Henrywood and Agarwal, Equation (12)

Percentage Accurate: 65.0% → 77.8%

Time: 21.9s

Alternatives: 22

Speedup: 1.0×

• 23.0% of points produce a very large (infinite) output. You may want to add a precondition.

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: 65.0% 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: 77.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ t_1 := \frac{M\_m}{\frac{d}{D\_m}}\\ \mathbf{if}\;d \leq -1.35 \cdot 10^{-194}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot \left(0.25 \cdot {t\_1}^{2}\right)}{\ell}\right)\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{-291}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.125, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}} \cdot {\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2}, t\_0\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}}\right)}^{3} \cdot \left(1 - 0.5 \cdot {\left(t\_0 \cdot \left(0.5 \cdot t\_1\right)\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ h l))) (t_1 (/ M_m (/ d D_m))))
   (if (<= d -1.35e-194)
     (*
      (* (/ (sqrt (- d)) (sqrt (- h))) (sqrt (/ d l)))
      (- 1.0 (* 0.5 (/ (* h (* 0.25 (pow t_1 2.0))) l))))
     (if (<= d 4.2e-291)
       (*
        d
        (/
         (fma
          -0.125
          (* (sqrt (pow (/ h l) 3.0)) (pow (* D_m (/ M_m d)) 2.0))
          t_0)
         h))
       (*
        (pow (cbrt (/ d (* (sqrt h) (sqrt l)))) 3.0)
        (- 1.0 (* 0.5 (pow (* t_0 (* 0.5 t_1)) 2.0))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((h / l));
	double t_1 = M_m / (d / D_m);
	double tmp;
	if (d <= -1.35e-194) {
		tmp = ((sqrt(-d) / sqrt(-h)) * sqrt((d / l))) * (1.0 - (0.5 * ((h * (0.25 * pow(t_1, 2.0))) / l)));
	} else if (d <= 4.2e-291) {
		tmp = d * (fma(-0.125, (sqrt(pow((h / l), 3.0)) * pow((D_m * (M_m / d)), 2.0)), t_0) / h);
	} else {
		tmp = pow(cbrt((d / (sqrt(h) * sqrt(l)))), 3.0) * (1.0 - (0.5 * pow((t_0 * (0.5 * t_1)), 2.0)));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(h / l))
	t_1 = Float64(M_m / Float64(d / D_m))
	tmp = 0.0
	if (d <= -1.35e-194)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * sqrt(Float64(d / l))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * Float64(0.25 * (t_1 ^ 2.0))) / l))));
	elseif (d <= 4.2e-291)
		tmp = Float64(d * Float64(fma(-0.125, Float64(sqrt((Float64(h / l) ^ 3.0)) * (Float64(D_m * Float64(M_m / d)) ^ 2.0)), t_0) / h));
	else
		tmp = Float64((cbrt(Float64(d / Float64(sqrt(h) * sqrt(l)))) ^ 3.0) * Float64(1.0 - Float64(0.5 * (Float64(t_0 * Float64(0.5 * t_1)) ^ 2.0))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(M$95$m / N[(d / D$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.35e-194], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[(0.25 * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.2e-291], N[(d * N[(N[(-0.125 * N[(N[Sqrt[N[Power[N[(h / l), $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Power[N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision] * N[(1.0 - N[(0.5 * N[Power[N[(t$95$0 * N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.35e-194

   1. Initial program 64.8%

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

   3. Simplified 63.8%

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

      1. associate-*r/ 68.1%

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

      2. associate-*l/ 68.1%

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

      3. div-inv 68.1%

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

      4. metadata-eval 68.1%

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

      5. *-commutative 68.1%

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

      6. unpow-prod-down 68.1%

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

      7. metadata-eval 68.1%

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

      8. clear-num 68.0%

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

      9. un-div-inv 68.0%

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

   6. Applied egg-rr 68.0%

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

      1. frac-2neg 68.0%

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

      2. sqrt-div 78.9%

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

   8. Applied egg-rr 78.9%

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

   if -1.35e-194 < d < 4.1999999999999999e-291

   1. Initial program 28.4%

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

   3. Simplified 28.4%

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

      1. add-cube-cbrt 28.2%

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

      2. pow3 28.2%

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

      3. sqrt-div 6.8%

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

      4. sqrt-div 6.9%

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

      5. frac-times 6.9%

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

      6. add-sqr-sqrt 6.9%

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

   6. Applied egg-rr 6.9%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}}\right)}^{3}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. pow1 6.9%

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

      2. rem-cube-cbrt 6.9%

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

      3. sqrt-unprod 7.8%

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

      4. cancel-sign-sub-inv 7.8%

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

      5. metadata-eval 7.8%

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

      6. *-commutative 7.8%

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

      7. div-inv 7.8%

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

      8. metadata-eval 7.8%

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

   8. Applied egg-rr 7.8%

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

      1. unpow1 7.8%

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

      2. associate-*l/ 7.7%

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

      3. associate-/l* 7.7%

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

   10. Simplified 7.7%

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

   11. Taylor expanded in h around 0 17.4%

       \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{h}{\ell}} + -0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right)}{h}} \]
   12. Step-by-step derivation

