Henrywood and Agarwal, Equation (12)

Percentage Accurate: 67.2% → 81.4%

Time: 24.9s

Alternatives: 19

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: 67.2% 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: 81.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(D \cdot \frac{M}{d}\right)\\ t_1 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{t\_1}{\sqrt{-h}} \cdot \left(\frac{t\_1}{\sqrt{-\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)\\ \mathbf{elif}\;\ell \leq 2.9 \cdot 10^{+158}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{-1}{\frac{\ell}{0.5 \cdot \left(h \cdot {t\_0}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \left(\sqrt{d} \cdot \sqrt{\frac{1}{\ell}}\right)\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot \left(\frac{M \cdot D}{d} \cdot \sqrt{0.5}\right)\right)}{\ell} \cdot \left(t\_0 \cdot \sqrt{0.5}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* 0.5 (* D (/ M d)))) (t_1 (sqrt (- d))))
   (if (<= l -1e-309)
     (*
      (/ t_1 (sqrt (- h)))
      (*
       (/ t_1 (sqrt (- l)))
       (fma (pow (* 0.5 (* M (/ D d))) 2.0) (* -0.5 (/ h l)) 1.0)))
     (if (<= l 2.9e+158)
       (*
        (* (/ (sqrt d) (sqrt h)) (pow (/ d l) 0.5))
        (+ 1.0 (/ -1.0 (/ l (* 0.5 (* h (pow t_0 2.0)))))))
       (*
        (* (pow (/ d h) 0.5) (* (sqrt d) (sqrt (/ 1.0 l))))
        (-
         1.0
         (*
          (/ (* h (* 0.5 (* (/ (* M D) d) (sqrt 0.5)))) l)
          (* t_0 (sqrt 0.5)))))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = 0.5 * (D * (M / d));
	double t_1 = sqrt(-d);
	double tmp;
	if (l <= -1e-309) {
		tmp = (t_1 / sqrt(-h)) * ((t_1 / sqrt(-l)) * fma(pow((0.5 * (M * (D / d))), 2.0), (-0.5 * (h / l)), 1.0));
	} else if (l <= 2.9e+158) {
		tmp = ((sqrt(d) / sqrt(h)) * pow((d / l), 0.5)) * (1.0 + (-1.0 / (l / (0.5 * (h * pow(t_0, 2.0))))));
	} else {
		tmp = (pow((d / h), 0.5) * (sqrt(d) * sqrt((1.0 / l)))) * (1.0 - (((h * (0.5 * (((M * D) / d) * sqrt(0.5)))) / l) * (t_0 * sqrt(0.5))));
	}
	return tmp;
}

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(0.5 * Float64(D * Float64(M / d)))
	t_1 = sqrt(Float64(-d))
	tmp = 0.0
	if (l <= -1e-309)
		tmp = Float64(Float64(t_1 / sqrt(Float64(-h))) * Float64(Float64(t_1 / sqrt(Float64(-l))) * fma((Float64(0.5 * Float64(M * Float64(D / d))) ^ 2.0), Float64(-0.5 * Float64(h / l)), 1.0)));
	elseif (l <= 2.9e+158)
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * (Float64(d / l) ^ 0.5)) * Float64(1.0 + Float64(-1.0 / Float64(l / Float64(0.5 * Float64(h * (t_0 ^ 2.0)))))));
	else
		tmp = Float64(Float64((Float64(d / h) ^ 0.5) * Float64(sqrt(d) * sqrt(Float64(1.0 / l)))) * Float64(1.0 - Float64(Float64(Float64(h * Float64(0.5 * Float64(Float64(Float64(M * D) / d) * sqrt(0.5)))) / l) * Float64(t_0 * sqrt(0.5)))));
	end
	return tmp
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(0.5 * N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[l, -1e-309], N[(N[(t$95$1 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(0.5 * N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.9e+158], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-1.0 / N[(l / N[(0.5 * N[(h * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] * N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(h * N[(0.5 * N[(N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$0 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.000000000000002e-309

   1. Initial program 72.4%

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

   3. Simplified 73.7%

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

      1. frac-2neg 73.7%

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

      2. sqrt-div 77.0%

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

   6. Applied egg-rr 77.0%

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

      1. frac-2neg 77.0%

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

      2. sqrt-div 87.2%

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

   8. Applied egg-rr 87.2%

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

   if -1.000000000000002e-309 < l < 2.90000000000000024e158

   1. Initial program 70.1%

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

      1. associate-*r/ 71.3%

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

      2. clear-num 71.3%

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

      3. metadata-eval 71.3%

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

      4. associate-*l* 71.3%

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

   4. Applied egg-rr 70.4%

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

      1. metadata-eval 70.4%

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

      2. pow1/2 70.4%

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

      3. sqrt-div 87.0%

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

   6. Applied egg-rr 87.0%

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

   if 2.90000000000000024e158 < l

   1. Initial program 54.2%

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

      1. *-commutative 54.2%

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

      2. metadata-eval 54.2%

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

      3. add-sqr-sqrt 54.2%

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

      4. associate-*r* 54.2%

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

   4. Applied egg-rr 54.9%

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

      1. associate-*l/ 64.0%

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

      2. metadata-eval 64.0%

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

      3. associate-*l* 64.0%

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

      4. metadata-eval 64.0%

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

      5. associate-*r/ 64.0%

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

   6. Applied egg-rr 64.0%

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

      1. div-inv 64.1%

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

      2. unpow-prod-down 82.6%

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

      3. metadata-eval 82.6%

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

      4. pow1/2 82.6%

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

      5. metadata-eval 82.6%

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

   8. Applied egg-rr 82.6%

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

      1. unpow1/2 82.6%

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

   10. Simplified 82.6%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)\\ \mathbf{elif}\;\ell \leq 2.9 \cdot 10^{+158}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{-1}{\frac{\ell}{0.5 \cdot \left(h \cdot {\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \left(\sqrt{d} \cdot \sqrt{\frac{1}{\ell}}\right)\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot \left(\frac{M \cdot D}{d} \cdot \sqrt{0.5}\right)\right)}{\ell} \cdot \left(\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \sqrt{0.5}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := {\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right)}^{2}\\ \mathbf{if}\;\ell \leq -11000000000:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 + t\_0 \cdot \frac{h \cdot -0.5}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{h \cdot \left({\left(\frac{M \cdot D}{d}\right)}^{2} \cdot 0.125\right)}{\ell} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{-1}{\frac{\ell}{0.5 \cdot \left(h \cdot t\_0\right)}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (pow (* 0.5 (* D (/ M d))) 2.0)))
   (if (<= l -11000000000.0)
     (*
      (sqrt (/ d h))
      (* (/ (sqrt (- d)) (sqrt (- l))) (+ 1.0 (* t_0 (/ (* h -0.5) l)))))
     (if (<= l -1e-309)
       (*
        (* d (sqrt (/ 1.0 (* l h))))
        (+ (/ (* h (* (pow (/ (* M D) d) 2.0) 0.125)) l) -1.0))
       (*
        (* (/ (sqrt d) (sqrt h)) (pow (/ d l) 0.5))
        (+ 1.0 (/ -1.0 (/ l (* 0.5 (* h t_0))))))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = pow((0.5 * (D * (M / d))), 2.0);
	double tmp;
	if (l <= -11000000000.0) {
		tmp = sqrt((d / h)) * ((sqrt(-d) / sqrt(-l)) * (1.0 + (t_0 * ((h * -0.5) / l))));
	} else if (l <= -1e-309) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (((h * (pow(((M * D) / d), 2.0) * 0.125)) / l) + -1.0);
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * pow((d / l), 0.5)) * (1.0 + (-1.0 / (l / (0.5 * (h * t_0)))));
	}
	return tmp;
}

Fortran

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (0.5d0 * (d_1 * (m / d))) ** 2.0d0
    if (l <= (-11000000000.0d0)) then
        tmp = sqrt((d / h)) * ((sqrt(-d) / sqrt(-l)) * (1.0d0 + (t_0 * ((h * (-0.5d0)) / l))))
    else if (l <= (-1d-309)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * (((h * ((((m * d_1) / d) ** 2.0d0) * 0.125d0)) / l) + (-1.0d0))
    else
        tmp = ((sqrt(d) / sqrt(h)) * ((d / l) ** 0.5d0)) * (1.0d0 + ((-1.0d0) / (l / (0.5d0 * (h * t_0)))))
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.pow((0.5 * (D * (M / d))), 2.0);
	double tmp;
	if (l <= -11000000000.0) {
		tmp = Math.sqrt((d / h)) * ((Math.sqrt(-d) / Math.sqrt(-l)) * (1.0 + (t_0 * ((h * -0.5) / l))));
	} else if (l <= -1e-309) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (((h * (Math.pow(((M * D) / d), 2.0) * 0.125)) / l) + -1.0);
	} else {
		tmp = ((Math.sqrt(d) / Math.sqrt(h)) * Math.pow((d / l), 0.5)) * (1.0 + (-1.0 / (l / (0.5 * (h * t_0)))));
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	t_0 = math.pow((0.5 * (D * (M / d))), 2.0)
	tmp = 0
	if l <= -11000000000.0:
		tmp = math.sqrt((d / h)) * ((math.sqrt(-d) / math.sqrt(-l)) * (1.0 + (t_0 * ((h * -0.5) / l))))
	elif l <= -1e-309:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (((h * (math.pow(((M * D) / d), 2.0) * 0.125)) / l) + -1.0)
	else:
		tmp = ((math.sqrt(d) / math.sqrt(h)) * math.pow((d / l), 0.5)) * (1.0 + (-1.0 / (l / (0.5 * (h * t_0)))))
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(0.5 * Float64(D * Float64(M / d))) ^ 2.0
	tmp = 0.0
	if (l <= -11000000000.0)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))) * Float64(1.0 + Float64(t_0 * Float64(Float64(h * -0.5) / l)))));
	elseif (l <= -1e-309)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(Float64(Float64(h * Float64((Float64(Float64(M * D) / d) ^ 2.0) * 0.125)) / l) + -1.0));
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * (Float64(d / l) ^ 0.5)) * Float64(1.0 + Float64(-1.0 / Float64(l / Float64(0.5 * Float64(h * t_0))))));
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = (0.5 * (D * (M / d))) ^ 2.0;
	tmp = 0.0;
	if (l <= -11000000000.0)
		tmp = sqrt((d / h)) * ((sqrt(-d) / sqrt(-l)) * (1.0 + (t_0 * ((h * -0.5) / l))));
	elseif (l <= -1e-309)
		tmp = (d * sqrt((1.0 / (l * h)))) * (((h * ((((M * D) / d) ^ 2.0) * 0.125)) / l) + -1.0);
	else
		tmp = ((sqrt(d) / sqrt(h)) * ((d / l) ^ 0.5)) * (1.0 + (-1.0 / (l / (0.5 * (h * t_0)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Power[N[(0.5 * N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[l, -11000000000.0], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(t$95$0 * N[(N[(h * -0.5), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1e-309], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(h * N[(N[Power[N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-1.0 / N[(l / N[(0.5 * N[(h * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.1e10

   1. Initial program 66.2%

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

   3. Simplified 69.0%

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

      1. frac-2neg 69.0%

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

      2. sqrt-div 74.2%

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

   6. Applied egg-rr 74.2%

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

      1. fma-undefine 74.2%

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

   8. Applied egg-rr 75.4%

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

   if -1.1e10 < l < -1.000000000000002e-309

   1. Initial program 78.3%

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

      1. associate-*r/ 82.5%

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

      2. clear-num 82.6%

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

      3. metadata-eval 82.6%

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

      4. associate-*l* 82.6%

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

   4. Applied egg-rr 82.6%

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

      1. *-commutative 82.6%

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

   6. Applied egg-rr 77.1%

      \[\leadsto \color{blue}{\left(1 - \frac{1}{\ell} \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 77.1%

         \[\leadsto \left(1 - \color{blue}{\frac{1 \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)}{\ell}}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      2. *-un-lft-identity 77.1%

         \[\leadsto \left(1 - \frac{\color{blue}{\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      3. *-commutative 77.1%

         \[\leadsto \left(1 - \frac{\color{blue}{h \cdot \left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right)}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      4. associate-*r* 77.1%

         \[\leadsto \left(1 - \frac{h \cdot \color{blue}{\left(\left(0.5 \cdot 0.25\right) \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      5. metadata-eval 77.1%

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

      6. associate-*r/ 77.1%

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

      7. *-commutative 77.1%

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

      8. associate-*r/ 77.1%

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

   8. Applied egg-rr 77.1%

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

   9. Taylor expanded in d around -inf 92.8%

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

       1. mul-1-neg 92.8%

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

       2. distribute-rgt-neg-in 92.8%

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

   11. Simplified 92.8%

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

   if -1.000000000000002e-309 < l

   1. Initial program 67.2%

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

      1. associate-*r/ 69.1%

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

      2. clear-num 69.1%

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

      3. metadata-eval 69.1%

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

      4. associate-*l* 69.1%

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

   4. Applied egg-rr 68.4%

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

      1. metadata-eval 68.4%

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

      2. pow1/2 68.4%

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

      3. sqrt-div 82.8%

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

   6. Applied egg-rr 82.8%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -11000000000:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 + {\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right)}^{2} \cdot \frac{h \cdot -0.5}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{h \cdot \left({\left(\frac{M \cdot D}{d}\right)}^{2} \cdot 0.125\right)}{\ell} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{-1}{\frac{\ell}{0.5 \cdot \left(h \cdot {\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right)}^{2}\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := D \cdot \frac{M}{d}\\ \mathbf{if}\;\ell \leq -30500000000:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 + {\left(0.5 \cdot t\_0\right)}^{2} \cdot \frac{h \cdot -0.5}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{h \cdot \left({\left(\frac{M \cdot D}{d}\right)}^{2} \cdot 0.125\right)}{\ell} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1 + h \cdot \frac{-0.125 \cdot {t\_0}^{2}}{\ell}}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* D (/ M d))))
   (if (<= l -30500000000.0)
     (*
      (sqrt (/ d h))
      (*
       (/ (sqrt (- d)) (sqrt (- l)))
       (+ 1.0 (* (pow (* 0.5 t_0) 2.0) (/ (* h -0.5) l)))))
     (if (<= l -1e-309)
       (*
        (* d (sqrt (/ 1.0 (* l h))))
        (+ (/ (* h (* (pow (/ (* M D) d) 2.0) 0.125)) l) -1.0))
       (*
        d
        (/ (+ 1.0 (* h (/ (* -0.125 (pow t_0 2.0)) l))) (sqrt (* l h))))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = D * (M / d);
	double tmp;
	if (l <= -30500000000.0) {
		tmp = sqrt((d / h)) * ((sqrt(-d) / sqrt(-l)) * (1.0 + (pow((0.5 * t_0), 2.0) * ((h * -0.5) / l))));
	} else if (l <= -1e-309) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (((h * (pow(((M * D) / d), 2.0) * 0.125)) / l) + -1.0);
	} else {
		tmp = d * ((1.0 + (h * ((-0.125 * pow(t_0, 2.0)) / l))) / sqrt((l * h)));
	}
	return tmp;
}

