        tmp = d * (sqrt((1.0d0 / l)) / sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = Math.pow((l * h), -0.5);
	double t_1 = d * -t_0;
	double tmp;
	if (l <= -2.8e-81) {
		tmp = t_1;
	} else if (l <= -6.3e-276) {
		tmp = d / Math.sqrt(0.0);
	} else if (l <= -1e-310) {
		tmp = t_1;
	} else if (l <= 1.5e+175) {
		tmp = (1.0 - (0.5 * ((h / l) * Math.pow(((D_m / d) * (M / 2.0)), 2.0)))) * (d * t_0);
	} else {
		tmp = d * (Math.sqrt((1.0 / l)) / Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = math.pow((l * h), -0.5)
	t_1 = d * -t_0
	tmp = 0
	if l <= -2.8e-81:
		tmp = t_1
	elif l <= -6.3e-276:
		tmp = d / math.sqrt(0.0)
	elif l <= -1e-310:
		tmp = t_1
	elif l <= 1.5e+175:
		tmp = (1.0 - (0.5 * ((h / l) * math.pow(((D_m / d) * (M / 2.0)), 2.0)))) * (d * t_0)
	else:
		tmp = d * (math.sqrt((1.0 / l)) / math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(l * h) ^ -0.5
	t_1 = Float64(d * Float64(-t_0))
	tmp = 0.0
	if (l <= -2.8e-81)
		tmp = t_1;
	elseif (l <= -6.3e-276)
		tmp = Float64(d / sqrt(0.0));
	elseif (l <= -1e-310)
		tmp = t_1;
	elseif (l <= 1.5e+175)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(M / 2.0)) ^ 2.0)))) * Float64(d * t_0));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / l)) / sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = (l * h) ^ -0.5;
	t_1 = d * -t_0;
	tmp = 0.0;
	if (l <= -2.8e-81)
		tmp = t_1;
	elseif (l <= -6.3e-276)
		tmp = d / sqrt(0.0);
	elseif (l <= -1e-310)
		tmp = t_1;
	elseif (l <= 1.5e+175)
		tmp = (1.0 - (0.5 * ((h / l) * (((D_m / d) * (M / 2.0)) ^ 2.0)))) * (d * t_0);
	else
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]}, Block[{t$95$1 = N[(d * (-t$95$0)), $MachinePrecision]}, If[LessEqual[l, -2.8e-81], t$95$1, If[LessEqual[l, -6.3e-276], N[(d / N[Sqrt[0.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1e-310], t$95$1, If[LessEqual[l, 1.5e+175], N[(N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d * t$95$0), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.7999999999999999e-81 or -6.29999999999999979e-276 < l < -9.999999999999969e-311

   1. Initial program 63.3%

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

   3. Simplified 62.3%

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

   5. Taylor expanded in d around inf 8.7%

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

      1. *-commutative 8.7%

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

   7. Simplified 8.7%

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

   8. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. unpow-1 0.0%

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

      3. metadata-eval 0.0%

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

      4. pow-sqr 0.0%

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

      5. rem-sqrt-square 0.0%

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

      6. rem-square-sqrt 0.0%

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

      7. fabs-sqr 0.0%

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

      8. rem-square-sqrt 0.0%

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

      9. *-commutative 0.0%

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

      10. unpow2 0.0%

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

      11. rem-square-sqrt 56.6%

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

      12. neg-mul-1 56.6%

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

   10. Simplified 56.6%

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

   if -2.7999999999999999e-81 < l < -6.29999999999999979e-276

   1. Initial program 91.4%

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

   3. Simplified 91.4%

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

   5. Taylor expanded in d around inf 26.1%

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

      1. *-commutative 26.1%

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

   7. Simplified 26.1%

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

      1. pow1 26.1%

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

      2. metadata-eval 26.1%

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

      3. sqrt-pow1 20.2%

         \[\leadsto \color{blue}{\sqrt{{d}^{2}}} \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      4. sqrt-prod 20.2%

         \[\leadsto \color{blue}{\sqrt{{d}^{2} \cdot \frac{1}{\ell \cdot h}}} \]

      5. div-inv 20.3%

         \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{\ell \cdot h}}} \]

      6. sqrt-div 20.3%

         \[\leadsto \color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{\ell \cdot h}}} \]

      7. sqrt-pow1 24.0%

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

      8. metadata-eval 24.0%

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

      9. pow1 24.0%

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

      10. *-commutative 24.0%

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

   9. Applied egg-rr 24.0%

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

       1. expm1-log1p-u 24.0%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(h \cdot \ell\right)\right)}}} \]

       2. expm1-undefine 51.0%

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

       3. *-commutative 51.0%

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

   11. Applied egg-rr 51.0%

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

       1. log1p-undefine 51.0%

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

       2. rem-exp-log 51.0%

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

       3. associate-+r- 51.0%

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

       4. *-commutative 51.0%

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

       5. fma-neg 51.0%

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

       6. metadata-eval 51.0%

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

   13. Simplified 51.0%

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

   14. Taylor expanded in h around 0 61.5%

       \[\leadsto \frac{d}{\sqrt{1 + \color{blue}{-1}}} \]