   13. Simplified 56.4%

       \[\leadsto d \cdot \color{blue}{\frac{\mathsf{fma}\left(-0.125, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}, \sqrt{\frac{h}{\ell}}\right)}{h}} \]

   if 4.1999999999999999e-291 < d

   1. Initial program 66.6%

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

   3. Simplified 68.2%

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

      1. add-cube-cbrt 67.7%

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

      2. pow3 67.7%

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

      3. sqrt-div 80.7%

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

      4. sqrt-div 84.6%

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

      5. frac-times 84.6%

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

      6. add-sqr-sqrt 84.6%

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

   6. Applied egg-rr 84.6%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}}\right)}^{3}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 84.6%

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

      2. pow2 84.6%

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

      3. sqrt-prod 84.6%

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

      4. sqrt-pow1 87.2%

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

      5. metadata-eval 87.2%

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

      6. associate-*l/ 87.2%

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

      7. div-inv 87.2%

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

      8. metadata-eval 87.2%

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

      9. *-commutative 87.2%

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

      10. pow1 87.2%

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

      11. clear-num 86.6%

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

      12. un-div-inv 86.6%

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

   8. Applied egg-rr 86.6%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.35 \cdot 10^{-194}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot \left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{-291}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.125, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}, \sqrt{\frac{h}{\ell}}\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}}\right)}^{3} \cdot \left(1 - 0.5 \cdot {\left(\sqrt{\frac{h}{\ell}} \cdot \left(0.5 \cdot \frac{M}{\frac{d}{D}}\right)\right)}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{M\_m}{\frac{d}{D\_m}}\\ t_1 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -7.5 \cdot 10^{+116}:\\ \;\;\;\;\left(1 - 0.5 \cdot {\left(\sqrt{\frac{h}{\ell}} \cdot \left(0.5 \cdot t\_0\right)\right)}^{2}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \frac{t\_1}{\sqrt{-\ell}}\right)\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{t\_1}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot \left(0.25 \cdot {t\_0}^{2}\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(M\_m \cdot \frac{D\_m}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (/ M_m (/ d D_m))) (t_1 (sqrt (- d))))
   (if (<= l -7.5e+116)
     (*
      (- 1.0 (* 0.5 (pow (* (sqrt (/ h l)) (* 0.5 t_0)) 2.0)))
      (* (sqrt (/ d h)) (/ t_1 (sqrt (- l)))))
     (if (<= l -4e-310)
       (*
        (* (/ t_1 (sqrt (- h))) (sqrt (/ d l)))
        (- 1.0 (* 0.5 (/ (* h (* 0.25 (pow t_0 2.0))) l))))
       (*
        d
        (/
         (fma (* -0.125 (pow (* M_m (/ D_m d)) 2.0)) (/ h l) 1.0)
         (* (sqrt h) (sqrt l))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = M_m / (d / D_m);
	double t_1 = sqrt(-d);
	double tmp;
	if (l <= -7.5e+116) {
		tmp = (1.0 - (0.5 * pow((sqrt((h / l)) * (0.5 * t_0)), 2.0))) * (sqrt((d / h)) * (t_1 / sqrt(-l)));
	} else if (l <= -4e-310) {
		tmp = ((t_1 / sqrt(-h)) * sqrt((d / l))) * (1.0 - (0.5 * ((h * (0.25 * pow(t_0, 2.0))) / l)));
	} else {
		tmp = d * (fma((-0.125 * pow((M_m * (D_m / d)), 2.0)), (h / l), 1.0) / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(M_m / Float64(d / D_m))
	t_1 = sqrt(Float64(-d))
	tmp = 0.0
	if (l <= -7.5e+116)
		tmp = Float64(Float64(1.0 - Float64(0.5 * (Float64(sqrt(Float64(h / l)) * Float64(0.5 * t_0)) ^ 2.0))) * Float64(sqrt(Float64(d / h)) * Float64(t_1 / sqrt(Float64(-l)))));
	elseif (l <= -4e-310)
		tmp = Float64(Float64(Float64(t_1 / sqrt(Float64(-h))) * sqrt(Float64(d / l))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * Float64(0.25 * (t_0 ^ 2.0))) / l))));
	else
		tmp = Float64(d * Float64(fma(Float64(-0.125 * (Float64(M_m * Float64(D_m / d)) ^ 2.0)), Float64(h / l), 1.0) / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(M$95$m / N[(d / D$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[l, -7.5e+116], N[(N[(1.0 - N[(0.5 * N[Power[N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -4e-310], N[(N[(N[(t$95$1 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[(0.25 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[(-0.125 * N[Power[N[(M$95$m * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -7.5e116