Fortran

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d_1 * (m / d)
    if (l <= (-30500000000.0d0)) then
        tmp = sqrt((d / h)) * ((sqrt(-d) / sqrt(-l)) * (1.0d0 + (((0.5d0 * t_0) ** 2.0d0) * ((h * (-0.5d0)) / l))))
    else if (l <= (-1d-309)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * (((h * ((((m * d_1) / d) ** 2.0d0) * 0.125d0)) / l) + (-1.0d0))
    else
        tmp = d * ((1.0d0 + (h * (((-0.125d0) * (t_0 ** 2.0d0)) / l))) / sqrt((l * h)))
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = D * (M / d);
	double tmp;
	if (l <= -30500000000.0) {
		tmp = Math.sqrt((d / h)) * ((Math.sqrt(-d) / Math.sqrt(-l)) * (1.0 + (Math.pow((0.5 * t_0), 2.0) * ((h * -0.5) / l))));
	} else if (l <= -1e-309) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (((h * (Math.pow(((M * D) / d), 2.0) * 0.125)) / l) + -1.0);
	} else {
		tmp = d * ((1.0 + (h * ((-0.125 * Math.pow(t_0, 2.0)) / l))) / Math.sqrt((l * h)));
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	t_0 = D * (M / d)
	tmp = 0
	if l <= -30500000000.0:
		tmp = math.sqrt((d / h)) * ((math.sqrt(-d) / math.sqrt(-l)) * (1.0 + (math.pow((0.5 * t_0), 2.0) * ((h * -0.5) / l))))
	elif l <= -1e-309:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (((h * (math.pow(((M * D) / d), 2.0) * 0.125)) / l) + -1.0)
	else:
		tmp = d * ((1.0 + (h * ((-0.125 * math.pow(t_0, 2.0)) / l))) / math.sqrt((l * h)))
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(D * Float64(M / d))
	tmp = 0.0
	if (l <= -30500000000.0)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))) * Float64(1.0 + Float64((Float64(0.5 * t_0) ^ 2.0) * Float64(Float64(h * -0.5) / l)))));
	elseif (l <= -1e-309)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(Float64(Float64(h * Float64((Float64(Float64(M * D) / d) ^ 2.0) * 0.125)) / l) + -1.0));
	else
		tmp = Float64(d * Float64(Float64(1.0 + Float64(h * Float64(Float64(-0.125 * (t_0 ^ 2.0)) / l))) / sqrt(Float64(l * h))));
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = D * (M / d);
	tmp = 0.0;
	if (l <= -30500000000.0)
		tmp = sqrt((d / h)) * ((sqrt(-d) / sqrt(-l)) * (1.0 + (((0.5 * t_0) ^ 2.0) * ((h * -0.5) / l))));
	elseif (l <= -1e-309)
		tmp = (d * sqrt((1.0 / (l * h)))) * (((h * ((((M * D) / d) ^ 2.0) * 0.125)) / l) + -1.0);
	else
		tmp = d * ((1.0 + (h * ((-0.125 * (t_0 ^ 2.0)) / l))) / sqrt((l * h)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -30500000000.0], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[Power[N[(0.5 * t$95$0), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h * -0.5), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1e-309], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(h * N[(N[Power[N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(1.0 + N[(h * N[(N[(-0.125 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.05e10

   1. Initial program 66.2%

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

   3. Simplified 69.0%

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

      1. frac-2neg 69.0%

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

      2. sqrt-div 74.2%

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

   6. Applied egg-rr 74.2%

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

      1. fma-undefine 74.2%

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

   8. Applied egg-rr 75.4%

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

   if -3.05e10 < l < -1.000000000000002e-309

   1. Initial program 78.3%

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

      1. associate-*r/ 82.5%

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

      2. clear-num 82.6%

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

      3. metadata-eval 82.6%

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

      4. associate-*l* 82.6%

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

   4. Applied egg-rr 82.6%

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

      1. *-commutative 82.6%

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

   6. Applied egg-rr 77.1%

      \[\leadsto \color{blue}{\left(1 - \frac{1}{\ell} \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 77.1%

         \[\leadsto \left(1 - \color{blue}{\frac{1 \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)}{\ell}}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      2. *-un-lft-identity 77.1%

         \[\leadsto \left(1 - \frac{\color{blue}{\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      3. *-commutative 77.1%

         \[\leadsto \left(1 - \frac{\color{blue}{h \cdot \left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right)}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      4. associate-*r* 77.1%

         \[\leadsto \left(1 - \frac{h \cdot \color{blue}{\left(\left(0.5 \cdot 0.25\right) \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      5. metadata-eval 77.1%

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

      6. associate-*r/ 77.1%

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

      7. *-commutative 77.1%

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

      8. associate-*r/ 77.1%

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

   8. Applied egg-rr 77.1%

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

   9. Taylor expanded in d around -inf 92.8%

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

       1. mul-1-neg 92.8%

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

       2. distribute-rgt-neg-in 92.8%

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

   11. Simplified 92.8%

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

   if -1.000000000000002e-309 < l

   1. Initial program 67.2%

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

      1. associate-*r/ 69.1%

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

      2. clear-num 69.1%

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

      3. metadata-eval 69.1%

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

      4. associate-*l* 69.1%

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

   4. Applied egg-rr 68.4%

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

      1. *-commutative 68.4%

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

   6. Applied egg-rr 59.8%

      \[\leadsto \color{blue}{\left(1 - \frac{1}{\ell} \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 59.8%

         \[\leadsto \left(1 - \color{blue}{\frac{1 \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)}{\ell}}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      2. *-un-lft-identity 59.8%

         \[\leadsto \left(1 - \frac{\color{blue}{\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      3. *-commutative 59.8%

         \[\leadsto \left(1 - \frac{\color{blue}{h \cdot \left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right)}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      4. associate-*r* 59.8%

         \[\leadsto \left(1 - \frac{h \cdot \color{blue}{\left(\left(0.5 \cdot 0.25\right) \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      5. metadata-eval 59.8%

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

      6. associate-*r/ 59.1%

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

      7. *-commutative 59.1%

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

      8. associate-*r/ 59.8%

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

   8. Applied egg-rr 59.8%

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

   10. Applied egg-rr 75.4%

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

       1. *-lft-identity 75.4%

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

       2. associate-*l/ 77.9%

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

       3. associate-/l* 77.1%

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

   12. Simplified 82.1%

       \[\leadsto \color{blue}{d \cdot \frac{1 + h \cdot \frac{-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}}{\sqrt{h \cdot \ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -30500000000:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 + {\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right)}^{2} \cdot \frac{h \cdot -0.5}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{h \cdot \left({\left(\frac{M \cdot D}{d}\right)}^{2} \cdot 0.125\right)}{\ell} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1 + h \cdot \frac{-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -6.5 \cdot 10^{-67}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(h, \frac{-0.125}{\ell} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, 1\right)\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{h \cdot \left({\left(\frac{M \cdot D}{d}\right)}^{2} \cdot 0.125\right)}{\ell} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1 + h \cdot \frac{-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= d -6.5e-67)
   (*
    (sqrt (/ d h))
    (* (sqrt (/ d l)) (fma h (* (/ -0.125 l) (pow (* M (/ D d)) 2.0)) 1.0)))
   (if (<= d -1e-309)
     (*
      (* d (sqrt (/ 1.0 (* l h))))
      (+ (/ (* h (* (pow (/ (* M D) d) 2.0) 0.125)) l) -1.0))
     (*
      d
      (/
       (+ 1.0 (* h (/ (* -0.125 (pow (* D (/ M d)) 2.0)) l)))
       (sqrt (* l h)))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -6.5e-67) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * fma(h, ((-0.125 / l) * pow((M * (D / d)), 2.0)), 1.0));
	} else if (d <= -1e-309) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (((h * (pow(((M * D) / d), 2.0) * 0.125)) / l) + -1.0);
	} else {
		tmp = d * ((1.0 + (h * ((-0.125 * pow((D * (M / d)), 2.0)) / l))) / sqrt((l * h)));
	}
	return tmp;
}

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -6.5e-67)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * fma(h, Float64(Float64(-0.125 / l) * (Float64(M * Float64(D / d)) ^ 2.0)), 1.0)));
	elseif (d <= -1e-309)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(Float64(Float64(h * Float64((Float64(Float64(M * D) / d) ^ 2.0) * 0.125)) / l) + -1.0));
	else
		tmp = Float64(d * Float64(Float64(1.0 + Float64(h * Float64(Float64(-0.125 * (Float64(D * Float64(M / d)) ^ 2.0)) / l))) / sqrt(Float64(l * h))));
	end
	return tmp
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[d, -6.5e-67], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(h * N[(N[(-0.125 / l), $MachinePrecision] * N[Power[N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1e-309], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(h * N[(N[Power[N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(1.0 + N[(h * N[(N[(-0.125 * N[Power[N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -6.4999999999999997e-67

   1. Initial program 86.0%

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

   3. Simplified 88.5%

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

   5. Taylor expanded in M around 0 55.3%

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

   7. Simplified 93.5%

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

   if -6.4999999999999997e-67 < d < -1.000000000000002e-309

   1. Initial program 53.7%

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

      1. associate-*r/ 54.0%

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

      2. clear-num 54.0%

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

      3. metadata-eval 54.0%

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

      4. associate-*l* 54.0%

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

   4. Applied egg-rr 54.0%

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

      1. *-commutative 54.0%

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

   6. Applied egg-rr 45.3%

      \[\leadsto \color{blue}{\left(1 - \frac{1}{\ell} \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 45.3%

         \[\leadsto \left(1 - \color{blue}{\frac{1 \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)}{\ell}}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      2. *-un-lft-identity 45.3%

         \[\leadsto \left(1 - \frac{\color{blue}{\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      3. *-commutative 45.3%

         \[\leadsto \left(1 - \frac{\color{blue}{h \cdot \left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right)}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      4. associate-*r* 45.3%

         \[\leadsto \left(1 - \frac{h \cdot \color{blue}{\left(\left(0.5 \cdot 0.25\right) \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      5. metadata-eval 45.3%

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

      6. associate-*r/ 45.3%

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

      7. *-commutative 45.3%

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

      8. associate-*r/ 45.3%

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

   8. Applied egg-rr 45.3%

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

   9. Taylor expanded in d around -inf 72.9%

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

       1. mul-1-neg 72.9%

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

       2. distribute-rgt-neg-in 72.9%

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

   11. Simplified 72.9%

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

   if -1.000000000000002e-309 < d

   1. Initial program 67.2%

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

      1. associate-*r/ 69.1%

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

      2. clear-num 69.1%

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

      3. metadata-eval 69.1%

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

      4. associate-*l* 69.1%

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

   4. Applied egg-rr 68.4%

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

      1. *-commutative 68.4%

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

   6. Applied egg-rr 59.8%

      \[\leadsto \color{blue}{\left(1 - \frac{1}{\ell} \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 59.8%

         \[\leadsto \left(1 - \color{blue}{\frac{1 \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)}{\ell}}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      2. *-un-lft-identity 59.8%

         \[\leadsto \left(1 - \frac{\color{blue}{\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      3. *-commutative 59.8%

         \[\leadsto \left(1 - \frac{\color{blue}{h \cdot \left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right)}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      4. associate-*r* 59.8%

         \[\leadsto \left(1 - \frac{h \cdot \color{blue}{\left(\left(0.5 \cdot 0.25\right) \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      5. metadata-eval 59.8%

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

      6. associate-*r/ 59.1%

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

      7. *-commutative 59.1%

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

      8. associate-*r/ 59.8%

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

   8. Applied egg-rr 59.8%

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

   10. Applied egg-rr 75.4%

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

       1. *-lft-identity 75.4%

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

       2. associate-*l/ 77.9%

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

       3. associate-/l* 77.1%

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

   12. Simplified 82.1%

       \[\leadsto \color{blue}{d \cdot \frac{1 + h \cdot \frac{-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}}{\sqrt{h \cdot \ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.5 \cdot 10^{-67}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(h, \frac{-0.125}{\ell} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, 1\right)\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{h \cdot \left({\left(\frac{M \cdot D}{d}\right)}^{2} \cdot 0.125\right)}{\ell} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1 + h \cdot \frac{-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := D \cdot \frac{M}{d}\\ \mathbf{if}\;d \leq -1 \cdot 10^{-85}:\\ \;\;\;\;\left(1 + \frac{-1}{\frac{\ell}{0.5 \cdot \left(h \cdot {\left(0.5 \cdot t\_0\right)}^{2}\right)}}\right) \cdot \left({\left(\frac{d}{h}\right)}^{0.5} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{h \cdot \left({\left(\frac{M \cdot D}{d}\right)}^{2} \cdot 0.125\right)}{\ell} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1 + h \cdot \frac{-0.125 \cdot {t\_0}^{2}}{\ell}}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* D (/ M d))))
   (if (<= d -1e-85)
     (*
      (+ 1.0 (/ -1.0 (/ l (* 0.5 (* h (pow (* 0.5 t_0) 2.0))))))
      (* (pow (/ d h) 0.5) (sqrt (/ d l))))
     (if (<= d -1e-309)
       (*
        (* d (sqrt (/ 1.0 (* l h))))
        (+ (/ (* h (* (pow (/ (* M D) d) 2.0) 0.125)) l) -1.0))
       (*
        d
        (/ (+ 1.0 (* h (/ (* -0.125 (pow t_0 2.0)) l))) (sqrt (* l h))))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = D * (M / d);
	double tmp;
	if (d <= -1e-85) {
		tmp = (1.0 + (-1.0 / (l / (0.5 * (h * pow((0.5 * t_0), 2.0)))))) * (pow((d / h), 0.5) * sqrt((d / l)));
	} else if (d <= -1e-309) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (((h * (pow(((M * D) / d), 2.0) * 0.125)) / l) + -1.0);
	} else {
		tmp = d * ((1.0 + (h * ((-0.125 * pow(t_0, 2.0)) / l))) / sqrt((l * h)));
	}
	return tmp;
}