   if -9.999999999999969e-311 < l < 1.5000000000000001e175

   1. Initial program 73.4%

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

   3. Simplified 73.4%

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

      1. sqrt-unprod 60.1%

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

      2. pow1/2 60.1%

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

      3. frac-times 53.2%

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

      4. pow2 53.2%

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

   6. Applied egg-rr 53.2%

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

      1. unpow1/2 53.2%

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

      2. *-commutative 53.2%

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

   8. Simplified 53.2%

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

      1. div-inv 53.1%

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

      2. sqrt-prod 61.1%

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

      3. sqrt-pow1 76.9%

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

      4. metadata-eval 76.9%

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

      5. pow1 76.9%

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

      6. *-commutative 76.9%

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

      7. inv-pow 76.9%

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

      8. sqrt-pow1 77.7%

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

      9. *-commutative 77.7%

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

      10. metadata-eval 77.7%

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

   10. Applied egg-rr 77.7%

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

   if 1.5000000000000001e175 < l

   1. Initial program 39.2%

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

   3. Simplified 39.2%

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

   5. Taylor expanded in d around inf 38.3%

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

      1. *-commutative 38.3%

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

   7. Simplified 38.3%

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

      1. associate-/r* 41.8%

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

      2. sqrt-div 60.7%

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

   9. Applied egg-rr 60.7%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.8 \cdot 10^{-81}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;\ell \leq -6.3 \cdot 10^{-276}:\\ \;\;\;\;\frac{d}{\sqrt{0}}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{+175}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{if}\;\ell \leq -8 \cdot 10^{-82}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -6.2 \cdot 10^{-276}:\\ \;\;\;\;\frac{d}{\sqrt{0}}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{+175}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (* d (- (pow (* l h) -0.5)))))
   (if (<= l -8e-82)
     t_0
     (if (<= l -6.2e-276)
       (/ d (sqrt 0.0))
       (if (<= l -1e-310)
         t_0
         (if (<= l 1.3e+175)
           (*
            (+ 1.0 (* -0.5 (* (/ h l) (pow (* (/ D_m d) (* M 0.5)) 2.0))))
            (/ d (sqrt (* l h))))
           (* d (/ (sqrt (/ 1.0 l)) (sqrt h)))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = d * -pow((l * h), -0.5);
	double tmp;
	if (l <= -8e-82) {
		tmp = t_0;
	} else if (l <= -6.2e-276) {
		tmp = d / sqrt(0.0);
	} else if (l <= -1e-310) {
		tmp = t_0;
	} else if (l <= 1.3e+175) {
		tmp = (1.0 + (-0.5 * ((h / l) * pow(((D_m / d) * (M * 0.5)), 2.0)))) * (d / sqrt((l * h)));
	} else {
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d * -((l * h) ** (-0.5d0))
    if (l <= (-8d-82)) then
        tmp = t_0
    else if (l <= (-6.2d-276)) then
        tmp = d / sqrt(0.0d0)
    else if (l <= (-1d-310)) then
        tmp = t_0
    else if (l <= 1.3d+175) then
        tmp = (1.0d0 + ((-0.5d0) * ((h / l) * (((d_m / d) * (m * 0.5d0)) ** 2.0d0)))) * (d / sqrt((l * h)))
    else
        tmp = d * (sqrt((1.0d0 / l)) / sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = d * -Math.pow((l * h), -0.5);
	double tmp;
	if (l <= -8e-82) {
		tmp = t_0;
	} else if (l <= -6.2e-276) {
		tmp = d / Math.sqrt(0.0);
	} else if (l <= -1e-310) {
		tmp = t_0;
	} else if (l <= 1.3e+175) {
		tmp = (1.0 + (-0.5 * ((h / l) * Math.pow(((D_m / d) * (M * 0.5)), 2.0)))) * (d / Math.sqrt((l * h)));
	} else {
		tmp = d * (Math.sqrt((1.0 / l)) / Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = d * -math.pow((l * h), -0.5)
	tmp = 0
	if l <= -8e-82:
		tmp = t_0
	elif l <= -6.2e-276:
		tmp = d / math.sqrt(0.0)
	elif l <= -1e-310:
		tmp = t_0
	elif l <= 1.3e+175:
		tmp = (1.0 + (-0.5 * ((h / l) * math.pow(((D_m / d) * (M * 0.5)), 2.0)))) * (d / math.sqrt((l * h)))
	else:
		tmp = d * (math.sqrt((1.0 / l)) / math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(d * Float64(-(Float64(l * h) ^ -0.5)))
	tmp = 0.0
	if (l <= -8e-82)
		tmp = t_0;
	elseif (l <= -6.2e-276)
		tmp = Float64(d / sqrt(0.0));
	elseif (l <= -1e-310)
		tmp = t_0;
	elseif (l <= 1.3e+175)
		tmp = Float64(Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(M * 0.5)) ^ 2.0)))) * Float64(d / sqrt(Float64(l * h))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / l)) / sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = d * -((l * h) ^ -0.5);
	tmp = 0.0;
	if (l <= -8e-82)
		tmp = t_0;
	elseif (l <= -6.2e-276)
		tmp = d / sqrt(0.0);
	elseif (l <= -1e-310)
		tmp = t_0;
	elseif (l <= 1.3e+175)
		tmp = (1.0 + (-0.5 * ((h / l) * (((D_m / d) * (M * 0.5)) ^ 2.0)))) * (d / sqrt((l * h)));
	else
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(d * (-N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision]}, If[LessEqual[l, -8e-82], t$95$0, If[LessEqual[l, -6.2e-276], N[(d / N[Sqrt[0.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1e-310], t$95$0, If[LessEqual[l, 1.3e+175], N[(N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(M * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -8e-82 or -6.19999999999999978e-276 < l < -9.999999999999969e-311

   1. Initial program 63.3%

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

   3. Simplified 62.3%

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

   5. Taylor expanded in d around inf 8.7%

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

      1. *-commutative 8.7%

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

   7. Simplified 8.7%

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

   8. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. unpow-1 0.0%

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

      3. metadata-eval 0.0%

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

      4. pow-sqr 0.0%

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

      5. rem-sqrt-square 0.0%

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

      6. rem-square-sqrt 0.0%

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

      7. fabs-sqr 0.0%

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

      8. rem-square-sqrt 0.0%

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

      9. *-commutative 0.0%

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

      10. unpow2 0.0%

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

      11. rem-square-sqrt 56.6%

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

      12. neg-mul-1 56.6%

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

   10. Simplified 56.6%

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

   if -8e-82 < l < -6.19999999999999978e-276

   1. Initial program 91.4%

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

   3. Simplified 91.4%

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

   5. Taylor expanded in d around inf 26.1%

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

      1. *-commutative 26.1%

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

   7. Simplified 26.1%

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

      1. pow1 26.1%

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

      2. metadata-eval 26.1%

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

      3. sqrt-pow1 20.2%

         \[\leadsto \color{blue}{\sqrt{{d}^{2}}} \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      4. sqrt-prod 20.2%

         \[\leadsto \color{blue}{\sqrt{{d}^{2} \cdot \frac{1}{\ell \cdot h}}} \]

      5. div-inv 20.3%

         \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{\ell \cdot h}}} \]

      6. sqrt-div 20.3%

         \[\leadsto \color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{\ell \cdot h}}} \]

      7. sqrt-pow1 24.0%

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

      8. metadata-eval 24.0%

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

      9. pow1 24.0%

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

      10. *-commutative 24.0%

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

   9. Applied egg-rr 24.0%

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

       1. expm1-log1p-u 24.0%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(h \cdot \ell\right)\right)}}} \]

       2. expm1-undefine 51.0%

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

       3. *-commutative 51.0%

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

   11. Applied egg-rr 51.0%

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

       1. log1p-undefine 51.0%

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

       2. rem-exp-log 51.0%

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

       3. associate-+r- 51.0%

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

       4. *-commutative 51.0%

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

       5. fma-neg 51.0%

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

       6. metadata-eval 51.0%

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

   13. Simplified 51.0%

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

   14. Taylor expanded in h around 0 61.5%

       \[\leadsto \frac{d}{\sqrt{1 + \color{blue}{-1}}} \]

   if -9.999999999999969e-311 < l < 1.3e175

   1. Initial program 73.4%

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

   3. Simplified 73.4%

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

      1. sqrt-unprod 60.1%

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

      2. pow1/2 60.1%

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

      3. frac-times 53.2%

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

      4. pow2 53.2%

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

   6. Applied egg-rr 53.2%

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

      1. unpow1/2 53.2%

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

      2. *-commutative 53.2%

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

   8. Simplified 53.2%

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

      1. pow1 53.2%

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

   10. Applied egg-rr 77.7%

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

       1. unpow1 77.7%

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

       2. *-commutative 77.7%

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

       3. *-commutative 77.7%

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

       4. *-commutative 77.7%

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

   12. Simplified 77.7%

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

   if 1.3e175 < l

   1. Initial program 39.2%

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

   3. Simplified 39.2%

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

   5. Taylor expanded in d around inf 38.3%

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

      1. *-commutative 38.3%

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

   7. Simplified 38.3%

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

      1. associate-/r* 41.8%

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

      2. sqrt-div 60.7%

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

   9. Applied egg-rr 60.7%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8 \cdot 10^{-82}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;\ell \leq -6.2 \cdot 10^{-276}:\\ \;\;\;\;\frac{d}{\sqrt{0}}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{+175}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 70.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \frac{M}{2}\right)}^{2}\right)\\ \mathbf{if}\;\ell \leq -7 \cdot 10^{-308}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + t\_0\right)\\ \mathbf{elif}\;\ell \leq 4.8 \cdot 10^{+175}:\\ \;\;\;\;\left(1 - t\_0\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (* (/ h l) (pow (* (/ D_m d) (/ M 2.0)) 2.0)))))
   (if (<= l -7e-308)
     (* (* d (sqrt (/ 1.0 (* l h)))) (+ -1.0 t_0))
     (if (<= l 4.8e+175)
       (* (- 1.0 t_0) (* d (pow (* l h) -0.5)))
       (* d (/ (sqrt (/ 1.0 l)) (sqrt h)))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = 0.5 * ((h / l) * pow(((D_m / d) * (M / 2.0)), 2.0));
	double tmp;
	if (l <= -7e-308) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + t_0);
	} else if (l <= 4.8e+175) {
		tmp = (1.0 - t_0) * (d * pow((l * h), -0.5));
	} else {
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * ((h / l) * (((d_m / d) * (m / 2.0d0)) ** 2.0d0))
    if (l <= (-7d-308)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * ((-1.0d0) + t_0)
    else if (l <= 4.8d+175) then
        tmp = (1.0d0 - t_0) * (d * ((l * h) ** (-0.5d0)))
    else
        tmp = d * (sqrt((1.0d0 / l)) / sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = 0.5 * ((h / l) * Math.pow(((D_m / d) * (M / 2.0)), 2.0));
	double tmp;
	if (l <= -7e-308) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (-1.0 + t_0);
	} else if (l <= 4.8e+175) {
		tmp = (1.0 - t_0) * (d * Math.pow((l * h), -0.5));
	} else {
		tmp = d * (Math.sqrt((1.0 / l)) / Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = 0.5 * ((h / l) * math.pow(((D_m / d) * (M / 2.0)), 2.0))
	tmp = 0
	if l <= -7e-308:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (-1.0 + t_0)
	elif l <= 4.8e+175:
		tmp = (1.0 - t_0) * (d * math.pow((l * h), -0.5))
	else:
		tmp = d * (math.sqrt((1.0 / l)) / math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(M / 2.0)) ^ 2.0)))
	tmp = 0.0
	if (l <= -7e-308)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(-1.0 + t_0));
	elseif (l <= 4.8e+175)
		tmp = Float64(Float64(1.0 - t_0) * Float64(d * (Float64(l * h) ^ -0.5)));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / l)) / sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = 0.5 * ((h / l) * (((D_m / d) * (M / 2.0)) ^ 2.0));
	tmp = 0.0;
	if (l <= -7e-308)
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + t_0);
	elseif (l <= 4.8e+175)
		tmp = (1.0 - t_0) * (d * ((l * h) ^ -0.5));
	else
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -7e-308], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.8e+175], N[(N[(1.0 - t$95$0), $MachinePrecision] * N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -7e-308

   1. Initial program 73.1%

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

   3. Simplified 72.4%

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

      1. sqrt-unprod 62.7%

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

      2. pow1/2 62.7%

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

      3. frac-times 46.9%

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

      4. pow2 46.9%

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

   6. Applied egg-rr 46.9%

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

      1. unpow1/2 46.9%

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

      2. *-commutative 46.9%

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

   8. Simplified 46.9%

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

   9. Taylor expanded in d around -inf 71.0%

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

   if -7e-308 < l < 4.8e175

   1. Initial program 72.6%

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

   3. Simplified 72.6%

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

      1. sqrt-unprod 59.4%

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

      2. pow1/2 59.4%

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

      3. frac-times 52.6%

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

      4. pow2 52.6%

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

   6. Applied egg-rr 52.6%

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

      1. unpow1/2 52.6%

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

      2. *-commutative 52.6%

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

   8. Simplified 52.6%

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

      1. div-inv 52.6%

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

      2. sqrt-prod 60.4%

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

      3. sqrt-pow1 76.1%

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

      4. metadata-eval 76.1%

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

      5. pow1 76.1%

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

      6. *-commutative 76.1%

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

      7. inv-pow 76.1%

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

      8. sqrt-pow1 76.9%

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

      9. *-commutative 76.9%

         \[\leadsto \left({\color{blue}{\left(h \cdot \ell\right)}}^{\left(\frac{-1}{2}\right)} \cdot d\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      10. metadata-eval 76.9%

          \[\leadsto \left({\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   10. Applied egg-rr 76.9%

       \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   if 4.8e175 < l

   1. Initial program 39.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 39.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 38.3%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative 38.3%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Simplified 38.3%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   8. Step-by-step derivation

      1. associate-/r* 41.8%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

      2. sqrt-div 60.7%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}} \]

   9. Applied egg-rr 60.7%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7 \cdot 10^{-308}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 4.8 \cdot 10^{+175}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 49.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{if}\;\ell \leq -2.9 \cdot 10^{-83}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -6.2 \cdot 10^{-276}:\\ \;\;\;\;\frac{d}{\sqrt{0}}\\ \mathbf{elif}\;\ell \leq 3.7 \cdot 10^{-284}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (* d (- (pow (* l h) -0.5)))))
   (if (<= l -2.9e-83)
     t_0
     (if (<= l -6.2e-276)
       (/ d (sqrt 0.0))
       (if (<= l 3.7e-284) t_0 (* d (/ (sqrt (/ 1.0 l)) (sqrt h))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = d * -pow((l * h), -0.5);
	double tmp;
	if (l <= -2.9e-83) {
		tmp = t_0;
	} else if (l <= -6.2e-276) {
		tmp = d / sqrt(0.0);
	} else if (l <= 3.7e-284) {
		tmp = t_0;
	} else {
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d * -((l * h) ** (-0.5d0))
    if (l <= (-2.9d-83)) then
        tmp = t_0
    else if (l <= (-6.2d-276)) then
        tmp = d / sqrt(0.0d0)
    else if (l <= 3.7d-284) then
        tmp = t_0
    else
        tmp = d * (sqrt((1.0d0 / l)) / sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = d * -Math.pow((l * h), -0.5);
	double tmp;
	if (l <= -2.9e-83) {
		tmp = t_0;
	} else if (l <= -6.2e-276) {
		tmp = d / Math.sqrt(0.0);
	} else if (l <= 3.7e-284) {
		tmp = t_0;
	} else {
		tmp = d * (Math.sqrt((1.0 / l)) / Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = d * -math.pow((l * h), -0.5)
	tmp = 0
	if l <= -2.9e-83:
		tmp = t_0
	elif l <= -6.2e-276:
		tmp = d / math.sqrt(0.0)
	elif l <= 3.7e-284:
		tmp = t_0
	else:
		tmp = d * (math.sqrt((1.0 / l)) / math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(d * Float64(-(Float64(l * h) ^ -0.5)))
	tmp = 0.0
	if (l <= -2.9e-83)
		tmp = t_0;
	elseif (l <= -6.2e-276)
		tmp = Float64(d / sqrt(0.0));
	elseif (l <= 3.7e-284)
		tmp = t_0;
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / l)) / sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = d * -((l * h) ^ -0.5);
	tmp = 0.0;
	if (l <= -2.9e-83)
		tmp = t_0;
	elseif (l <= -6.2e-276)
		tmp = d / sqrt(0.0);
	elseif (l <= 3.7e-284)
		tmp = t_0;
	else
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(d * (-N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision]}, If[LessEqual[l, -2.9e-83], t$95$0, If[LessEqual[l, -6.2e-276], N[(d / N[Sqrt[0.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.7e-284], t$95$0, N[(d * N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.8999999999999999e-83 or -6.19999999999999978e-276 < l < 3.7e-284