Fortran

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d_1 * (m / d)
    if (d <= (-1d-85)) then
        tmp = (1.0d0 + ((-1.0d0) / (l / (0.5d0 * (h * ((0.5d0 * t_0) ** 2.0d0)))))) * (((d / h) ** 0.5d0) * sqrt((d / l)))
    else if (d <= (-1d-309)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * (((h * ((((m * d_1) / d) ** 2.0d0) * 0.125d0)) / l) + (-1.0d0))
    else
        tmp = d * ((1.0d0 + (h * (((-0.125d0) * (t_0 ** 2.0d0)) / l))) / sqrt((l * h)))
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = D * (M / d);
	double tmp;
	if (d <= -1e-85) {
		tmp = (1.0 + (-1.0 / (l / (0.5 * (h * Math.pow((0.5 * t_0), 2.0)))))) * (Math.pow((d / h), 0.5) * Math.sqrt((d / l)));
	} else if (d <= -1e-309) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (((h * (Math.pow(((M * D) / d), 2.0) * 0.125)) / l) + -1.0);
	} else {
		tmp = d * ((1.0 + (h * ((-0.125 * Math.pow(t_0, 2.0)) / l))) / Math.sqrt((l * h)));
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	t_0 = D * (M / d)
	tmp = 0
	if d <= -1e-85:
		tmp = (1.0 + (-1.0 / (l / (0.5 * (h * math.pow((0.5 * t_0), 2.0)))))) * (math.pow((d / h), 0.5) * math.sqrt((d / l)))
	elif d <= -1e-309:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (((h * (math.pow(((M * D) / d), 2.0) * 0.125)) / l) + -1.0)
	else:
		tmp = d * ((1.0 + (h * ((-0.125 * math.pow(t_0, 2.0)) / l))) / math.sqrt((l * h)))
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(D * Float64(M / d))
	tmp = 0.0
	if (d <= -1e-85)
		tmp = Float64(Float64(1.0 + Float64(-1.0 / Float64(l / Float64(0.5 * Float64(h * (Float64(0.5 * t_0) ^ 2.0)))))) * Float64((Float64(d / h) ^ 0.5) * sqrt(Float64(d / l))));
	elseif (d <= -1e-309)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(Float64(Float64(h * Float64((Float64(Float64(M * D) / d) ^ 2.0) * 0.125)) / l) + -1.0));
	else
		tmp = Float64(d * Float64(Float64(1.0 + Float64(h * Float64(Float64(-0.125 * (t_0 ^ 2.0)) / l))) / sqrt(Float64(l * h))));
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = D * (M / d);
	tmp = 0.0;
	if (d <= -1e-85)
		tmp = (1.0 + (-1.0 / (l / (0.5 * (h * ((0.5 * t_0) ^ 2.0)))))) * (((d / h) ^ 0.5) * sqrt((d / l)));
	elseif (d <= -1e-309)
		tmp = (d * sqrt((1.0 / (l * h)))) * (((h * ((((M * D) / d) ^ 2.0) * 0.125)) / l) + -1.0);
	else
		tmp = d * ((1.0 + (h * ((-0.125 * (t_0 ^ 2.0)) / l))) / sqrt((l * h)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1e-85], N[(N[(1.0 + N[(-1.0 / N[(l / N[(0.5 * N[(h * N[Power[N[(0.5 * t$95$0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1e-309], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(h * N[(N[Power[N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(1.0 + N[(h * N[(N[(-0.125 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -9.9999999999999998e-86

   1. Initial program 86.5%

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

      1. associate-*r/ 90.2%

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

      2. clear-num 90.2%

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

      3. metadata-eval 90.2%

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

      4. associate-*l* 90.2%

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

   4. Applied egg-rr 91.2%

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

   5. Taylor expanded in d around 0 91.2%

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

   if -9.9999999999999998e-86 < d < -1.000000000000002e-309

   1. Initial program 51.1%

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

      1. associate-*r/ 51.4%

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

      2. clear-num 51.4%

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

      3. metadata-eval 51.4%

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

      4. associate-*l* 51.4%

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

   4. Applied egg-rr 51.4%

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

      1. *-commutative 51.4%

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

   6. Applied egg-rr 42.3%

      \[\leadsto \color{blue}{\left(1 - \frac{1}{\ell} \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 42.3%

         \[\leadsto \left(1 - \color{blue}{\frac{1 \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)}{\ell}}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      2. *-un-lft-identity 42.3%

         \[\leadsto \left(1 - \frac{\color{blue}{\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      3. *-commutative 42.3%

         \[\leadsto \left(1 - \frac{\color{blue}{h \cdot \left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right)}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      4. associate-*r* 42.3%

         \[\leadsto \left(1 - \frac{h \cdot \color{blue}{\left(\left(0.5 \cdot 0.25\right) \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      5. metadata-eval 42.3%

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

      6. associate-*r/ 42.3%

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

      7. *-commutative 42.3%

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

      8. associate-*r/ 42.3%

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

   8. Applied egg-rr 42.3%

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

   9. Taylor expanded in d around -inf 73.1%

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

       1. mul-1-neg 73.1%

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

       2. distribute-rgt-neg-in 73.1%

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

   11. Simplified 73.1%

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

   if -1.000000000000002e-309 < d

   1. Initial program 67.2%

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

      1. associate-*r/ 69.1%

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

      2. clear-num 69.1%

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

      3. metadata-eval 69.1%

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

      4. associate-*l* 69.1%

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

   4. Applied egg-rr 68.4%

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

      1. *-commutative 68.4%

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

   6. Applied egg-rr 59.8%

      \[\leadsto \color{blue}{\left(1 - \frac{1}{\ell} \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 59.8%

         \[\leadsto \left(1 - \color{blue}{\frac{1 \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)}{\ell}}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      2. *-un-lft-identity 59.8%

         \[\leadsto \left(1 - \frac{\color{blue}{\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      3. *-commutative 59.8%

         \[\leadsto \left(1 - \frac{\color{blue}{h \cdot \left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right)}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      4. associate-*r* 59.8%

         \[\leadsto \left(1 - \frac{h \cdot \color{blue}{\left(\left(0.5 \cdot 0.25\right) \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      5. metadata-eval 59.8%

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

      6. associate-*r/ 59.1%

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

      7. *-commutative 59.1%

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

      8. associate-*r/ 59.8%

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

   8. Applied egg-rr 59.8%

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

   10. Applied egg-rr 75.4%

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

       1. *-lft-identity 75.4%

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

       2. associate-*l/ 77.9%

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

       3. associate-/l* 77.1%

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

   12. Simplified 82.1%

       \[\leadsto \color{blue}{d \cdot \frac{1 + h \cdot \frac{-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}}{\sqrt{h \cdot \ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1 \cdot 10^{-85}:\\ \;\;\;\;\left(1 + \frac{-1}{\frac{\ell}{0.5 \cdot \left(h \cdot {\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right)}^{2}\right)}}\right) \cdot \left({\left(\frac{d}{h}\right)}^{0.5} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{h \cdot \left({\left(\frac{M \cdot D}{d}\right)}^{2} \cdot 0.125\right)}{\ell} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1 + h \cdot \frac{-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := {\left(\frac{M \cdot D}{d}\right)}^{2}\\ \mathbf{if}\;d \leq -9.2 \cdot 10^{-87}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(0.5 \cdot \left(t\_0 \cdot 0.25\right)\right)\right) \cdot \frac{-1}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{h \cdot \left(t\_0 \cdot 0.125\right)}{\ell} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1 + h \cdot \frac{-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (pow (/ (* M D) d) 2.0)))
   (if (<= d -9.2e-87)
     (*
      (sqrt (/ d h))
      (* (sqrt (/ d l)) (+ 1.0 (* (* h (* 0.5 (* t_0 0.25))) (/ -1.0 l)))))
     (if (<= d -1e-309)
       (* (* d (sqrt (/ 1.0 (* l h)))) (+ (/ (* h (* t_0 0.125)) l) -1.0))
       (*
        d
        (/
         (+ 1.0 (* h (/ (* -0.125 (pow (* D (/ M d)) 2.0)) l)))
         (sqrt (* l h))))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = pow(((M * D) / d), 2.0);
	double tmp;
	if (d <= -9.2e-87) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 + ((h * (0.5 * (t_0 * 0.25))) * (-1.0 / l))));
	} else if (d <= -1e-309) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (((h * (t_0 * 0.125)) / l) + -1.0);
	} else {
		tmp = d * ((1.0 + (h * ((-0.125 * pow((D * (M / d)), 2.0)) / l))) / sqrt((l * h)));
	}
	return tmp;
}

Fortran

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((m * d_1) / d) ** 2.0d0
    if (d <= (-9.2d-87)) then
        tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 + ((h * (0.5d0 * (t_0 * 0.25d0))) * ((-1.0d0) / l))))
    else if (d <= (-1d-309)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * (((h * (t_0 * 0.125d0)) / l) + (-1.0d0))
    else
        tmp = d * ((1.0d0 + (h * (((-0.125d0) * ((d_1 * (m / d)) ** 2.0d0)) / l))) / sqrt((l * h)))
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.pow(((M * D) / d), 2.0);
	double tmp;
	if (d <= -9.2e-87) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 + ((h * (0.5 * (t_0 * 0.25))) * (-1.0 / l))));
	} else if (d <= -1e-309) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (((h * (t_0 * 0.125)) / l) + -1.0);
	} else {
		tmp = d * ((1.0 + (h * ((-0.125 * Math.pow((D * (M / d)), 2.0)) / l))) / Math.sqrt((l * h)));
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	t_0 = math.pow(((M * D) / d), 2.0)
	tmp = 0
	if d <= -9.2e-87:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 + ((h * (0.5 * (t_0 * 0.25))) * (-1.0 / l))))
	elif d <= -1e-309:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (((h * (t_0 * 0.125)) / l) + -1.0)
	else:
		tmp = d * ((1.0 + (h * ((-0.125 * math.pow((D * (M / d)), 2.0)) / l))) / math.sqrt((l * h)))
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(M * D) / d) ^ 2.0
	tmp = 0.0
	if (d <= -9.2e-87)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(Float64(h * Float64(0.5 * Float64(t_0 * 0.25))) * Float64(-1.0 / l)))));
	elseif (d <= -1e-309)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(Float64(Float64(h * Float64(t_0 * 0.125)) / l) + -1.0));
	else
		tmp = Float64(d * Float64(Float64(1.0 + Float64(h * Float64(Float64(-0.125 * (Float64(D * Float64(M / d)) ^ 2.0)) / l))) / sqrt(Float64(l * h))));
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = ((M * D) / d) ^ 2.0;
	tmp = 0.0;
	if (d <= -9.2e-87)
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 + ((h * (0.5 * (t_0 * 0.25))) * (-1.0 / l))));
	elseif (d <= -1e-309)
		tmp = (d * sqrt((1.0 / (l * h)))) * (((h * (t_0 * 0.125)) / l) + -1.0);
	else
		tmp = d * ((1.0 + (h * ((-0.125 * ((D * (M / d)) ^ 2.0)) / l))) / sqrt((l * h)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Power[N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[d, -9.2e-87], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h * N[(0.5 * N[(t$95$0 * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1e-309], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(h * N[(t$95$0 * 0.125), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(1.0 + N[(h * N[(N[(-0.125 * N[Power[N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -9.2000000000000006e-87

   1. Initial program 86.5%

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

      1. associate-*r/ 90.2%

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

      2. clear-num 90.2%

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

      3. metadata-eval 90.2%

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

      4. associate-*l* 90.2%

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

   4. Applied egg-rr 91.2%

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

      1. metadata-eval 91.2%

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

      2. pow1/2 91.2%

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

      3. metadata-eval 91.2%

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

      4. pow1/2 91.2%

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

      5. associate-*l* 91.2%

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

      6. associate-/r/ 91.2%

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

      7. metadata-eval 91.2%

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

      8. associate-*r* 91.2%

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

   6. Applied egg-rr 90.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{1}{\ell} \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)\right)\right)} \]

   if -9.2000000000000006e-87 < d < -1.000000000000002e-309

   1. Initial program 51.1%

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

      1. associate-*r/ 51.4%

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

      2. clear-num 51.4%

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

      3. metadata-eval 51.4%

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

      4. associate-*l* 51.4%

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

   4. Applied egg-rr 51.4%

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

      1. *-commutative 51.4%

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

   6. Applied egg-rr 42.3%

      \[\leadsto \color{blue}{\left(1 - \frac{1}{\ell} \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 42.3%

         \[\leadsto \left(1 - \color{blue}{\frac{1 \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)}{\ell}}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      2. *-un-lft-identity 42.3%

         \[\leadsto \left(1 - \frac{\color{blue}{\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      3. *-commutative 42.3%

         \[\leadsto \left(1 - \frac{\color{blue}{h \cdot \left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right)}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      4. associate-*r* 42.3%

         \[\leadsto \left(1 - \frac{h \cdot \color{blue}{\left(\left(0.5 \cdot 0.25\right) \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      5. metadata-eval 42.3%

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

      6. associate-*r/ 42.3%

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

      7. *-commutative 42.3%

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

      8. associate-*r/ 42.3%

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

   8. Applied egg-rr 42.3%

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

   9. Taylor expanded in d around -inf 73.1%

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

       1. mul-1-neg 73.1%

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

       2. distribute-rgt-neg-in 73.1%

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

   11. Simplified 73.1%

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

   if -1.000000000000002e-309 < d

   1. Initial program 67.2%

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

      1. associate-*r/ 69.1%

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

      2. clear-num 69.1%

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

      3. metadata-eval 69.1%

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

      4. associate-*l* 69.1%

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

   4. Applied egg-rr 68.4%

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

      1. *-commutative 68.4%

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

   6. Applied egg-rr 59.8%

      \[\leadsto \color{blue}{\left(1 - \frac{1}{\ell} \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 59.8%

         \[\leadsto \left(1 - \color{blue}{\frac{1 \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)}{\ell}}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      2. *-un-lft-identity 59.8%

         \[\leadsto \left(1 - \frac{\color{blue}{\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      3. *-commutative 59.8%

         \[\leadsto \left(1 - \frac{\color{blue}{h \cdot \left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right)}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      4. associate-*r* 59.8%

         \[\leadsto \left(1 - \frac{h \cdot \color{blue}{\left(\left(0.5 \cdot 0.25\right) \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      5. metadata-eval 59.8%