   1. Initial program 63.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 62.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 8.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative 8.6%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Simplified 8.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   8. Taylor expanded in l around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. unpow-1 0.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      3. metadata-eval 0.0%

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      4. pow-sqr 0.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      5. rem-sqrt-square 0.0%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      6. rem-square-sqrt 0.0%

         \[\leadsto \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      7. fabs-sqr 0.0%

         \[\leadsto \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      8. rem-square-sqrt 0.0%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      9. *-commutative 0.0%

         \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      10. unpow2 0.0%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      11. rem-square-sqrt 57.0%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      12. neg-mul-1 57.0%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\left(-d\right)} \]

   10. Simplified 57.0%

       \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-d\right)} \]

   if -2.8999999999999999e-83 < l < -6.19999999999999978e-276

   1. Initial program 91.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 91.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 26.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative 26.1%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Simplified 26.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   8. Step-by-step derivation

      1. pow1 26.1%

         \[\leadsto \color{blue}{{d}^{1}} \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      2. metadata-eval 26.1%

         \[\leadsto {d}^{\color{blue}{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      3. sqrt-pow1 20.2%

         \[\leadsto \color{blue}{\sqrt{{d}^{2}}} \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      4. sqrt-prod 20.2%

         \[\leadsto \color{blue}{\sqrt{{d}^{2} \cdot \frac{1}{\ell \cdot h}}} \]

      5. div-inv 20.3%

         \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{\ell \cdot h}}} \]

      6. sqrt-div 20.3%

         \[\leadsto \color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{\ell \cdot h}}} \]

      7. sqrt-pow1 24.0%

         \[\leadsto \frac{\color{blue}{{d}^{\left(\frac{2}{2}\right)}}}{\sqrt{\ell \cdot h}} \]

      8. metadata-eval 24.0%

         \[\leadsto \frac{{d}^{\color{blue}{1}}}{\sqrt{\ell \cdot h}} \]

      9. pow1 24.0%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell \cdot h}} \]

      10. *-commutative 24.0%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   9. Applied egg-rr 24.0%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 24.0%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(h \cdot \ell\right)\right)}}} \]

       2. expm1-undefine 51.0%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{e^{\mathsf{log1p}\left(h \cdot \ell\right)} - 1}}} \]

       3. *-commutative 51.0%

          \[\leadsto \frac{d}{\sqrt{e^{\mathsf{log1p}\left(\color{blue}{\ell \cdot h}\right)} - 1}} \]

   11. Applied egg-rr 51.0%

       \[\leadsto \frac{d}{\sqrt{\color{blue}{e^{\mathsf{log1p}\left(\ell \cdot h\right)} - 1}}} \]
   12. Step-by-step derivation

       1. log1p-undefine 51.0%

          \[\leadsto \frac{d}{\sqrt{e^{\color{blue}{\log \left(1 + \ell \cdot h\right)}} - 1}} \]

       2. rem-exp-log 51.0%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\left(1 + \ell \cdot h\right)} - 1}} \]

       3. associate-+r- 51.0%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{1 + \left(\ell \cdot h - 1\right)}}} \]

       4. *-commutative 51.0%

          \[\leadsto \frac{d}{\sqrt{1 + \left(\color{blue}{h \cdot \ell} - 1\right)}} \]

       5. fma-neg 51.0%

          \[\leadsto \frac{d}{\sqrt{1 + \color{blue}{\mathsf{fma}\left(h, \ell, -1\right)}}} \]

       6. metadata-eval 51.0%

          \[\leadsto \frac{d}{\sqrt{1 + \mathsf{fma}\left(h, \ell, \color{blue}{-1}\right)}} \]

   13. Simplified 51.0%

       \[\leadsto \frac{d}{\sqrt{\color{blue}{1 + \mathsf{fma}\left(h, \ell, -1\right)}}} \]

   14. Taylor expanded in h around 0 61.5%

       \[\leadsto \frac{d}{\sqrt{1 + \color{blue}{-1}}} \]

   if 3.7e-284 < l

   1. Initial program 65.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 65.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 34.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative 34.8%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Simplified 34.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   8. Step-by-step derivation

      1. associate-/r* 35.5%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

      2. sqrt-div 43.7%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}} \]

   9. Applied egg-rr 43.7%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.9 \cdot 10^{-83}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;\ell \leq -6.2 \cdot 10^{-276}:\\ \;\;\;\;\frac{d}{\sqrt{0}}\\ \mathbf{elif}\;\ell \leq 3.7 \cdot 10^{-284}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 48.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{if}\;\ell \leq -1.5 \cdot 10^{-79}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -2.05 \cdot 10^{-274}:\\ \;\;\;\;\frac{d}{\sqrt{0}}\\ \mathbf{elif}\;\ell \leq 3.9 \cdot 10^{-277}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (* d (- (pow (* l h) -0.5)))))
   (if (<= l -1.5e-79)
     t_0
     (if (<= l -2.05e-274)
       (/ d (sqrt 0.0))
       (if (<= l 3.9e-277) t_0 (/ d (* (sqrt l) (sqrt h))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = d * -pow((l * h), -0.5);
	double tmp;
	if (l <= -1.5e-79) {
		tmp = t_0;
	} else if (l <= -2.05e-274) {
		tmp = d / sqrt(0.0);
	} else if (l <= 3.9e-277) {
		tmp = t_0;
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d * -((l * h) ** (-0.5d0))
    if (l <= (-1.5d-79)) then
        tmp = t_0
    else if (l <= (-2.05d-274)) then
        tmp = d / sqrt(0.0d0)
    else if (l <= 3.9d-277) then
        tmp = t_0
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = d * -Math.pow((l * h), -0.5);
	double tmp;
	if (l <= -1.5e-79) {
		tmp = t_0;
	} else if (l <= -2.05e-274) {
		tmp = d / Math.sqrt(0.0);
	} else if (l <= 3.9e-277) {
		tmp = t_0;
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = d * -math.pow((l * h), -0.5)
	tmp = 0
	if l <= -1.5e-79:
		tmp = t_0
	elif l <= -2.05e-274:
		tmp = d / math.sqrt(0.0)
	elif l <= 3.9e-277:
		tmp = t_0
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(d * Float64(-(Float64(l * h) ^ -0.5)))
	tmp = 0.0
	if (l <= -1.5e-79)
		tmp = t_0;
	elseif (l <= -2.05e-274)
		tmp = Float64(d / sqrt(0.0));
	elseif (l <= 3.9e-277)
		tmp = t_0;
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = d * -((l * h) ^ -0.5);
	tmp = 0.0;
	if (l <= -1.5e-79)
		tmp = t_0;
	elseif (l <= -2.05e-274)
		tmp = d / sqrt(0.0);
	elseif (l <= 3.9e-277)
		tmp = t_0;
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(d * (-N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision]}, If[LessEqual[l, -1.5e-79], t$95$0, If[LessEqual[l, -2.05e-274], N[(d / N[Sqrt[0.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.9e-277], t$95$0, N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.5e-79 or -2.04999999999999994e-274 < l < 3.89999999999999987e-277

   1. Initial program 63.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 62.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 8.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative 8.6%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Simplified 8.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   8. Taylor expanded in l around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. unpow-1 0.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      3. metadata-eval 0.0%