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

      6. associate-*r/ 59.1%

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

      7. *-commutative 59.1%

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

      8. associate-*r/ 59.8%

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

   8. Applied egg-rr 59.8%

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

   10. Applied egg-rr 75.4%

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

       1. *-lft-identity 75.4%

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

       2. associate-*l/ 77.9%

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

       3. associate-/l* 77.1%

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

   12. Simplified 82.1%

       \[\leadsto \color{blue}{d \cdot \frac{1 + h \cdot \frac{-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}}{\sqrt{h \cdot \ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.2 \cdot 10^{-87}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(0.5 \cdot \left({\left(\frac{M \cdot D}{d}\right)}^{2} \cdot 0.25\right)\right)\right) \cdot \frac{-1}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{h \cdot \left({\left(\frac{M \cdot D}{d}\right)}^{2} \cdot 0.125\right)}{\ell} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1 + h \cdot \frac{-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -4 \cdot 10^{-86}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot \left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{h \cdot \left({\left(\frac{M \cdot D}{d}\right)}^{2} \cdot 0.125\right)}{\ell} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1 + h \cdot \frac{-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= d -4e-86)
   (*
    (sqrt (/ d h))
    (*
     (sqrt (/ d l))
     (- 1.0 (* 0.5 (* (/ h l) (* 0.25 (pow (/ M (/ d D)) 2.0)))))))
   (if (<= d -1e-309)
     (*
      (* d (sqrt (/ 1.0 (* l h))))
      (+ (/ (* h (* (pow (/ (* M D) d) 2.0) 0.125)) l) -1.0))
     (*
      d
      (/
       (+ 1.0 (* h (/ (* -0.125 (pow (* D (/ M d)) 2.0)) l)))
       (sqrt (* l h)))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -4e-86) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * (0.25 * pow((M / (d / D)), 2.0))))));
	} else if (d <= -1e-309) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (((h * (pow(((M * D) / d), 2.0) * 0.125)) / l) + -1.0);
	} else {
		tmp = d * ((1.0 + (h * ((-0.125 * pow((D * (M / d)), 2.0)) / l))) / sqrt((l * h)));
	}
	return tmp;
}

Fortran

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= (-4d-86)) then
        tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 - (0.5d0 * ((h / l) * (0.25d0 * ((m / (d / d_1)) ** 2.0d0))))))
    else if (d <= (-1d-309)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * (((h * ((((m * d_1) / d) ** 2.0d0) * 0.125d0)) / l) + (-1.0d0))
    else
        tmp = d * ((1.0d0 + (h * (((-0.125d0) * ((d_1 * (m / d)) ** 2.0d0)) / l))) / sqrt((l * h)))
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -4e-86) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * (0.25 * Math.pow((M / (d / D)), 2.0))))));
	} else if (d <= -1e-309) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (((h * (Math.pow(((M * D) / d), 2.0) * 0.125)) / l) + -1.0);
	} else {
		tmp = d * ((1.0 + (h * ((-0.125 * Math.pow((D * (M / d)), 2.0)) / l))) / Math.sqrt((l * h)));
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	tmp = 0
	if d <= -4e-86:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * (0.25 * math.pow((M / (d / D)), 2.0))))))
	elif d <= -1e-309:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (((h * (math.pow(((M * D) / d), 2.0) * 0.125)) / l) + -1.0)
	else:
		tmp = d * ((1.0 + (h * ((-0.125 * math.pow((D * (M / d)), 2.0)) / l))) / math.sqrt((l * h)))
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -4e-86)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * Float64(0.25 * (Float64(M / Float64(d / D)) ^ 2.0)))))));
	elseif (d <= -1e-309)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(Float64(Float64(h * Float64((Float64(Float64(M * D) / d) ^ 2.0) * 0.125)) / l) + -1.0));
	else
		tmp = Float64(d * Float64(Float64(1.0 + Float64(h * Float64(Float64(-0.125 * (Float64(D * Float64(M / d)) ^ 2.0)) / l))) / sqrt(Float64(l * h))));
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -4e-86)
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * (0.25 * ((M / (d / D)) ^ 2.0))))));
	elseif (d <= -1e-309)
		tmp = (d * sqrt((1.0 / (l * h)))) * (((h * ((((M * D) / d) ^ 2.0) * 0.125)) / l) + -1.0);
	else
		tmp = d * ((1.0 + (h * ((-0.125 * ((D * (M / d)) ^ 2.0)) / l))) / sqrt((l * h)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[d, -4e-86], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[(0.25 * N[Power[N[(M / N[(d / D), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1e-309], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(h * N[(N[Power[N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(1.0 + N[(h * N[(N[(-0.125 * N[Power[N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.00000000000000034e-86

   1. Initial program 86.5%

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

      1. times-frac 88.9%

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

      2. div-inv 88.9%

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

      3. metadata-eval 88.9%

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

      4. *-commutative 88.9%

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

      5. clear-num 88.9%

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

      6. un-div-inv 88.9%

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

   4. Applied egg-rr 88.9%

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

      1. metadata-eval 88.9%

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

      2. pow1/2 88.9%

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

      3. metadata-eval 88.9%

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

      4. pow1/2 88.9%

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

      5. associate-*l* 88.9%

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

      6. associate-*l* 88.9%

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

      7. metadata-eval 88.9%

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

   6. Applied egg-rr 88.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

   if -4.00000000000000034e-86 < d < -1.000000000000002e-309

   1. Initial program 51.1%

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

      1. associate-*r/ 51.4%

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

      2. clear-num 51.4%

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

      3. metadata-eval 51.4%

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

      4. associate-*l* 51.4%

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

   4. Applied egg-rr 51.4%

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

      1. *-commutative 51.4%

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

   6. Applied egg-rr 42.3%

      \[\leadsto \color{blue}{\left(1 - \frac{1}{\ell} \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 42.3%

         \[\leadsto \left(1 - \color{blue}{\frac{1 \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)}{\ell}}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      2. *-un-lft-identity 42.3%

         \[\leadsto \left(1 - \frac{\color{blue}{\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      3. *-commutative 42.3%

         \[\leadsto \left(1 - \frac{\color{blue}{h \cdot \left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right)}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      4. associate-*r* 42.3%

         \[\leadsto \left(1 - \frac{h \cdot \color{blue}{\left(\left(0.5 \cdot 0.25\right) \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      5. metadata-eval 42.3%

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

      6. associate-*r/ 42.3%

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

      7. *-commutative 42.3%

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

      8. associate-*r/ 42.3%

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

   8. Applied egg-rr 42.3%

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

   9. Taylor expanded in d around -inf 73.1%

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

       1. mul-1-neg 73.1%

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

       2. distribute-rgt-neg-in 73.1%

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

   11. Simplified 73.1%

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

   if -1.000000000000002e-309 < d

   1. Initial program 67.2%

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

      1. associate-*r/ 69.1%

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

      2. clear-num 69.1%

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

      3. metadata-eval 69.1%

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

      4. associate-*l* 69.1%

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

   4. Applied egg-rr 68.4%

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

      1. *-commutative 68.4%

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

   6. Applied egg-rr 59.8%

      \[\leadsto \color{blue}{\left(1 - \frac{1}{\ell} \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 59.8%

         \[\leadsto \left(1 - \color{blue}{\frac{1 \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)}{\ell}}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      2. *-un-lft-identity 59.8%

         \[\leadsto \left(1 - \frac{\color{blue}{\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      3. *-commutative 59.8%

         \[\leadsto \left(1 - \frac{\color{blue}{h \cdot \left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right)}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      4. associate-*r* 59.8%

         \[\leadsto \left(1 - \frac{h \cdot \color{blue}{\left(\left(0.5 \cdot 0.25\right) \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      5. metadata-eval 59.8%

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

      6. associate-*r/ 59.1%

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

      7. *-commutative 59.1%

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

      8. associate-*r/ 59.8%

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

   8. Applied egg-rr 59.8%

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

   10. Applied egg-rr 75.4%

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

       1. *-lft-identity 75.4%

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

       2. associate-*l/ 77.9%

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

       3. associate-/l* 77.1%

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

   12. Simplified 82.1%

       \[\leadsto \color{blue}{d \cdot \frac{1 + h \cdot \frac{-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}}{\sqrt{h \cdot \ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4 \cdot 10^{-86}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot \left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{h \cdot \left({\left(\frac{M \cdot D}{d}\right)}^{2} \cdot 0.125\right)}{\ell} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1 + h \cdot \frac{-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := {\left(D \cdot \frac{M}{d}\right)}^{2}\\ \mathbf{if}\;\ell \leq -9.5 \cdot 10^{+150}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot t\_0\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{h \cdot \left({\left(\frac{M \cdot D}{d}\right)}^{2} \cdot 0.125\right)}{\ell} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1 + h \cdot \frac{-0.125 \cdot t\_0}{\ell}}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (pow (* D (/ M d)) 2.0)))
   (if (<= l -9.5e+150)
     (* (sqrt (/ d l)) (* (sqrt (/ d h)) (+ 1.0 (* -0.125 (* (/ h l) t_0)))))
     (if (<= l -1e-309)
       (*
        (* d (sqrt (/ 1.0 (* l h))))
        (+ (/ (* h (* (pow (/ (* M D) d) 2.0) 0.125)) l) -1.0))
       (* d (/ (+ 1.0 (* h (/ (* -0.125 t_0) l))) (sqrt (* l h))))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = pow((D * (M / d)), 2.0);
	double tmp;
	if (l <= -9.5e+150) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + (-0.125 * ((h / l) * t_0))));
	} else if (l <= -1e-309) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (((h * (pow(((M * D) / d), 2.0) * 0.125)) / l) + -1.0);
	} else {
		tmp = d * ((1.0 + (h * ((-0.125 * t_0) / l))) / sqrt((l * h)));
	}
	return tmp;
}

Fortran

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (d_1 * (m / d)) ** 2.0d0
    if (l <= (-9.5d+150)) then
        tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0d0 + ((-0.125d0) * ((h / l) * t_0))))
    else if (l <= (-1d-309)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * (((h * ((((m * d_1) / d) ** 2.0d0) * 0.125d0)) / l) + (-1.0d0))
    else
        tmp = d * ((1.0d0 + (h * (((-0.125d0) * t_0) / l))) / sqrt((l * h)))
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.pow((D * (M / d)), 2.0);
	double tmp;
	if (l <= -9.5e+150) {
		tmp = Math.sqrt((d / l)) * (Math.sqrt((d / h)) * (1.0 + (-0.125 * ((h / l) * t_0))));
	} else if (l <= -1e-309) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (((h * (Math.pow(((M * D) / d), 2.0) * 0.125)) / l) + -1.0);
	} else {
		tmp = d * ((1.0 + (h * ((-0.125 * t_0) / l))) / Math.sqrt((l * h)));
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	t_0 = math.pow((D * (M / d)), 2.0)
	tmp = 0
	if l <= -9.5e+150:
		tmp = math.sqrt((d / l)) * (math.sqrt((d / h)) * (1.0 + (-0.125 * ((h / l) * t_0))))
	elif l <= -1e-309:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (((h * (math.pow(((M * D) / d), 2.0) * 0.125)) / l) + -1.0)
	else:
		tmp = d * ((1.0 + (h * ((-0.125 * t_0) / l))) / math.sqrt((l * h)))
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(D * Float64(M / d)) ^ 2.0
	tmp = 0.0
	if (l <= -9.5e+150)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(-0.125 * Float64(Float64(h / l) * t_0)))));
	elseif (l <= -1e-309)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(Float64(Float64(h * Float64((Float64(Float64(M * D) / d) ^ 2.0) * 0.125)) / l) + -1.0));
	else
		tmp = Float64(d * Float64(Float64(1.0 + Float64(h * Float64(Float64(-0.125 * t_0) / l))) / sqrt(Float64(l * h))));
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = (D * (M / d)) ^ 2.0;
	tmp = 0.0;
	if (l <= -9.5e+150)
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + (-0.125 * ((h / l) * t_0))));
	elseif (l <= -1e-309)
		tmp = (d * sqrt((1.0 / (l * h)))) * (((h * ((((M * D) / d) ^ 2.0) * 0.125)) / l) + -1.0);
	else
		tmp = d * ((1.0 + (h * ((-0.125 * t_0) / l))) / sqrt((l * h)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Power[N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[l, -9.5e+150], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.125 * N[(N[(h / l), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1e-309], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(h * N[(N[Power[N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(1.0 + N[(h * N[(N[(-0.125 * t$95$0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -9.5000000000000001e150

   1. Initial program 64.9%

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

      1. associate-*r/ 65.5%

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

      2. clear-num 65.5%

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

      3. metadata-eval 65.5%

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

      4. associate-*l* 65.5%

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

   4. Applied egg-rr 65.5%

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

      1. metadata-eval 65.5%

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

      2. pow1/2 65.4%

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

      3. metadata-eval 65.4%

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

      4. pow1/2 65.4%

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

      5. associate-*l* 65.4%

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

      6. associate-/r/ 65.4%

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

      7. metadata-eval 65.4%

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

      8. associate-*r* 65.4%

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

   6. Applied egg-rr 65.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{1}{\ell} \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)\right)\right)} \]

   8. Simplified 67.5%

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

   if -9.5000000000000001e150 < l < -1.000000000000002e-309

   1. Initial program 75.1%

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

      1. associate-*r/ 78.0%

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

      2. clear-num 78.0%

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

      3. metadata-eval 78.0%

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

      4. associate-*l* 78.0%

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

   4. Applied egg-rr 78.9%

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

      1. *-commutative 78.9%

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

   6. Applied egg-rr 70.2%

      \[\leadsto \color{blue}{\left(1 - \frac{1}{\ell} \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 70.2%

         \[\leadsto \left(1 - \color{blue}{\frac{1 \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)}{\ell}}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      2. *-un-lft-identity 70.2%

         \[\leadsto \left(1 - \frac{\color{blue}{\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      3. *-commutative 70.2%

         \[\leadsto \left(1 - \frac{\color{blue}{h \cdot \left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right)}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      4. associate-*r* 70.2%

         \[\leadsto \left(1 - \frac{h \cdot \color{blue}{\left(\left(0.5 \cdot 0.25\right) \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      5. metadata-eval 70.2%

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

      6. associate-*r/ 71.0%

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

      7. *-commutative 71.0%

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

      8. associate-*r/ 70.2%

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

   8. Applied egg-rr 70.2%

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

   9. Taylor expanded in d around -inf 88.0%

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

       1. mul-1-neg 88.0%

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

       2. distribute-rgt-neg-in 88.0%

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

   11. Simplified 88.0%

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

   if -1.000000000000002e-309 < l

   1. Initial program 67.2%

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

      1. associate-*r/ 69.1%

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

      2. clear-num 69.1%

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

      3. metadata-eval 69.1%

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

      4. associate-*l* 69.1%

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

   4. Applied egg-rr 68.4%

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

      1. *-commutative 68.4%

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

   6. Applied egg-rr 59.8%

      \[\leadsto \color{blue}{\left(1 - \frac{1}{\ell} \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 59.8%

         \[\leadsto \left(1 - \color{blue}{\frac{1 \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)}{\ell}}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      2. *-un-lft-identity 59.8%

         \[\leadsto \left(1 - \frac{\color{blue}{\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      3. *-commutative 59.8%

         \[\leadsto \left(1 - \frac{\color{blue}{h \cdot \left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right)}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      4. associate-*r* 59.8%

         \[\leadsto \left(1 - \frac{h \cdot \color{blue}{\left(\left(0.5 \cdot 0.25\right) \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      5. metadata-eval 59.8%