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      4. pow-sqr 0.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      5. rem-sqrt-square 0.0%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      6. rem-square-sqrt 0.0%

         \[\leadsto \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      7. fabs-sqr 0.0%

         \[\leadsto \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      8. rem-square-sqrt 0.0%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      9. *-commutative 0.0%

         \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      10. unpow2 0.0%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      11. rem-square-sqrt 57.0%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      12. neg-mul-1 57.0%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\left(-d\right)} \]

   10. Simplified 57.0%

       \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-d\right)} \]

   if -1.5e-79 < l < -2.04999999999999994e-274

   1. Initial program 91.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 91.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 26.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative 26.1%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Simplified 26.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   8. Step-by-step derivation

      1. pow1 26.1%

         \[\leadsto \color{blue}{{d}^{1}} \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      2. metadata-eval 26.1%

         \[\leadsto {d}^{\color{blue}{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      3. sqrt-pow1 20.2%

         \[\leadsto \color{blue}{\sqrt{{d}^{2}}} \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      4. sqrt-prod 20.2%

         \[\leadsto \color{blue}{\sqrt{{d}^{2} \cdot \frac{1}{\ell \cdot h}}} \]

      5. div-inv 20.3%

         \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{\ell \cdot h}}} \]

      6. sqrt-div 20.3%

         \[\leadsto \color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{\ell \cdot h}}} \]

      7. sqrt-pow1 24.0%

         \[\leadsto \frac{\color{blue}{{d}^{\left(\frac{2}{2}\right)}}}{\sqrt{\ell \cdot h}} \]

      8. metadata-eval 24.0%

         \[\leadsto \frac{{d}^{\color{blue}{1}}}{\sqrt{\ell \cdot h}} \]

      9. pow1 24.0%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell \cdot h}} \]

      10. *-commutative 24.0%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   9. Applied egg-rr 24.0%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 24.0%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(h \cdot \ell\right)\right)}}} \]

       2. expm1-undefine 51.0%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{e^{\mathsf{log1p}\left(h \cdot \ell\right)} - 1}}} \]

       3. *-commutative 51.0%

          \[\leadsto \frac{d}{\sqrt{e^{\mathsf{log1p}\left(\color{blue}{\ell \cdot h}\right)} - 1}} \]

   11. Applied egg-rr 51.0%

       \[\leadsto \frac{d}{\sqrt{\color{blue}{e^{\mathsf{log1p}\left(\ell \cdot h\right)} - 1}}} \]
   12. Step-by-step derivation

       1. log1p-undefine 51.0%

          \[\leadsto \frac{d}{\sqrt{e^{\color{blue}{\log \left(1 + \ell \cdot h\right)}} - 1}} \]

       2. rem-exp-log 51.0%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\left(1 + \ell \cdot h\right)} - 1}} \]

       3. associate-+r- 51.0%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{1 + \left(\ell \cdot h - 1\right)}}} \]

       4. *-commutative 51.0%

          \[\leadsto \frac{d}{\sqrt{1 + \left(\color{blue}{h \cdot \ell} - 1\right)}} \]

       5. fma-neg 51.0%

          \[\leadsto \frac{d}{\sqrt{1 + \color{blue}{\mathsf{fma}\left(h, \ell, -1\right)}}} \]

       6. metadata-eval 51.0%

          \[\leadsto \frac{d}{\sqrt{1 + \mathsf{fma}\left(h, \ell, \color{blue}{-1}\right)}} \]

   13. Simplified 51.0%

       \[\leadsto \frac{d}{\sqrt{\color{blue}{1 + \mathsf{fma}\left(h, \ell, -1\right)}}} \]

   14. Taylor expanded in h around 0 61.5%

       \[\leadsto \frac{d}{\sqrt{1 + \color{blue}{-1}}} \]

   if 3.89999999999999987e-277 < l

   1. Initial program 65.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 65.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 34.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative 34.8%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Simplified 34.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   8. Step-by-step derivation

      1. pow1 34.8%

         \[\leadsto \color{blue}{{d}^{1}} \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      2. metadata-eval 34.8%

         \[\leadsto {d}^{\color{blue}{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      3. sqrt-pow1 25.9%

         \[\leadsto \color{blue}{\sqrt{{d}^{2}}} \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      4. sqrt-prod 21.1%

         \[\leadsto \color{blue}{\sqrt{{d}^{2} \cdot \frac{1}{\ell \cdot h}}} \]

      5. div-inv 21.2%

         \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{\ell \cdot h}}} \]

      6. sqrt-div 25.9%

         \[\leadsto \color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{\ell \cdot h}}} \]

      7. sqrt-pow1 35.1%

         \[\leadsto \frac{\color{blue}{{d}^{\left(\frac{2}{2}\right)}}}{\sqrt{\ell \cdot h}} \]

      8. metadata-eval 35.1%

         \[\leadsto \frac{{d}^{\color{blue}{1}}}{\sqrt{\ell \cdot h}} \]

      9. pow1 35.1%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell \cdot h}} \]

      10. *-commutative 35.1%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   9. Applied egg-rr 35.1%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. *-commutative 35.1%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       2. sqrt-prod 43.6%

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

   11. Applied egg-rr 43.6%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.5 \cdot 10^{-79}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;\ell \leq -2.05 \cdot 10^{-274}:\\ \;\;\;\;\frac{d}{\sqrt{0}}\\ \mathbf{elif}\;\ell \leq 3.9 \cdot 10^{-277}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 44.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{if}\;\ell \leq -2.45 \cdot 10^{-82}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -1.68 \cdot 10^{-273}:\\ \;\;\;\;\frac{d}{\sqrt{0}}\\ \mathbf{elif}\;\ell \leq 1.28 \cdot 10^{-270}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (* d (- (pow (* l h) -0.5)))))
   (if (<= l -2.45e-82)
     t_0
     (if (<= l -1.68e-273)
       (/ d (sqrt 0.0))
       (if (<= l 1.28e-270) t_0 (* d (sqrt (/ (/ 1.0 h) l))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = d * -pow((l * h), -0.5);
	double tmp;
	if (l <= -2.45e-82) {
		tmp = t_0;
	} else if (l <= -1.68e-273) {
		tmp = d / sqrt(0.0);
	} else if (l <= 1.28e-270) {
		tmp = t_0;
	} else {
		tmp = d * sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d * -((l * h) ** (-0.5d0))
    if (l <= (-2.45d-82)) then
        tmp = t_0
    else if (l <= (-1.68d-273)) then
        tmp = d / sqrt(0.0d0)
    else if (l <= 1.28d-270) then
        tmp = t_0
    else
        tmp = d * sqrt(((1.0d0 / h) / l))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = d * -Math.pow((l * h), -0.5);
	double tmp;
	if (l <= -2.45e-82) {
		tmp = t_0;
	} else if (l <= -1.68e-273) {
		tmp = d / Math.sqrt(0.0);
	} else if (l <= 1.28e-270) {
		tmp = t_0;
	} else {
		tmp = d * Math.sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = d * -math.pow((l * h), -0.5)
	tmp = 0
	if l <= -2.45e-82:
		tmp = t_0
	elif l <= -1.68e-273:
		tmp = d / math.sqrt(0.0)
	elif l <= 1.28e-270:
		tmp = t_0
	else:
		tmp = d * math.sqrt(((1.0 / h) / l))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(d * Float64(-(Float64(l * h) ^ -0.5)))
	tmp = 0.0
	if (l <= -2.45e-82)
		tmp = t_0;
	elseif (l <= -1.68e-273)
		tmp = Float64(d / sqrt(0.0));
	elseif (l <= 1.28e-270)
		tmp = t_0;
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / h) / l)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = d * -((l * h) ^ -0.5);
	tmp = 0.0;
	if (l <= -2.45e-82)
		tmp = t_0;
	elseif (l <= -1.68e-273)
		tmp = d / sqrt(0.0);
	elseif (l <= 1.28e-270)
		tmp = t_0;
	else
		tmp = d * sqrt(((1.0 / h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(d * (-N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision]}, If[LessEqual[l, -2.45e-82], t$95$0, If[LessEqual[l, -1.68e-273], N[(d / N[Sqrt[0.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.28e-270], t$95$0, N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.4500000000000001e-82 or -1.67999999999999995e-273 < l < 1.27999999999999991e-270