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

      6. associate-*r/ 59.1%

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

      7. *-commutative 59.1%

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

      8. associate-*r/ 59.8%

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

   8. Applied egg-rr 59.8%

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

   10. Applied egg-rr 75.4%

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

       1. *-lft-identity 75.4%

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

       2. associate-*l/ 77.9%

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

       3. associate-/l* 77.1%

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

   12. Simplified 82.1%

       \[\leadsto \color{blue}{d \cdot \frac{1 + h \cdot \frac{-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}}{\sqrt{h \cdot \ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9.5 \cdot 10^{+150}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{h \cdot \left({\left(\frac{M \cdot D}{d}\right)}^{2} \cdot 0.125\right)}{\ell} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1 + h \cdot \frac{-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := {\left(\frac{M \cdot D}{d}\right)}^{2}\\ \mathbf{if}\;\ell \leq -8.5 \cdot 10^{+147}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - t\_0 \cdot \left(\frac{h}{\ell} \cdot 0.125\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{h \cdot \left(t\_0 \cdot 0.125\right)}{\ell} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1 + h \cdot \frac{-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (pow (/ (* M D) d) 2.0)))
   (if (<= l -8.5e+147)
     (* (sqrt (/ d h)) (* (sqrt (/ d l)) (- 1.0 (* t_0 (* (/ h l) 0.125)))))
     (if (<= l -1e-309)
       (* (* d (sqrt (/ 1.0 (* l h)))) (+ (/ (* h (* t_0 0.125)) l) -1.0))
       (*
        d
        (/
         (+ 1.0 (* h (/ (* -0.125 (pow (* D (/ M d)) 2.0)) l)))
         (sqrt (* l h))))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = pow(((M * D) / d), 2.0);
	double tmp;
	if (l <= -8.5e+147) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (t_0 * ((h / l) * 0.125))));
	} else if (l <= -1e-309) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (((h * (t_0 * 0.125)) / l) + -1.0);
	} else {
		tmp = d * ((1.0 + (h * ((-0.125 * pow((D * (M / d)), 2.0)) / l))) / sqrt((l * h)));
	}
	return tmp;
}

Fortran

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((m * d_1) / d) ** 2.0d0
    if (l <= (-8.5d+147)) then
        tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 - (t_0 * ((h / l) * 0.125d0))))
    else if (l <= (-1d-309)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * (((h * (t_0 * 0.125d0)) / l) + (-1.0d0))
    else
        tmp = d * ((1.0d0 + (h * (((-0.125d0) * ((d_1 * (m / d)) ** 2.0d0)) / l))) / sqrt((l * h)))
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.pow(((M * D) / d), 2.0);
	double tmp;
	if (l <= -8.5e+147) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 - (t_0 * ((h / l) * 0.125))));
	} else if (l <= -1e-309) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (((h * (t_0 * 0.125)) / l) + -1.0);
	} else {
		tmp = d * ((1.0 + (h * ((-0.125 * Math.pow((D * (M / d)), 2.0)) / l))) / Math.sqrt((l * h)));
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	t_0 = math.pow(((M * D) / d), 2.0)
	tmp = 0
	if l <= -8.5e+147:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 - (t_0 * ((h / l) * 0.125))))
	elif l <= -1e-309:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (((h * (t_0 * 0.125)) / l) + -1.0)
	else:
		tmp = d * ((1.0 + (h * ((-0.125 * math.pow((D * (M / d)), 2.0)) / l))) / math.sqrt((l * h)))
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(M * D) / d) ^ 2.0
	tmp = 0.0
	if (l <= -8.5e+147)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(t_0 * Float64(Float64(h / l) * 0.125)))));
	elseif (l <= -1e-309)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(Float64(Float64(h * Float64(t_0 * 0.125)) / l) + -1.0));
	else
		tmp = Float64(d * Float64(Float64(1.0 + Float64(h * Float64(Float64(-0.125 * (Float64(D * Float64(M / d)) ^ 2.0)) / l))) / sqrt(Float64(l * h))));
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = ((M * D) / d) ^ 2.0;
	tmp = 0.0;
	if (l <= -8.5e+147)
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (t_0 * ((h / l) * 0.125))));
	elseif (l <= -1e-309)
		tmp = (d * sqrt((1.0 / (l * h)))) * (((h * (t_0 * 0.125)) / l) + -1.0);
	else
		tmp = d * ((1.0 + (h * ((-0.125 * ((D * (M / d)) ^ 2.0)) / l))) / sqrt((l * h)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Power[N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[l, -8.5e+147], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(t$95$0 * N[(N[(h / l), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1e-309], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(h * N[(t$95$0 * 0.125), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(1.0 + N[(h * N[(N[(-0.125 * N[Power[N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -8.5000000000000007e147

   1. Initial program 64.9%

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

      1. associate-*r/ 65.5%

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

      2. clear-num 65.5%

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

      3. metadata-eval 65.5%

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

      4. associate-*l* 65.5%

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

   4. Applied egg-rr 65.5%

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

      1. *-commutative 65.5%

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

   6. Applied egg-rr 57.4%

      \[\leadsto \color{blue}{\left(1 - \frac{1}{\ell} \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   8. Applied egg-rr 64.9%

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

   if -8.5000000000000007e147 < l < -1.000000000000002e-309

   1. Initial program 75.1%

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

      1. associate-*r/ 78.0%

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

      2. clear-num 78.0%

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

      3. metadata-eval 78.0%

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

      4. associate-*l* 78.0%

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

   4. Applied egg-rr 78.9%

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

      1. *-commutative 78.9%

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

   6. Applied egg-rr 70.2%

      \[\leadsto \color{blue}{\left(1 - \frac{1}{\ell} \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 70.2%

         \[\leadsto \left(1 - \color{blue}{\frac{1 \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)}{\ell}}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      2. *-un-lft-identity 70.2%

         \[\leadsto \left(1 - \frac{\color{blue}{\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      3. *-commutative 70.2%

         \[\leadsto \left(1 - \frac{\color{blue}{h \cdot \left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right)}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      4. associate-*r* 70.2%

         \[\leadsto \left(1 - \frac{h \cdot \color{blue}{\left(\left(0.5 \cdot 0.25\right) \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      5. metadata-eval 70.2%

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

      6. associate-*r/ 71.0%

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

      7. *-commutative 71.0%

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

      8. associate-*r/ 70.2%

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

   8. Applied egg-rr 70.2%

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

   9. Taylor expanded in d around -inf 88.0%

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

       1. mul-1-neg 88.0%

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

       2. distribute-rgt-neg-in 88.0%

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

   11. Simplified 88.0%

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

   if -1.000000000000002e-309 < l

   1. Initial program 67.2%

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

      1. associate-*r/ 69.1%

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

      2. clear-num 69.1%

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

      3. metadata-eval 69.1%

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

      4. associate-*l* 69.1%

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

   4. Applied egg-rr 68.4%

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

      1. *-commutative 68.4%

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

   6. Applied egg-rr 59.8%

      \[\leadsto \color{blue}{\left(1 - \frac{1}{\ell} \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 59.8%

         \[\leadsto \left(1 - \color{blue}{\frac{1 \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)}{\ell}}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      2. *-un-lft-identity 59.8%

         \[\leadsto \left(1 - \frac{\color{blue}{\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      3. *-commutative 59.8%

         \[\leadsto \left(1 - \frac{\color{blue}{h \cdot \left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right)}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      4. associate-*r* 59.8%

         \[\leadsto \left(1 - \frac{h \cdot \color{blue}{\left(\left(0.5 \cdot 0.25\right) \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      5. metadata-eval 59.8%

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

      6. associate-*r/ 59.1%

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

      7. *-commutative 59.1%

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

      8. associate-*r/ 59.8%

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

   8. Applied egg-rr 59.8%

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

   10. Applied egg-rr 75.4%

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

       1. *-lft-identity 75.4%

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

       2. associate-*l/ 77.9%

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

       3. associate-/l* 77.1%

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

   12. Simplified 82.1%

       \[\leadsto \color{blue}{d \cdot \frac{1 + h \cdot \frac{-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}}{\sqrt{h \cdot \ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8.5 \cdot 10^{+147}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - {\left(\frac{M \cdot D}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{h \cdot \left({\left(\frac{M \cdot D}{d}\right)}^{2} \cdot 0.125\right)}{\ell} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1 + h \cdot \frac{-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6 \cdot 10^{+259}:\\ \;\;\;\;\left(1 - \left(h \cdot \frac{0.125}{\ell}\right) \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{-2}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{h \cdot \left({\left(\frac{M \cdot D}{d}\right)}^{2} \cdot 0.125\right)}{\ell} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1 + h \cdot \frac{-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l -6e+259)
   (*
    (- 1.0 (* (* h (/ 0.125 l)) (pow (/ (/ d D) M) -2.0)))
    (sqrt (* (/ d l) (/ d h))))
   (if (<= l -1e-309)
     (*
      (* d (sqrt (/ 1.0 (* l h))))
      (+ (/ (* h (* (pow (/ (* M D) d) 2.0) 0.125)) l) -1.0))
     (*
      d
      (/
       (+ 1.0 (* h (/ (* -0.125 (pow (* D (/ M d)) 2.0)) l)))
       (sqrt (* l h)))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -6e+259) {
		tmp = (1.0 - ((h * (0.125 / l)) * pow(((d / D) / M), -2.0))) * sqrt(((d / l) * (d / h)));
	} else if (l <= -1e-309) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (((h * (pow(((M * D) / d), 2.0) * 0.125)) / l) + -1.0);
	} else {
		tmp = d * ((1.0 + (h * ((-0.125 * pow((D * (M / d)), 2.0)) / l))) / sqrt((l * h)));
	}
	return tmp;
}

Fortran

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-6d+259)) then
        tmp = (1.0d0 - ((h * (0.125d0 / l)) * (((d / d_1) / m) ** (-2.0d0)))) * sqrt(((d / l) * (d / h)))
    else if (l <= (-1d-309)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * (((h * ((((m * d_1) / d) ** 2.0d0) * 0.125d0)) / l) + (-1.0d0))
    else
        tmp = d * ((1.0d0 + (h * (((-0.125d0) * ((d_1 * (m / d)) ** 2.0d0)) / l))) / sqrt((l * h)))
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -6e+259) {
		tmp = (1.0 - ((h * (0.125 / l)) * Math.pow(((d / D) / M), -2.0))) * Math.sqrt(((d / l) * (d / h)));
	} else if (l <= -1e-309) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (((h * (Math.pow(((M * D) / d), 2.0) * 0.125)) / l) + -1.0);
	} else {
		tmp = d * ((1.0 + (h * ((-0.125 * Math.pow((D * (M / d)), 2.0)) / l))) / Math.sqrt((l * h)));
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= -6e+259:
		tmp = (1.0 - ((h * (0.125 / l)) * math.pow(((d / D) / M), -2.0))) * math.sqrt(((d / l) * (d / h)))
	elif l <= -1e-309:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (((h * (math.pow(((M * D) / d), 2.0) * 0.125)) / l) + -1.0)
	else:
		tmp = d * ((1.0 + (h * ((-0.125 * math.pow((D * (M / d)), 2.0)) / l))) / math.sqrt((l * h)))
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -6e+259)
		tmp = Float64(Float64(1.0 - Float64(Float64(h * Float64(0.125 / l)) * (Float64(Float64(d / D) / M) ^ -2.0))) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	elseif (l <= -1e-309)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(Float64(Float64(h * Float64((Float64(Float64(M * D) / d) ^ 2.0) * 0.125)) / l) + -1.0));
	else
		tmp = Float64(d * Float64(Float64(1.0 + Float64(h * Float64(Float64(-0.125 * (Float64(D * Float64(M / d)) ^ 2.0)) / l))) / sqrt(Float64(l * h))));
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -6e+259)
		tmp = (1.0 - ((h * (0.125 / l)) * (((d / D) / M) ^ -2.0))) * sqrt(((d / l) * (d / h)));
	elseif (l <= -1e-309)
		tmp = (d * sqrt((1.0 / (l * h)))) * (((h * ((((M * D) / d) ^ 2.0) * 0.125)) / l) + -1.0);
	else
		tmp = d * ((1.0 + (h * ((-0.125 * ((D * (M / d)) ^ 2.0)) / l))) / sqrt((l * h)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, -6e+259], N[(N[(1.0 - N[(N[(h * N[(0.125 / l), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(d / D), $MachinePrecision] / M), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1e-309], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(h * N[(N[Power[N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(1.0 + N[(h * N[(N[(-0.125 * N[Power[N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.00000000000000026e259

   1. Initial program 67.0%

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

      1. associate-*r/ 67.0%

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

      2. clear-num 67.0%

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

      3. metadata-eval 67.0%

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

      4. associate-*l* 67.0%

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

   4. Applied egg-rr 67.0%

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

      1. *-commutative 67.0%

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

   6. Applied egg-rr 55.8%

      \[\leadsto \color{blue}{\left(1 - \frac{1}{\ell} \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   7. Taylor expanded in l around 0 22.0%

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

      1. associate-*r* 22.0%

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

      2. times-frac 22.1%

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

      3. associate-/l* 33.2%

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

      4. unpow2 33.2%

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

      5. unpow2 33.2%

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

      6. unpow2 33.2%

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

      7. times-frac 55.4%

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

      8. swap-sqr 66.5%

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

      9. associate-*r/ 55.8%

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

      10. associate-*r/ 55.8%

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

      11. unpow2 55.8%

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

      12. *-commutative 55.8%

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

      13. associate-*l* 55.8%

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

      14. *-commutative 55.8%

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

      15. associate-*l/ 55.8%

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

      16. associate-/l* 55.8%

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

      17. unpow2 55.8%

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

   9. Simplified 66.5%

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

   if -6.00000000000000026e259 < l < -1.000000000000002e-309

   1. Initial program 72.7%

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

      1. associate-*r/ 75.2%

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

      2. clear-num 75.3%

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

      3. metadata-eval 75.3%

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

      4. associate-*l* 75.3%

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

   4. Applied egg-rr 75.9%

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

      1. *-commutative 75.9%

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

   6. Applied egg-rr 67.6%

      \[\leadsto \color{blue}{\left(1 - \frac{1}{\ell} \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 67.6%

         \[\leadsto \left(1 - \color{blue}{\frac{1 \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)}{\ell}}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      2. *-un-lft-identity 67.6%

         \[\leadsto \left(1 - \frac{\color{blue}{\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      3. *-commutative 67.6%

         \[\leadsto \left(1 - \frac{\color{blue}{h \cdot \left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right)}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      4. associate-*r* 67.6%

         \[\leadsto \left(1 - \frac{h \cdot \color{blue}{\left(\left(0.5 \cdot 0.25\right) \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      5. metadata-eval 67.6%