   1. Initial program 64.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 63.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 8.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative 8.4%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Simplified 8.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   8. Taylor expanded in l around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. unpow-1 0.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      3. metadata-eval 0.0%

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      4. pow-sqr 0.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      5. rem-sqrt-square 0.0%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      6. rem-square-sqrt 0.0%

         \[\leadsto \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      7. fabs-sqr 0.0%

         \[\leadsto \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      8. rem-square-sqrt 0.0%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      9. *-commutative 0.0%

         \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      10. unpow2 0.0%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      11. rem-square-sqrt 55.9%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      12. neg-mul-1 55.9%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\left(-d\right)} \]

   10. Simplified 55.9%

       \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-d\right)} \]

   if -2.4500000000000001e-82 < l < -1.67999999999999995e-273

   1. Initial program 91.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 91.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 26.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative 26.1%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Simplified 26.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   8. Step-by-step derivation

      1. pow1 26.1%

         \[\leadsto \color{blue}{{d}^{1}} \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      2. metadata-eval 26.1%

         \[\leadsto {d}^{\color{blue}{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      3. sqrt-pow1 20.2%

         \[\leadsto \color{blue}{\sqrt{{d}^{2}}} \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      4. sqrt-prod 20.2%

         \[\leadsto \color{blue}{\sqrt{{d}^{2} \cdot \frac{1}{\ell \cdot h}}} \]

      5. div-inv 20.3%

         \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{\ell \cdot h}}} \]

      6. sqrt-div 20.3%

         \[\leadsto \color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{\ell \cdot h}}} \]

      7. sqrt-pow1 24.0%

         \[\leadsto \frac{\color{blue}{{d}^{\left(\frac{2}{2}\right)}}}{\sqrt{\ell \cdot h}} \]

      8. metadata-eval 24.0%

         \[\leadsto \frac{{d}^{\color{blue}{1}}}{\sqrt{\ell \cdot h}} \]

      9. pow1 24.0%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell \cdot h}} \]

      10. *-commutative 24.0%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   9. Applied egg-rr 24.0%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 24.0%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(h \cdot \ell\right)\right)}}} \]

       2. expm1-undefine 51.0%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{e^{\mathsf{log1p}\left(h \cdot \ell\right)} - 1}}} \]

       3. *-commutative 51.0%

          \[\leadsto \frac{d}{\sqrt{e^{\mathsf{log1p}\left(\color{blue}{\ell \cdot h}\right)} - 1}} \]

   11. Applied egg-rr 51.0%

       \[\leadsto \frac{d}{\sqrt{\color{blue}{e^{\mathsf{log1p}\left(\ell \cdot h\right)} - 1}}} \]
   12. Step-by-step derivation

       1. log1p-undefine 51.0%

          \[\leadsto \frac{d}{\sqrt{e^{\color{blue}{\log \left(1 + \ell \cdot h\right)}} - 1}} \]

       2. rem-exp-log 51.0%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\left(1 + \ell \cdot h\right)} - 1}} \]

       3. associate-+r- 51.0%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{1 + \left(\ell \cdot h - 1\right)}}} \]

       4. *-commutative 51.0%

          \[\leadsto \frac{d}{\sqrt{1 + \left(\color{blue}{h \cdot \ell} - 1\right)}} \]

       5. fma-neg 51.0%

          \[\leadsto \frac{d}{\sqrt{1 + \color{blue}{\mathsf{fma}\left(h, \ell, -1\right)}}} \]

       6. metadata-eval 51.0%

          \[\leadsto \frac{d}{\sqrt{1 + \mathsf{fma}\left(h, \ell, \color{blue}{-1}\right)}} \]

   13. Simplified 51.0%

       \[\leadsto \frac{d}{\sqrt{\color{blue}{1 + \mathsf{fma}\left(h, \ell, -1\right)}}} \]

   14. Taylor expanded in h around 0 61.5%

       \[\leadsto \frac{d}{\sqrt{1 + \color{blue}{-1}}} \]

   if 1.27999999999999991e-270 < l

   1. Initial program 65.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 65.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 35.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative 35.4%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Simplified 35.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   8. Taylor expanded in l around 0 35.4%

      \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. associate-/r* 36.1%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   10. Simplified 36.1%

       \[\leadsto d \cdot \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.45 \cdot 10^{-82}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;\ell \leq -1.68 \cdot 10^{-273}:\\ \;\;\;\;\frac{d}{\sqrt{0}}\\ \mathbf{elif}\;\ell \leq 1.28 \cdot 10^{-270}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 29.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 6.4 \cdot 10^{-115}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{0}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= M 6.4e-115) (* d (sqrt (/ (/ 1.0 h) l))) (/ d (sqrt 0.0))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (M <= 6.4e-115) {
		tmp = d * sqrt(((1.0 / h) / l));
	} else {
		tmp = d / sqrt(0.0);
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: tmp
    if (m <= 6.4d-115) then
        tmp = d * sqrt(((1.0d0 / h) / l))
    else
        tmp = d / sqrt(0.0d0)
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (M <= 6.4e-115) {
		tmp = d * Math.sqrt(((1.0 / h) / l));
	} else {
		tmp = d / Math.sqrt(0.0);
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if M <= 6.4e-115:
		tmp = d * math.sqrt(((1.0 / h) / l))
	else:
		tmp = d / math.sqrt(0.0)
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (M <= 6.4e-115)
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / h) / l)));
	else
		tmp = Float64(d / sqrt(0.0));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (M <= 6.4e-115)
		tmp = d * sqrt(((1.0 / h) / l));
	else
		tmp = d / sqrt(0.0);
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[M, 6.4e-115], N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[Sqrt[0.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 6.4e-115

   1. Initial program 69.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 68.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 24.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative 24.5%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Simplified 24.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   8. Taylor expanded in l around 0 24.5%

      \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. associate-/r* 25.0%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   10. Simplified 25.0%

       \[\leadsto d \cdot \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}}} \]

   if 6.4e-115 < M

   1. Initial program 70.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 70.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 22.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative 22.1%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Simplified 22.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   8. Step-by-step derivation

      1. pow1 22.1%

         \[\leadsto \color{blue}{{d}^{1}} \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      2. metadata-eval 22.1%

         \[\leadsto {d}^{\color{blue}{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      3. sqrt-pow1 23.9%

         \[\leadsto \color{blue}{\sqrt{{d}^{2}}} \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      4. sqrt-prod 20.1%

         \[\leadsto \color{blue}{\sqrt{{d}^{2} \cdot \frac{1}{\ell \cdot h}}} \]

      5. div-inv 20.1%

         \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{\ell \cdot h}}} \]

      6. sqrt-div 24.0%

         \[\leadsto \color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{\ell \cdot h}}} \]

      7. sqrt-pow1 21.2%

         \[\leadsto \frac{\color{blue}{{d}^{\left(\frac{2}{2}\right)}}}{\sqrt{\ell \cdot h}} \]

      8. metadata-eval 21.2%

         \[\leadsto \frac{{d}^{\color{blue}{1}}}{\sqrt{\ell \cdot h}} \]

      9. pow1 21.2%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell \cdot h}} \]

      10. *-commutative 21.2%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   9. Applied egg-rr 21.2%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 21.0%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(h \cdot \ell\right)\right)}}} \]

       2. expm1-undefine 28.0%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{e^{\mathsf{log1p}\left(h \cdot \ell\right)} - 1}}} \]

       3. *-commutative 28.0%

          \[\leadsto \frac{d}{\sqrt{e^{\mathsf{log1p}\left(\color{blue}{\ell \cdot h}\right)} - 1}} \]

   11. Applied egg-rr 28.0%

       \[\leadsto \frac{d}{\sqrt{\color{blue}{e^{\mathsf{log1p}\left(\ell \cdot h\right)} - 1}}} \]
   12. Step-by-step derivation

       1. log1p-undefine 28.0%

          \[\leadsto \frac{d}{\sqrt{e^{\color{blue}{\log \left(1 + \ell \cdot h\right)}} - 1}} \]

       2. rem-exp-log 28.2%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\left(1 + \ell \cdot h\right)} - 1}} \]

       3. associate-+r- 28.2%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{1 + \left(\ell \cdot h - 1\right)}}} \]

       4. *-commutative 28.2%

          \[\leadsto \frac{d}{\sqrt{1 + \left(\color{blue}{h \cdot \ell} - 1\right)}} \]

       5. fma-neg 28.2%

          \[\leadsto \frac{d}{\sqrt{1 + \color{blue}{\mathsf{fma}\left(h, \ell, -1\right)}}} \]

       6. metadata-eval 28.2%

          \[\leadsto \frac{d}{\sqrt{1 + \mathsf{fma}\left(h, \ell, \color{blue}{-1}\right)}} \]

   13. Simplified 28.2%

       \[\leadsto \frac{d}{\sqrt{\color{blue}{1 + \mathsf{fma}\left(h, \ell, -1\right)}}} \]

   14. Taylor expanded in h around 0 31.8%

       \[\leadsto \frac{d}{\sqrt{1 + \color{blue}{-1}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 6.4 \cdot 10^{-115}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{0}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 29.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 2.7 \cdot 10^{-115}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{0}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= M 2.7e-115) (* d (pow (* l h) -0.5)) (/ d (sqrt 0.0))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (M <= 2.7e-115) {
		tmp = d * pow((l * h), -0.5);
	} else {
		tmp = d / sqrt(0.0);
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: tmp
    if (m <= 2.7d-115) then
        tmp = d * ((l * h) ** (-0.5d0))
    else
        tmp = d / sqrt(0.0d0)
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (M <= 2.7e-115) {
		tmp = d * Math.pow((l * h), -0.5);
	} else {
		tmp = d / Math.sqrt(0.0);
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if M <= 2.7e-115:
		tmp = d * math.pow((l * h), -0.5)
	else:
		tmp = d / math.sqrt(0.0)
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (M <= 2.7e-115)
		tmp = Float64(d * (Float64(l * h) ^ -0.5));
	else
		tmp = Float64(d / sqrt(0.0));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (M <= 2.7e-115)
		tmp = d * ((l * h) ^ -0.5);
	else
		tmp = d / sqrt(0.0);
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[M, 2.7e-115], N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(d / N[Sqrt[0.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.7e-115

   1. Initial program 69.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 68.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 24.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative 24.5%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Simplified 24.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   8. Taylor expanded in l around 0 24.5%

      \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. unpow-1 24.5%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

      2. metadata-eval 24.5%

         \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      3. pow-sqr 24.5%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

      4. rem-sqrt-square 24.7%

         \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

      5. rem-square-sqrt 24.6%

         \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

      6. fabs-sqr 24.6%

         \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

      7. rem-square-sqrt 24.7%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Simplified 24.7%

       \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   if 2.7e-115 < M

   1. Initial program 70.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 70.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 22.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative 22.1%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Simplified 22.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   8. Step-by-step derivation

      1. pow1 22.1%

         \[\leadsto \color{blue}{{d}^{1}} \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      2. metadata-eval 22.1%

         \[\leadsto {d}^{\color{blue}{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      3. sqrt-pow1 23.9%

         \[\leadsto \color{blue}{\sqrt{{d}^{2}}} \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      4. sqrt-prod 20.1%

         \[\leadsto \color{blue}{\sqrt{{d}^{2} \cdot \frac{1}{\ell \cdot h}}} \]

      5. div-inv 20.1%

         \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{\ell \cdot h}}} \]

      6. sqrt-div 24.0%

         \[\leadsto \color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{\ell \cdot h}}} \]

      7. sqrt-pow1 21.2%

         \[\leadsto \frac{\color{blue}{{d}^{\left(\frac{2}{2}\right)}}}{\sqrt{\ell \cdot h}} \]

      8. metadata-eval 21.2%

         \[\leadsto \frac{{d}^{\color{blue}{1}}}{\sqrt{\ell \cdot h}} \]

      9. pow1 21.2%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell \cdot h}} \]

      10. *-commutative 21.2%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   9. Applied egg-rr 21.2%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 21.0%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(h \cdot \ell\right)\right)}}} \]

       2. expm1-undefine 28.0%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{e^{\mathsf{log1p}\left(h \cdot \ell\right)} - 1}}} \]

       3. *-commutative 28.0%

          \[\leadsto \frac{d}{\sqrt{e^{\mathsf{log1p}\left(\color{blue}{\ell \cdot h}\right)} - 1}} \]

   11. Applied egg-rr 28.0%

       \[\leadsto \frac{d}{\sqrt{\color{blue}{e^{\mathsf{log1p}\left(\ell \cdot h\right)} - 1}}} \]
   12. Step-by-step derivation

       1. log1p-undefine 28.0%

          \[\leadsto \frac{d}{\sqrt{e^{\color{blue}{\log \left(1 + \ell \cdot h\right)}} - 1}} \]

       2. rem-exp-log 28.2%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\left(1 + \ell \cdot h\right)} - 1}} \]

       3. associate-+r- 28.2%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{1 + \left(\ell \cdot h - 1\right)}}} \]

       4. *-commutative 28.2%

          \[\leadsto \frac{d}{\sqrt{1 + \left(\color{blue}{h \cdot \ell} - 1\right)}} \]

       5. fma-neg 28.2%

          \[\leadsto \frac{d}{\sqrt{1 + \color{blue}{\mathsf{fma}\left(h, \ell, -1\right)}}} \]

       6. metadata-eval 28.2%

          \[\leadsto \frac{d}{\sqrt{1 + \mathsf{fma}\left(h, \ell, \color{blue}{-1}\right)}} \]

   13. Simplified 28.2%

       \[\leadsto \frac{d}{\sqrt{\color{blue}{1 + \mathsf{fma}\left(h, \ell, -1\right)}}} \]