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

      6. associate-*r/ 68.2%

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

      7. *-commutative 68.2%

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

      8. associate-*r/ 67.6%

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

   8. Applied egg-rr 67.6%

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

   9. Taylor expanded in d around -inf 81.4%

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

       1. mul-1-neg 81.4%

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

       2. distribute-rgt-neg-in 81.4%

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

   11. Simplified 81.4%

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

   if -1.000000000000002e-309 < l

   1. Initial program 67.2%

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

      1. associate-*r/ 69.1%

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

      2. clear-num 69.1%

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

      3. metadata-eval 69.1%

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

      4. associate-*l* 69.1%

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

   4. Applied egg-rr 68.4%

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

      1. *-commutative 68.4%

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

   6. Applied egg-rr 59.8%

      \[\leadsto \color{blue}{\left(1 - \frac{1}{\ell} \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 59.8%

         \[\leadsto \left(1 - \color{blue}{\frac{1 \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)}{\ell}}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      2. *-un-lft-identity 59.8%

         \[\leadsto \left(1 - \frac{\color{blue}{\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      3. *-commutative 59.8%

         \[\leadsto \left(1 - \frac{\color{blue}{h \cdot \left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right)}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      4. associate-*r* 59.8%

         \[\leadsto \left(1 - \frac{h \cdot \color{blue}{\left(\left(0.5 \cdot 0.25\right) \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      5. metadata-eval 59.8%

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

      6. associate-*r/ 59.1%

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

      7. *-commutative 59.1%

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

      8. associate-*r/ 59.8%

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

   8. Applied egg-rr 59.8%

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

   10. Applied egg-rr 75.4%

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

       1. *-lft-identity 75.4%

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

       2. associate-*l/ 77.9%

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

       3. associate-/l* 77.1%

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

   12. Simplified 82.1%

       \[\leadsto \color{blue}{d \cdot \frac{1 + h \cdot \frac{-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}}{\sqrt{h \cdot \ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6 \cdot 10^{+259}:\\ \;\;\;\;\left(1 - \left(h \cdot \frac{0.125}{\ell}\right) \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{-2}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{h \cdot \left({\left(\frac{M \cdot D}{d}\right)}^{2} \cdot 0.125\right)}{\ell} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1 + h \cdot \frac{-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;\ell \leq -1.5 \cdot 10^{-226}:\\ \;\;\;\;{\left(\frac{d}{h}\right)}^{0.5} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{d}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1 + h \cdot \frac{-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (* l h))))
   (if (<= l -1.5e-226)
     (* (pow (/ d h) 0.5) (sqrt (/ d l)))
     (if (<= l -1e-309)
       (/ d t_0)
       (* d (/ (+ 1.0 (* h (/ (* -0.125 (pow (* D (/ M d)) 2.0)) l))) t_0))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((l * h));
	double tmp;
	if (l <= -1.5e-226) {
		tmp = pow((d / h), 0.5) * sqrt((d / l));
	} else if (l <= -1e-309) {
		tmp = d / t_0;
	} else {
		tmp = d * ((1.0 + (h * ((-0.125 * pow((D * (M / d)), 2.0)) / l))) / t_0);
	}
	return tmp;
}

Fortran

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((l * h))
    if (l <= (-1.5d-226)) then
        tmp = ((d / h) ** 0.5d0) * sqrt((d / l))
    else if (l <= (-1d-309)) then
        tmp = d / t_0
    else
        tmp = d * ((1.0d0 + (h * (((-0.125d0) * ((d_1 * (m / d)) ** 2.0d0)) / l))) / t_0)
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((l * h));
	double tmp;
	if (l <= -1.5e-226) {
		tmp = Math.pow((d / h), 0.5) * Math.sqrt((d / l));
	} else if (l <= -1e-309) {
		tmp = d / t_0;
	} else {
		tmp = d * ((1.0 + (h * ((-0.125 * Math.pow((D * (M / d)), 2.0)) / l))) / t_0);
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	t_0 = math.sqrt((l * h))
	tmp = 0
	if l <= -1.5e-226:
		tmp = math.pow((d / h), 0.5) * math.sqrt((d / l))
	elif l <= -1e-309:
		tmp = d / t_0
	else:
		tmp = d * ((1.0 + (h * ((-0.125 * math.pow((D * (M / d)), 2.0)) / l))) / t_0)
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(l * h))
	tmp = 0.0
	if (l <= -1.5e-226)
		tmp = Float64((Float64(d / h) ^ 0.5) * sqrt(Float64(d / l)));
	elseif (l <= -1e-309)
		tmp = Float64(d / t_0);
	else
		tmp = Float64(d * Float64(Float64(1.0 + Float64(h * Float64(Float64(-0.125 * (Float64(D * Float64(M / d)) ^ 2.0)) / l))) / t_0));
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((l * h));
	tmp = 0.0;
	if (l <= -1.5e-226)
		tmp = ((d / h) ^ 0.5) * sqrt((d / l));
	elseif (l <= -1e-309)
		tmp = d / t_0;
	else
		tmp = d * ((1.0 + (h * ((-0.125 * ((D * (M / d)) ^ 2.0)) / l))) / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1.5e-226], N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1e-309], N[(d / t$95$0), $MachinePrecision], N[(d * N[(N[(1.0 + N[(h * N[(N[(-0.125 * N[Power[N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.49999999999999998e-226

   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 71.2%

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

   5. Taylor expanded in M around 0 47.5%

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

      1. pow1/2 47.5%

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

   7. Applied egg-rr 47.5%

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

   if -1.49999999999999998e-226 < l < -1.000000000000002e-309

   1. Initial program 89.3%

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

   3. Simplified 89.3%

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

   5. Taylor expanded in M around 0 21.4%

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

      1. pow1/2 21.4%

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

      2. metadata-eval 21.4%

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

      3. *-rgt-identity 21.4%

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

      4. pow1/2 21.4%

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

      5. metadata-eval 21.4%

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

      6. pow-prod-down 21.4%

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

      7. metadata-eval 21.4%

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

      8. unpow1/2 21.4%

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

   7. Applied egg-rr 21.4%

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

      1. associate-*l/ 21.4%

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

   9. Applied egg-rr 21.4%

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

       1. associate-*l/ 21.4%

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

       2. frac-times 21.3%

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

       3. unpow2 21.3%

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

       4. sqrt-div 21.3%

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

       5. sqrt-pow1 58.5%

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

       6. metadata-eval 58.5%

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

       7. pow1 58.5%

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

       8. *-commutative 58.5%

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

   11. Applied egg-rr 58.5%

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

   if -1.000000000000002e-309 < l

   1. Initial program 67.2%

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

      1. associate-*r/ 69.1%

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

      2. clear-num 69.1%

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

      3. metadata-eval 69.1%

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

      4. associate-*l* 69.1%

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

   4. Applied egg-rr 68.4%

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

      1. *-commutative 68.4%

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

   6. Applied egg-rr 59.8%

      \[\leadsto \color{blue}{\left(1 - \frac{1}{\ell} \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 59.8%

         \[\leadsto \left(1 - \color{blue}{\frac{1 \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)}{\ell}}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      2. *-un-lft-identity 59.8%

         \[\leadsto \left(1 - \frac{\color{blue}{\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      3. *-commutative 59.8%

         \[\leadsto \left(1 - \frac{\color{blue}{h \cdot \left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right)}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      4. associate-*r* 59.8%

         \[\leadsto \left(1 - \frac{h \cdot \color{blue}{\left(\left(0.5 \cdot 0.25\right) \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      5. metadata-eval 59.8%

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

      6. associate-*r/ 59.1%

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

      7. *-commutative 59.1%

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

      8. associate-*r/ 59.8%

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

   8. Applied egg-rr 59.8%

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

   10. Applied egg-rr 75.4%

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

       1. *-lft-identity 75.4%

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

       2. associate-*l/ 77.9%

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

       3. associate-/l* 77.1%

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

   12. Simplified 82.1%

       \[\leadsto \color{blue}{d \cdot \frac{1 + h \cdot \frac{-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}}{\sqrt{h \cdot \ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.5 \cdot 10^{-226}:\\ \;\;\;\;{\left(\frac{d}{h}\right)}^{0.5} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1 + h \cdot \frac{-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq 5.6 \cdot 10^{-146}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left({\left(\frac{M \cdot D}{d}\right)}^{2} \cdot 0.125\right)}{\ell}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1 + h \cdot \frac{-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= h 5.6e-146)
   (*
    (- 1.0 (/ (* h (* (pow (/ (* M D) d) 2.0) 0.125)) l))
    (sqrt (* (/ d l) (/ d h))))
   (*
    d
    (/
     (+ 1.0 (* h (/ (* -0.125 (pow (* D (/ M d)) 2.0)) l)))
     (sqrt (* l h))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= 5.6e-146) {
		tmp = (1.0 - ((h * (pow(((M * D) / d), 2.0) * 0.125)) / l)) * sqrt(((d / l) * (d / h)));
	} else {
		tmp = d * ((1.0 + (h * ((-0.125 * pow((D * (M / d)), 2.0)) / l))) / sqrt((l * h)));
	}
	return tmp;
}

Fortran

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (h <= 5.6d-146) then
        tmp = (1.0d0 - ((h * ((((m * d_1) / d) ** 2.0d0) * 0.125d0)) / l)) * sqrt(((d / l) * (d / h)))
    else
        tmp = d * ((1.0d0 + (h * (((-0.125d0) * ((d_1 * (m / d)) ** 2.0d0)) / l))) / sqrt((l * h)))
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= 5.6e-146) {
		tmp = (1.0 - ((h * (Math.pow(((M * D) / d), 2.0) * 0.125)) / l)) * Math.sqrt(((d / l) * (d / h)));
	} else {
		tmp = d * ((1.0 + (h * ((-0.125 * Math.pow((D * (M / d)), 2.0)) / l))) / Math.sqrt((l * h)));
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	tmp = 0
	if h <= 5.6e-146:
		tmp = (1.0 - ((h * (math.pow(((M * D) / d), 2.0) * 0.125)) / l)) * math.sqrt(((d / l) * (d / h)))
	else:
		tmp = d * ((1.0 + (h * ((-0.125 * math.pow((D * (M / d)), 2.0)) / l))) / math.sqrt((l * h)))
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= 5.6e-146)
		tmp = Float64(Float64(1.0 - Float64(Float64(h * Float64((Float64(Float64(M * D) / d) ^ 2.0) * 0.125)) / l)) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	else
		tmp = Float64(d * Float64(Float64(1.0 + Float64(h * Float64(Float64(-0.125 * (Float64(D * Float64(M / d)) ^ 2.0)) / l))) / sqrt(Float64(l * h))));
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= 5.6e-146)
		tmp = (1.0 - ((h * ((((M * D) / d) ^ 2.0) * 0.125)) / l)) * sqrt(((d / l) * (d / h)));
	else
		tmp = d * ((1.0 + (h * ((-0.125 * ((D * (M / d)) ^ 2.0)) / l))) / sqrt((l * h)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[h, 5.6e-146], N[(N[(1.0 - N[(N[(h * N[(N[Power[N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(1.0 + N[(h * N[(N[(-0.125 * N[Power[N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 5.60000000000000006e-146

   1. Initial program 72.8%

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

      1. associate-*r/ 75.3%

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

      2. clear-num 75.3%

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

      3. metadata-eval 75.3%

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

      4. associate-*l* 75.3%

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

   4. Applied egg-rr 75.2%

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

      1. *-commutative 75.2%

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

   6. Applied egg-rr 69.0%

      \[\leadsto \color{blue}{\left(1 - \frac{1}{\ell} \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 69.0%

         \[\leadsto \left(1 - \color{blue}{\frac{1 \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)}{\ell}}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      2. *-un-lft-identity 69.0%

         \[\leadsto \left(1 - \frac{\color{blue}{\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      3. *-commutative 69.0%

         \[\leadsto \left(1 - \frac{\color{blue}{h \cdot \left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right)}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      4. associate-*r* 69.0%

         \[\leadsto \left(1 - \frac{h \cdot \color{blue}{\left(\left(0.5 \cdot 0.25\right) \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      5. metadata-eval 69.0%

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

      6. associate-*r/ 68.9%

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

      7. *-commutative 68.9%

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

      8. associate-*r/ 69.0%

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

   8. Applied egg-rr 69.0%

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

   if 5.60000000000000006e-146 < h

   1. Initial program 64.4%

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

      1. associate-*r/ 65.8%

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

      2. clear-num 65.8%

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

      3. metadata-eval 65.8%

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

      4. associate-*l* 65.8%

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

   4. Applied egg-rr 65.8%

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

      1. *-commutative 65.8%

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

   6. Applied egg-rr 52.8%

      \[\leadsto \color{blue}{\left(1 - \frac{1}{\ell} \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 52.8%

         \[\leadsto \left(1 - \color{blue}{\frac{1 \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)}{\ell}}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      2. *-un-lft-identity 52.8%

         \[\leadsto \left(1 - \frac{\color{blue}{\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      3. *-commutative 52.8%

         \[\leadsto \left(1 - \frac{\color{blue}{h \cdot \left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right)}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      4. associate-*r* 52.8%

         \[\leadsto \left(1 - \frac{h \cdot \color{blue}{\left(\left(0.5 \cdot 0.25\right) \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      5. metadata-eval 52.8%

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

      6. associate-*r/ 52.8%

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

      7. *-commutative 52.8%

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

      8. associate-*r/ 52.8%

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

   8. Applied egg-rr 52.8%

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

   10. Applied egg-rr 76.3%

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

       1. *-lft-identity 76.3%

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

       2. associate-*l/ 79.7%

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

       3. associate-/l* 78.6%

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

   12. Simplified 84.4%

       \[\leadsto \color{blue}{d \cdot \frac{1 + h \cdot \frac{-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}}{\sqrt{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 5.6 \cdot 10^{-146}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left({\left(\frac{M \cdot D}{d}\right)}^{2} \cdot 0.125\right)}{\ell}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1 + h \cdot \frac{-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 65.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1.75 \cdot 10^{-289}:\\ \;\;\;\;\left(1 - \left(h \cdot \frac{0.125}{\ell}\right) \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{-2}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1 + h \cdot \frac{-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= d -1.75e-289)
   (*
    (- 1.0 (* (* h (/ 0.125 l)) (pow (/ (/ d D) M) -2.0)))
    (sqrt (* (/ d l) (/ d h))))
   (*
    d
    (/
     (+ 1.0 (* h (/ (* -0.125 (pow (* D (/ M d)) 2.0)) l)))
     (sqrt (* l h))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -1.75e-289) {
		tmp = (1.0 - ((h * (0.125 / l)) * pow(((d / D) / M), -2.0))) * sqrt(((d / l) * (d / h)));
	} else {
		tmp = d * ((1.0 + (h * ((-0.125 * pow((D * (M / d)), 2.0)) / l))) / sqrt((l * h)));
	}
	return tmp;
}

Fortran

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= (-1.75d-289)) then
        tmp = (1.0d0 - ((h * (0.125d0 / l)) * (((d / d_1) / m) ** (-2.0d0)))) * sqrt(((d / l) * (d / h)))
    else
        tmp = d * ((1.0d0 + (h * (((-0.125d0) * ((d_1 * (m / d)) ** 2.0d0)) / l))) / sqrt((l * h)))
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -1.75e-289) {
		tmp = (1.0 - ((h * (0.125 / l)) * Math.pow(((d / D) / M), -2.0))) * Math.sqrt(((d / l) * (d / h)));
	} else {
		tmp = d * ((1.0 + (h * ((-0.125 * Math.pow((D * (M / d)), 2.0)) / l))) / Math.sqrt((l * h)));
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	tmp = 0
	if d <= -1.75e-289:
		tmp = (1.0 - ((h * (0.125 / l)) * math.pow(((d / D) / M), -2.0))) * math.sqrt(((d / l) * (d / h)))
	else:
		tmp = d * ((1.0 + (h * ((-0.125 * math.pow((D * (M / d)), 2.0)) / l))) / math.sqrt((l * h)))
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -1.75e-289)
		tmp = Float64(Float64(1.0 - Float64(Float64(h * Float64(0.125 / l)) * (Float64(Float64(d / D) / M) ^ -2.0))) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	else
		tmp = Float64(d * Float64(Float64(1.0 + Float64(h * Float64(Float64(-0.125 * (Float64(D * Float64(M / d)) ^ 2.0)) / l))) / sqrt(Float64(l * h))));
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -1.75e-289)
		tmp = (1.0 - ((h * (0.125 / l)) * (((d / D) / M) ^ -2.0))) * sqrt(((d / l) * (d / h)));
	else
		tmp = d * ((1.0 + (h * ((-0.125 * ((D * (M / d)) ^ 2.0)) / l))) / sqrt((l * h)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[d, -1.75e-289], N[(N[(1.0 - N[(N[(h * N[(0.125 / l), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(d / D), $MachinePrecision] / M), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(1.0 + N[(h * N[(N[(-0.125 * N[Power[N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.75e-289

   1. Initial program 76.9%

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

      1. associate-*r/ 79.3%

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

      2. clear-num 79.3%

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

      3. metadata-eval 79.3%

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

      4. associate-*l* 79.3%

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

   4. Applied egg-rr 80.0%

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

      1. *-commutative 80.0%

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

   6. Applied egg-rr 70.9%

      \[\leadsto \color{blue}{\left(1 - \frac{1}{\ell} \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   7. Taylor expanded in l around 0 38.3%

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

      1. associate-*r* 41.5%

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

      2. times-frac 42.9%

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

      3. associate-/l* 43.7%

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

      4. unpow2 43.7%

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

      5. unpow2 43.7%

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

      6. unpow2 43.7%

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

      7. times-frac 54.3%

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

      8. swap-sqr 69.1%

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

      9. associate-*r/ 67.6%

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

      10. associate-*r/ 67.7%

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

      11. unpow2 67.7%

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

      12. *-commutative 67.7%

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

      13. associate-*l* 67.7%

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

      14. *-commutative 67.7%

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

      15. associate-*l/ 68.5%

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

      16. associate-/l* 68.5%

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

      17. unpow2 68.5%

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

   9. Simplified 70.1%

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

   if -1.75e-289 < d

   1. Initial program 63.1%

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

      1. associate-*r/ 64.9%

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

      2. clear-num 64.9%

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

      3. metadata-eval 64.9%

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

      4. associate-*l* 64.9%

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

   4. Applied egg-rr 64.2%

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

      1. *-commutative 64.2%

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

   6. Applied egg-rr 56.2%

      \[\leadsto \color{blue}{\left(1 - \frac{1}{\ell} \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 56.2%

         \[\leadsto \left(1 - \color{blue}{\frac{1 \cdot \left(\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h\right)}{\ell}}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      2. *-un-lft-identity 56.2%

         \[\leadsto \left(1 - \frac{\color{blue}{\left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot h}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      3. *-commutative 56.2%

         \[\leadsto \left(1 - \frac{\color{blue}{h \cdot \left(0.5 \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right)}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      4. associate-*r* 56.2%

         \[\leadsto \left(1 - \frac{h \cdot \color{blue}{\left(\left(0.5 \cdot 0.25\right) \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      5. metadata-eval 56.2%

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

      6. associate-*r/ 55.5%

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

      7. *-commutative 55.5%

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

      8. associate-*r/ 56.2%

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

   8. Applied egg-rr 56.2%

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

   10. Applied egg-rr 70.8%

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

       1. *-lft-identity 70.8%

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

       2. associate-*l/ 73.1%

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

       3. associate-/l* 72.3%

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

   12. Simplified 77.1%

       \[\leadsto \color{blue}{d \cdot \frac{1 + h \cdot \frac{-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}}{\sqrt{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.75 \cdot 10^{-289}:\\ \;\;\;\;\left(1 - \left(h \cdot \frac{0.125}{\ell}\right) \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{-2}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1 + h \cdot \frac{-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 44.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.1 \cdot 10^{+155}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -2.5 \cdot 10^{-229}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{-298}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\ell \cdot h}}{d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l -2.1e+155)
   (sqrt (* (/ d l) (/ d h)))
   (if (<= l -2.5e-229)
     (* d (- (sqrt (/ 1.0 (* l h)))))
     (if (<= l 3.8e-298)
       (/ 1.0 (/ (sqrt (* l h)) d))
       (/ (/ d (sqrt l)) (sqrt h))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -2.1e+155) {
		tmp = sqrt(((d / l) * (d / h)));
	} else if (l <= -2.5e-229) {
		tmp = d * -sqrt((1.0 / (l * h)));
	} else if (l <= 3.8e-298) {
		tmp = 1.0 / (sqrt((l * h)) / d);
	} else {
		tmp = (d / sqrt(l)) / sqrt(h);
	}
	return tmp;
}

Fortran

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-2.1d+155)) then
        tmp = sqrt(((d / l) * (d / h)))
    else if (l <= (-2.5d-229)) then
        tmp = d * -sqrt((1.0d0 / (l * h)))
    else if (l <= 3.8d-298) then
        tmp = 1.0d0 / (sqrt((l * h)) / d)
    else
        tmp = (d / sqrt(l)) / sqrt(h)
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -2.1e+155) {
		tmp = Math.sqrt(((d / l) * (d / h)));
	} else if (l <= -2.5e-229) {
		tmp = d * -Math.sqrt((1.0 / (l * h)));
	} else if (l <= 3.8e-298) {
		tmp = 1.0 / (Math.sqrt((l * h)) / d);
	} else {
		tmp = (d / Math.sqrt(l)) / Math.sqrt(h);
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= -2.1e+155:
		tmp = math.sqrt(((d / l) * (d / h)))
	elif l <= -2.5e-229:
		tmp = d * -math.sqrt((1.0 / (l * h)))
	elif l <= 3.8e-298:
		tmp = 1.0 / (math.sqrt((l * h)) / d)
	else:
		tmp = (d / math.sqrt(l)) / math.sqrt(h)
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -2.1e+155)
		tmp = sqrt(Float64(Float64(d / l) * Float64(d / h)));
	elseif (l <= -2.5e-229)
		tmp = Float64(d * Float64(-sqrt(Float64(1.0 / Float64(l * h)))));
	elseif (l <= 3.8e-298)
		tmp = Float64(1.0 / Float64(sqrt(Float64(l * h)) / d));
	else
		tmp = Float64(Float64(d / sqrt(l)) / sqrt(h));
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -2.1e+155)
		tmp = sqrt(((d / l) * (d / h)));
	elseif (l <= -2.5e-229)
		tmp = d * -sqrt((1.0 / (l * h)));
	elseif (l <= 3.8e-298)
		tmp = 1.0 / (sqrt((l * h)) / d);
	else
		tmp = (d / sqrt(l)) / sqrt(h);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, -2.1e+155], N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, -2.5e-229], N[(d * (-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, 3.8e-298], N[(1.0 / N[(N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.1e155

   1. Initial program 62.8%

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

   3. Simplified 65.6%

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

   5. Taylor expanded in M around 0 48.9%

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

      1. pow1/2 49.0%

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

      2. metadata-eval 49.0%

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

      3. *-rgt-identity 49.0%

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

      4. pow1/2 49.0%

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

      5. metadata-eval 49.0%

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

      6. pow-prod-down 46.4%

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

      7. metadata-eval 46.4%

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

      8. unpow1/2 46.4%

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

   7. Applied egg-rr 46.4%

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

   if -2.1e155 < l < -2.50000000000000008e-229

   1. Initial program 71.5%

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

      1. associate-*r/ 75.0%

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

      2. clear-num 75.1%

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

      3. metadata-eval 75.1%

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

      4. associate-*l* 75.1%

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

   4. Applied egg-rr 76.0%

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

   5. Taylor expanded in l around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. 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. *-commutative 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 48.7%

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

      6. neg-mul-1 48.7%

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

   7. Simplified 48.7%

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

   if -2.50000000000000008e-229 < l < 3.8e-298

   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 89.9%

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

   5. Taylor expanded in M around 0 25.2%

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

      1. pow1/2 25.2%

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

      2. metadata-eval 25.2%

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

      3. *-rgt-identity 25.2%

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

      4. pow1/2 25.2%

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

      5. metadata-eval 25.2%

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

      6. pow-prod-down 25.2%

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

      7. metadata-eval 25.2%

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

      8. unpow1/2 25.2%

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

   7. Applied egg-rr 25.2%

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

      1. associate-*l/ 25.2%

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

   9. Applied egg-rr 25.2%

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

       1. associate-*l/ 25.2%

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

       2. frac-times 25.1%

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

       3. unpow2 25.1%

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

       4. sqrt-div 25.1%

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

       5. sqrt-pow1 60.5%

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

       6. metadata-eval 60.5%

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

       7. pow1 60.5%

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

       8. clear-num 60.5%

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

       9. *-commutative 60.5%

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

   11. Applied egg-rr 60.5%

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

   if 3.8e-298 < l

   1. Initial program 67.5%

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

   3. Simplified 66.7%

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

   5. Taylor expanded in M around 0 37.5%

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

      1. pow1/2 37.5%

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

      2. metadata-eval 37.5%

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

      3. *-rgt-identity 37.5%

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

      4. pow1/2 37.5%

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

      5. metadata-eval 37.5%

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

      6. pow-prod-down 30.5%

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

      7. metadata-eval 30.5%

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

      8. unpow1/2 30.5%

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

   7. Applied egg-rr 30.5%

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

      1. associate-*l/ 25.7%

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

   9. Applied egg-rr 25.7%

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

       1. sqrt-div 30.1%

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

       2. clear-num 30.1%

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

       3. associate-*r/ 23.7%

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

       4. unpow2 23.7%

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

       5. sqrt-div 32.7%

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

       6. sqrt-pow1 46.2%

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

       7. metadata-eval 46.2%

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

       8. pow1 46.2%

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

   11. Applied egg-rr 46.2%

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

       1. associate-/r/ 46.9%

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

       2. times-frac 47.8%

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

       3. *-lft-identity 47.8%

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

       4. associate-/l/ 47.0%

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

   13. Simplified 47.0%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.1 \cdot 10^{+155}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -2.5 \cdot 10^{-229}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{-298}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\ell \cdot h}}{d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 44.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -9 \cdot 10^{-227}:\\ \;\;\;\;{\left(\frac{d}{h}\right)}^{0.5} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l -9e-227)
   (* (pow (/ d h) 0.5) (sqrt (/ d l)))
   (if (<= l -1e-309) (/ d (sqrt (* l h))) (/ (/ d (sqrt l)) (sqrt h)))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -9e-227) {
		tmp = pow((d / h), 0.5) * sqrt((d / l));
	} else if (l <= -1e-309) {
		tmp = d / sqrt((l * h));
	} else {
		tmp = (d / sqrt(l)) / sqrt(h);
	}
	return tmp;
}

Fortran

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-9d-227)) then
        tmp = ((d / h) ** 0.5d0) * sqrt((d / l))
    else if (l <= (-1d-309)) then
        tmp = d / sqrt((l * h))
    else
        tmp = (d / sqrt(l)) / sqrt(h)
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -9e-227) {
		tmp = Math.pow((d / h), 0.5) * Math.sqrt((d / l));
	} else if (l <= -1e-309) {
		tmp = d / Math.sqrt((l * h));
	} else {
		tmp = (d / Math.sqrt(l)) / Math.sqrt(h);
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= -9e-227:
		tmp = math.pow((d / h), 0.5) * math.sqrt((d / l))
	elif l <= -1e-309:
		tmp = d / math.sqrt((l * h))
	else:
		tmp = (d / math.sqrt(l)) / math.sqrt(h)
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -9e-227)
		tmp = Float64((Float64(d / h) ^ 0.5) * sqrt(Float64(d / l)));
	elseif (l <= -1e-309)
		tmp = Float64(d / sqrt(Float64(l * h)));
	else
		tmp = Float64(Float64(d / sqrt(l)) / sqrt(h));
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -9e-227)
		tmp = ((d / h) ^ 0.5) * sqrt((d / l));
	elseif (l <= -1e-309)
		tmp = d / sqrt((l * h));
	else
		tmp = (d / sqrt(l)) / sqrt(h);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, -9e-227], N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1e-309], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -8.99999999999999986e-227