   14. Taylor expanded in h around 0 31.8%

       \[\leadsto \frac{d}{\sqrt{1 + \color{blue}{-1}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.7 \cdot 10^{-115}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{0}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 29.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 2.75 \cdot 10^{-115}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{0}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= M 2.75e-115) (/ d (sqrt (* l h))) (/ d (sqrt 0.0))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (M <= 2.75e-115) {
		tmp = d / sqrt((l * h));
	} else {
		tmp = d / sqrt(0.0);
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: tmp
    if (m <= 2.75d-115) then
        tmp = d / sqrt((l * h))
    else
        tmp = d / sqrt(0.0d0)
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (M <= 2.75e-115) {
		tmp = d / Math.sqrt((l * h));
	} else {
		tmp = d / Math.sqrt(0.0);
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if M <= 2.75e-115:
		tmp = d / math.sqrt((l * h))
	else:
		tmp = d / math.sqrt(0.0)
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (M <= 2.75e-115)
		tmp = Float64(d / sqrt(Float64(l * h)));
	else
		tmp = Float64(d / sqrt(0.0));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (M <= 2.75e-115)
		tmp = d / sqrt((l * h));
	else
		tmp = d / sqrt(0.0);
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[M, 2.75e-115], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[Sqrt[0.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.75000000000000014e-115

   1. Initial program 69.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 68.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 24.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative 24.5%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Simplified 24.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   8. Step-by-step derivation

      1. pow1 24.5%

         \[\leadsto \color{blue}{{d}^{1}} \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      2. metadata-eval 24.5%

         \[\leadsto {d}^{\color{blue}{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      3. sqrt-pow1 32.0%

         \[\leadsto \color{blue}{\sqrt{{d}^{2}}} \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      4. sqrt-prod 28.1%

         \[\leadsto \color{blue}{\sqrt{{d}^{2} \cdot \frac{1}{\ell \cdot h}}} \]

      5. div-inv 28.6%

         \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{\ell \cdot h}}} \]

      6. sqrt-div 32.4%

         \[\leadsto \color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{\ell \cdot h}}} \]

      7. sqrt-pow1 24.7%

         \[\leadsto \frac{\color{blue}{{d}^{\left(\frac{2}{2}\right)}}}{\sqrt{\ell \cdot h}} \]

      8. metadata-eval 24.7%

         \[\leadsto \frac{{d}^{\color{blue}{1}}}{\sqrt{\ell \cdot h}} \]

      9. pow1 24.7%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell \cdot h}} \]

      10. *-commutative 24.7%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   9. Applied egg-rr 24.7%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

   if 2.75000000000000014e-115 < M

   1. Initial program 70.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 70.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 22.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative 22.1%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Simplified 22.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   8. Step-by-step derivation

      1. pow1 22.1%

         \[\leadsto \color{blue}{{d}^{1}} \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      2. metadata-eval 22.1%

         \[\leadsto {d}^{\color{blue}{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      3. sqrt-pow1 23.9%

         \[\leadsto \color{blue}{\sqrt{{d}^{2}}} \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      4. sqrt-prod 20.1%

         \[\leadsto \color{blue}{\sqrt{{d}^{2} \cdot \frac{1}{\ell \cdot h}}} \]

      5. div-inv 20.1%

         \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{\ell \cdot h}}} \]

      6. sqrt-div 24.0%

         \[\leadsto \color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{\ell \cdot h}}} \]

      7. sqrt-pow1 21.2%

         \[\leadsto \frac{\color{blue}{{d}^{\left(\frac{2}{2}\right)}}}{\sqrt{\ell \cdot h}} \]

      8. metadata-eval 21.2%

         \[\leadsto \frac{{d}^{\color{blue}{1}}}{\sqrt{\ell \cdot h}} \]

      9. pow1 21.2%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell \cdot h}} \]

      10. *-commutative 21.2%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   9. Applied egg-rr 21.2%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 21.0%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(h \cdot \ell\right)\right)}}} \]

       2. expm1-undefine 28.0%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{e^{\mathsf{log1p}\left(h \cdot \ell\right)} - 1}}} \]

       3. *-commutative 28.0%

          \[\leadsto \frac{d}{\sqrt{e^{\mathsf{log1p}\left(\color{blue}{\ell \cdot h}\right)} - 1}} \]

   11. Applied egg-rr 28.0%

       \[\leadsto \frac{d}{\sqrt{\color{blue}{e^{\mathsf{log1p}\left(\ell \cdot h\right)} - 1}}} \]
   12. Step-by-step derivation

       1. log1p-undefine 28.0%

          \[\leadsto \frac{d}{\sqrt{e^{\color{blue}{\log \left(1 + \ell \cdot h\right)}} - 1}} \]

       2. rem-exp-log 28.2%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\left(1 + \ell \cdot h\right)} - 1}} \]

       3. associate-+r- 28.2%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{1 + \left(\ell \cdot h - 1\right)}}} \]

       4. *-commutative 28.2%

          \[\leadsto \frac{d}{\sqrt{1 + \left(\color{blue}{h \cdot \ell} - 1\right)}} \]

       5. fma-neg 28.2%

          \[\leadsto \frac{d}{\sqrt{1 + \color{blue}{\mathsf{fma}\left(h, \ell, -1\right)}}} \]

       6. metadata-eval 28.2%

          \[\leadsto \frac{d}{\sqrt{1 + \mathsf{fma}\left(h, \ell, \color{blue}{-1}\right)}} \]

   13. Simplified 28.2%

       \[\leadsto \frac{d}{\sqrt{\color{blue}{1 + \mathsf{fma}\left(h, \ell, -1\right)}}} \]

   14. Taylor expanded in h around 0 31.8%

       \[\leadsto \frac{d}{\sqrt{1 + \color{blue}{-1}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.75 \cdot 10^{-115}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{0}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 26.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \frac{d}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m) :precision binary64 (/ d (sqrt (* l h))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	return d / sqrt((l * h));
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    code = d / sqrt((l * h))
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	return d / Math.sqrt((l * h));
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	return d / math.sqrt((l * h))

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	return Float64(d / sqrt(Float64(l * h)))
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp = code(d, h, l, M, D_m)
	tmp = d / sqrt((l * h));
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.6%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 69.3%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in d around inf 23.6%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
6. Step-by-step derivation

   1. *-commutative 23.6%

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

7. Simplified 23.6%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
8. Step-by-step derivation

   1. pow1 23.6%

      \[\leadsto \color{blue}{{d}^{1}} \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

   2. metadata-eval 23.6%

      \[\leadsto {d}^{\color{blue}{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

   3. sqrt-pow1 29.0%

      \[\leadsto \color{blue}{\sqrt{{d}^{2}}} \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

   4. sqrt-prod 25.1%

      \[\leadsto \color{blue}{\sqrt{{d}^{2} \cdot \frac{1}{\ell \cdot h}}} \]

   5. div-inv 25.4%

      \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{\ell \cdot h}}} \]

   6. sqrt-div 29.2%

      \[\leadsto \color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{\ell \cdot h}}} \]

   7. sqrt-pow1 23.3%

      \[\leadsto \frac{\color{blue}{{d}^{\left(\frac{2}{2}\right)}}}{\sqrt{\ell \cdot h}} \]

   8. metadata-eval 23.3%

      \[\leadsto \frac{{d}^{\color{blue}{1}}}{\sqrt{\ell \cdot h}} \]

   9. pow1 23.3%

      \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell \cdot h}} \]

   10. *-commutative 23.3%

       \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

9. Applied egg-rr 23.3%

   \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

10. Final simplification 23.3%

    \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024135 
(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)))))