   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 71.2%

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

   5. Taylor expanded in M around 0 47.5%

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

      1. pow1/2 47.5%

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

   7. Applied egg-rr 47.5%

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

   if -8.99999999999999986e-227 < l < -1.000000000000002e-309

   1. Initial program 89.3%

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

   3. Simplified 89.3%

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

   5. Taylor expanded in M around 0 21.4%

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

      1. pow1/2 21.4%

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

      2. metadata-eval 21.4%

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

      3. *-rgt-identity 21.4%

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

      4. pow1/2 21.4%

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

      5. metadata-eval 21.4%

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

      6. pow-prod-down 21.4%

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

      7. metadata-eval 21.4%

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

      8. unpow1/2 21.4%

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

   7. Applied egg-rr 21.4%

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

      1. associate-*l/ 21.4%

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

   9. Applied egg-rr 21.4%

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

       1. associate-*l/ 21.4%

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

       2. frac-times 21.3%

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

       3. unpow2 21.3%

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

       4. sqrt-div 21.3%

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

       5. sqrt-pow1 58.5%

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

       6. metadata-eval 58.5%

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

       7. pow1 58.5%

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

       8. *-commutative 58.5%

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

   11. Applied egg-rr 58.5%

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

   if -1.000000000000002e-309 < l

   1. Initial program 67.2%

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

   3. Simplified 66.4%

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

   5. Taylor expanded in M around 0 37.7%

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

      1. pow1/2 37.7%

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

      2. metadata-eval 37.7%

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

      3. *-rgt-identity 37.7%

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

      4. pow1/2 37.7%

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

      5. metadata-eval 37.7%

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

      6. pow-prod-down 30.9%

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

      7. metadata-eval 30.9%

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

      8. unpow1/2 30.9%

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

   7. Applied egg-rr 30.9%

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

      1. associate-*l/ 26.1%

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

   9. Applied egg-rr 26.1%

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

       1. sqrt-div 30.5%

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

       2. clear-num 30.4%

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

       3. associate-*r/ 24.1%

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

       4. unpow2 24.1%

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

       5. sqrt-div 33.0%

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

       6. sqrt-pow1 46.3%

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

       7. metadata-eval 46.3%

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

       8. pow1 46.3%

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

   11. Applied egg-rr 46.3%

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

       1. associate-/r/ 47.0%

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

       2. times-frac 47.8%

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

       3. *-lft-identity 47.8%

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

       4. associate-/l/ 47.0%

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

   13. Simplified 47.0%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9 \cdot 10^{-227}:\\ \;\;\;\;{\left(\frac{d}{h}\right)}^{0.5} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 44.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.2 \cdot 10^{-227}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{-309}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l -2.2e-227)
   (* (sqrt (/ d h)) (sqrt (/ d l)))
   (if (<= l 2e-309) (/ d (sqrt (* l h))) (/ (/ d (sqrt l)) (sqrt h)))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -2.2e-227) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (l <= 2e-309) {
		tmp = d / sqrt((l * h));
	} else {
		tmp = (d / sqrt(l)) / sqrt(h);
	}
	return tmp;
}

Fortran

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-2.2d-227)) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else if (l <= 2d-309) then
        tmp = d / sqrt((l * h))
    else
        tmp = (d / sqrt(l)) / sqrt(h)
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -2.2e-227) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (l <= 2e-309) {
		tmp = d / Math.sqrt((l * h));
	} else {
		tmp = (d / Math.sqrt(l)) / Math.sqrt(h);
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= -2.2e-227:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	elif l <= 2e-309:
		tmp = d / math.sqrt((l * h))
	else:
		tmp = (d / math.sqrt(l)) / math.sqrt(h)
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -2.2e-227)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (l <= 2e-309)
		tmp = Float64(d / sqrt(Float64(l * h)));
	else
		tmp = Float64(Float64(d / sqrt(l)) / sqrt(h));
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -2.2e-227)
		tmp = sqrt((d / h)) * sqrt((d / l));
	elseif (l <= 2e-309)
		tmp = d / sqrt((l * h));
	else
		tmp = (d / sqrt(l)) / sqrt(h);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, -2.2e-227], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2e-309], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.19999999999999981e-227

   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 71.2%

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

   5. Taylor expanded in M around 0 47.5%

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

      1. *-rgt-identity 47.5%

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

   7. Applied egg-rr 47.5%

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

      1. *-commutative 47.5%

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

   9. Simplified 47.5%

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

   if -2.19999999999999981e-227 < l < 1.9999999999999988e-309

   1. Initial program 89.3%

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

   3. Simplified 89.3%

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

   5. Taylor expanded in M around 0 21.4%

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

      1. pow1/2 21.4%

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

      2. metadata-eval 21.4%

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

      3. *-rgt-identity 21.4%

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

      4. pow1/2 21.4%

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

      5. metadata-eval 21.4%

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

      6. pow-prod-down 21.4%

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

      7. metadata-eval 21.4%

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

      8. unpow1/2 21.4%

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

   7. Applied egg-rr 21.4%

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

      1. associate-*l/ 21.4%

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

   9. Applied egg-rr 21.4%

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

       1. associate-*l/ 21.4%

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

       2. frac-times 21.3%

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

       3. unpow2 21.3%

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

       4. sqrt-div 21.3%

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

       5. sqrt-pow1 58.5%

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

       6. metadata-eval 58.5%

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

       7. pow1 58.5%

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

       8. *-commutative 58.5%

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

   11. Applied egg-rr 58.5%

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

   if 1.9999999999999988e-309 < l

   1. Initial program 67.2%

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

   3. Simplified 66.4%

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

   5. Taylor expanded in M around 0 37.7%

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

      1. pow1/2 37.7%

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

      2. metadata-eval 37.7%

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

      3. *-rgt-identity 37.7%

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

      4. pow1/2 37.7%

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

      5. metadata-eval 37.7%

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

      6. pow-prod-down 30.9%

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

      7. metadata-eval 30.9%

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

      8. unpow1/2 30.9%

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

   7. Applied egg-rr 30.9%

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

      1. associate-*l/ 26.1%

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

   9. Applied egg-rr 26.1%

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

       1. sqrt-div 30.5%

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

       2. clear-num 30.4%

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

       3. associate-*r/ 24.1%

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

       4. unpow2 24.1%

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

       5. sqrt-div 33.0%

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

       6. sqrt-pow1 46.3%

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

       7. metadata-eval 46.3%

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

       8. pow1 46.3%

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

   11. Applied egg-rr 46.3%

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

       1. associate-/r/ 47.0%

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

       2. times-frac 47.8%

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

       3. *-lft-identity 47.8%

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

       4. associate-/l/ 47.0%

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

   13. Simplified 47.0%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.2 \cdot 10^{-227}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{-309}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 41.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{+156}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -3.2 \cdot 10^{-227}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1e+156)
   (sqrt (* (/ d l) (/ d h)))
   (if (<= l -3.2e-227)
     (* d (- (sqrt (/ 1.0 (* l h)))))
     (/ d (sqrt (* l h))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1e+156) {
		tmp = sqrt(((d / l) * (d / h)));
	} else if (l <= -3.2e-227) {
		tmp = d * -sqrt((1.0 / (l * h)));
	} else {
		tmp = d / sqrt((l * h));
	}
	return tmp;
}

Fortran

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-1d+156)) then
        tmp = sqrt(((d / l) * (d / h)))
    else if (l <= (-3.2d-227)) then
        tmp = d * -sqrt((1.0d0 / (l * h)))
    else
        tmp = d / sqrt((l * h))
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1e+156) {
		tmp = Math.sqrt(((d / l) * (d / h)));
	} else if (l <= -3.2e-227) {
		tmp = d * -Math.sqrt((1.0 / (l * h)));
	} else {
		tmp = d / Math.sqrt((l * h));
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= -1e+156:
		tmp = math.sqrt(((d / l) * (d / h)))
	elif l <= -3.2e-227:
		tmp = d * -math.sqrt((1.0 / (l * h)))
	else:
		tmp = d / math.sqrt((l * h))
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1e+156)
		tmp = sqrt(Float64(Float64(d / l) * Float64(d / h)));
	elseif (l <= -3.2e-227)
		tmp = Float64(d * Float64(-sqrt(Float64(1.0 / Float64(l * h)))));
	else
		tmp = Float64(d / sqrt(Float64(l * h)));
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -1e+156)
		tmp = sqrt(((d / l) * (d / h)));
	elseif (l <= -3.2e-227)
		tmp = d * -sqrt((1.0 / (l * h)));
	else
		tmp = d / sqrt((l * h));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1e+156], N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, -3.2e-227], N[(d * (-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -9.9999999999999998e155

   1. Initial program 62.8%

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

   3. Simplified 65.6%

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

   5. Taylor expanded in M around 0 48.9%

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

      1. pow1/2 49.0%

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

      2. metadata-eval 49.0%

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

      3. *-rgt-identity 49.0%

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

      4. pow1/2 49.0%

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

      5. metadata-eval 49.0%

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

      6. pow-prod-down 46.4%

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

      7. metadata-eval 46.4%

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

      8. unpow1/2 46.4%

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

   7. Applied egg-rr 46.4%

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

   if -9.9999999999999998e155 < l < -3.2000000000000001e-227

   1. Initial program 72.4%

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

      1. associate-*r/ 75.9%

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

      2. clear-num 76.0%

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

      3. metadata-eval 76.0%

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

      4. associate-*l* 76.0%

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

   4. Applied egg-rr 77.0%

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

   5. Taylor expanded in l around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. 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. *-commutative 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 49.3%

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

      6. neg-mul-1 49.3%

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

   7. Simplified 49.3%

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

   if -3.2000000000000001e-227 < l

   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 69.6%

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

   5. Taylor expanded in M around 0 35.5%

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

      1. pow1/2 35.5%

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

      2. metadata-eval 35.5%

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

      3. *-rgt-identity 35.5%

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

      4. pow1/2 35.5%

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

      5. metadata-eval 35.5%

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

      6. pow-prod-down 29.6%

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

      7. metadata-eval 29.6%

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

      8. unpow1/2 29.6%

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

   7. Applied egg-rr 29.6%

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

      1. associate-*l/ 25.5%

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

   9. Applied egg-rr 25.5%

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

       1. associate-*l/ 29.6%

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

       2. frac-times 23.7%

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

       3. unpow2 23.7%

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

       4. sqrt-div 29.2%

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

       5. sqrt-pow1 45.7%

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

       6. metadata-eval 45.7%

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

       7. pow1 45.7%

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

       8. *-commutative 45.7%

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

   11. Applied egg-rr 45.7%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{+156}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -3.2 \cdot 10^{-227}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 37.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.48 \cdot 10^{-227}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1.48e-227) (sqrt (* (/ d l) (/ d h))) (/ d (sqrt (* l h)))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.48e-227) {
		tmp = sqrt(((d / l) * (d / h)));
	} else {
		tmp = d / sqrt((l * h));
	}
	return tmp;
}

Fortran

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-1.48d-227)) then
        tmp = sqrt(((d / l) * (d / h)))
    else
        tmp = d / sqrt((l * h))
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.48e-227) {
		tmp = Math.sqrt(((d / l) * (d / h)));
	} else {
		tmp = d / Math.sqrt((l * h));
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= -1.48e-227:
		tmp = math.sqrt(((d / l) * (d / h)))
	else:
		tmp = d / math.sqrt((l * h))
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1.48e-227)
		tmp = sqrt(Float64(Float64(d / l) * Float64(d / h)));
	else
		tmp = Float64(d / sqrt(Float64(l * h)));
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -1.48e-227)
		tmp = sqrt(((d / l) * (d / h)));
	else
		tmp = d / sqrt((l * h));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1.48e-227], N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.48e-227

   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 71.2%

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

   5. Taylor expanded in M around 0 47.5%

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

      1. pow1/2 47.5%

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

      2. metadata-eval 47.5%

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

      3. *-rgt-identity 47.5%

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

      4. pow1/2 47.5%

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

      5. metadata-eval 47.5%

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

      6. pow-prod-down 43.6%

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

      7. metadata-eval 43.6%

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

      8. unpow1/2 43.6%

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

   7. Applied egg-rr 43.6%

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

   if -1.48e-227 < l

   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 69.6%

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

   5. Taylor expanded in M around 0 35.5%

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

      1. pow1/2 35.5%

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

      2. metadata-eval 35.5%

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

      3. *-rgt-identity 35.5%

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

      4. pow1/2 35.5%

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

      5. metadata-eval 35.5%

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

      6. pow-prod-down 29.6%

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

      7. metadata-eval 29.6%

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

      8. unpow1/2 29.6%

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

   7. Applied egg-rr 29.6%

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

      1. associate-*l/ 25.5%

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

   9. Applied egg-rr 25.5%

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

       1. associate-*l/ 29.6%

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

       2. frac-times 23.7%

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

       3. unpow2 23.7%

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

       4. sqrt-div 29.2%

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

       5. sqrt-pow1 45.7%

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

       6. metadata-eval 45.7%

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

       7. pow1 45.7%

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

       8. *-commutative 45.7%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\ell \cdot h}}} \]

   11. Applied egg-rr 45.7%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.48 \cdot 10^{-227}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 26.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \frac{d}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D) :precision binary64 (/ d (sqrt (* l h))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	return d / sqrt((l * h));
}

Fortran

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = d / sqrt((l * h))
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	return d / Math.sqrt((l * h));
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	return d / math.sqrt((l * h))

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	return Float64(d / sqrt(Float64(l * h)))
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp = code(d, h, l, M, D)
	tmp = d / sqrt((l * h));
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.9%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 70.3%

   \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in M around 0 40.9%

   \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]
6. Step-by-step derivation

   1. pow1/2 40.9%

      \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{0.5}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot 1\right) \]

   2. metadata-eval 40.9%

      \[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot 1\right) \]

   3. *-rgt-identity 40.9%

      \[\leadsto {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

   4. pow1/2 40.9%

      \[\leadsto {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{0.5}} \]

   5. metadata-eval 40.9%

      \[\leadsto {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \]

   6. pow-prod-down 35.9%

      \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

   7. metadata-eval 35.9%

      \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{0.5}} \]

   8. unpow1/2 35.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

7. Applied egg-rr 35.9%

   \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]
8. Step-by-step derivation

   1. associate-*l/ 29.7%

      \[\leadsto \sqrt{\color{blue}{\frac{d \cdot \frac{d}{\ell}}{h}}} \]

9. Applied egg-rr 29.7%

   \[\leadsto \sqrt{\color{blue}{\frac{d \cdot \frac{d}{\ell}}{h}}} \]
10. Step-by-step derivation

    1. associate-*l/ 35.9%

       \[\leadsto \sqrt{\color{blue}{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

    2. frac-times 24.9%

       \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \]

    3. unpow2 24.9%

       \[\leadsto \sqrt{\frac{\color{blue}{{d}^{2}}}{h \cdot \ell}} \]

    4. sqrt-div 28.6%

       \[\leadsto \color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}}} \]

    5. sqrt-pow1 27.4%

       \[\leadsto \frac{\color{blue}{{d}^{\left(\frac{2}{2}\right)}}}{\sqrt{h \cdot \ell}} \]

    6. metadata-eval 27.4%

       \[\leadsto \frac{{d}^{\color{blue}{1}}}{\sqrt{h \cdot \ell}} \]

    7. pow1 27.4%

       \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

    8. *-commutative 27.4%

       \[\leadsto \frac{d}{\sqrt{\color{blue}{\ell \cdot h}}} \]

11. Applied egg-rr 27.4%

    \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(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)))))