   1. Initial program 59.4%

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

   3. Simplified 59.4%

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

      1. add-sqr-sqrt 0.0%

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

      2. pow2 0.0%

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

      3. sqrt-prod 0.0%

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

      4. sqrt-pow1 0.0%

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

      5. metadata-eval 0.0%

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

      6. associate-*l/ 0.0%

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

      7. div-inv 0.0%

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

      8. metadata-eval 0.0%

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

      9. *-commutative 0.0%

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

      10. pow1 0.0%

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

      11. clear-num 0.0%

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

      12. un-div-inv 0.0%

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

   6. Applied egg-rr 67.0%

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

      1. frac-2neg 67.0%

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

      2. sqrt-div 77.2%

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

   8. Applied egg-rr 77.2%

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

   if -7.5e116 < l < -3.999999999999988e-310

   1. Initial program 56.2%

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

   3. Simplified 55.0%

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

      1. associate-*r/ 60.0%

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

      2. associate-*l/ 60.0%

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

      3. div-inv 60.0%

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

      4. metadata-eval 60.0%

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

      5. *-commutative 60.0%

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

      6. unpow-prod-down 60.0%

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

      7. metadata-eval 60.0%

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

      8. clear-num 60.0%

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

      9. un-div-inv 60.0%

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

   6. Applied egg-rr 60.0%

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

      1. frac-2neg 60.0%

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

      2. sqrt-div 73.5%

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

   8. Applied egg-rr 73.5%

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

   if -3.999999999999988e-310 < l

   1. Initial program 65.6%

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

   3. Simplified 67.1%

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

      1. pow1 67.1%

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

      2. associate-*r* 67.1%

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

      3. sqrt-div 79.6%

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

      4. sqrt-div 83.4%

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

      5. frac-times 83.3%

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

      6. add-sqr-sqrt 83.4%

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

      7. unpow-prod-down 83.4%

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

      8. metadata-eval 83.4%

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

      9. clear-num 82.9%

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

      10. un-div-inv 82.9%

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

   6. Applied egg-rr 82.9%

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

      1. unpow1 82.9%

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

      2. associate-*l/ 84.4%

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

      3. associate-/l* 85.1%

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

      4. fma-define 85.1%

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

      5. associate-*r* 85.1%

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

      6. fma-define 85.1%

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

      7. *-commutative 85.1%

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

      8. associate-*l* 85.1%

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

      9. metadata-eval 85.1%

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

      10. *-commutative 85.1%

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

      11. associate-/r/ 85.1%

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

      12. associate-*l/ 83.7%

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

      13. associate-/l* 85.7%

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

   8. Simplified 85.7%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7.5 \cdot 10^{+116}:\\ \;\;\;\;\left(1 - 0.5 \cdot {\left(\sqrt{\frac{h}{\ell}} \cdot \left(0.5 \cdot \frac{M}{\frac{d}{D}}\right)\right)}^{2}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right)\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot \left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -1.5 \cdot 10^{-179}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{t\_0}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D\_m \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot \left(0.25 \cdot {\left(\frac{M\_m}{\frac{d}{D\_m}}\right)}^{2}\right)}{\ell}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \frac{t\_0}{\sqrt{-\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(M\_m \cdot \frac{D\_m}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (- d))))
   (if (<= l -1.5e-179)
     (*
      (sqrt (/ d l))
      (*
       (/ t_0 (sqrt (- h)))
       (+ 1.0 (* (/ h l) (* (pow (* D_m (/ (/ M_m 2.0) d)) 2.0) -0.5)))))
     (if (<= l -4e-310)
       (*
        (- 1.0 (* 0.5 (/ (* h (* 0.25 (pow (/ M_m (/ d D_m)) 2.0))) l)))
        (* (sqrt (/ d h)) (/ t_0 (sqrt (- l)))))
       (*
        d
        (/
         (fma (* -0.125 (pow (* M_m (/ D_m d)) 2.0)) (/ h l) 1.0)
         (* (sqrt h) (sqrt l))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt(-d);
	double tmp;
	if (l <= -1.5e-179) {
		tmp = sqrt((d / l)) * ((t_0 / sqrt(-h)) * (1.0 + ((h / l) * (pow((D_m * ((M_m / 2.0) / d)), 2.0) * -0.5))));
	} else if (l <= -4e-310) {
		tmp = (1.0 - (0.5 * ((h * (0.25 * pow((M_m / (d / D_m)), 2.0))) / l))) * (sqrt((d / h)) * (t_0 / sqrt(-l)));
	} else {
		tmp = d * (fma((-0.125 * pow((M_m * (D_m / d)), 2.0)), (h / l), 1.0) / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(-d))
	tmp = 0.0
	if (l <= -1.5e-179)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(Float64(t_0 / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D_m * Float64(Float64(M_m / 2.0) / d)) ^ 2.0) * -0.5)))));
	elseif (l <= -4e-310)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h * Float64(0.25 * (Float64(M_m / Float64(d / D_m)) ^ 2.0))) / l))) * Float64(sqrt(Float64(d / h)) * Float64(t_0 / sqrt(Float64(-l)))));
	else
		tmp = Float64(d * Float64(fma(Float64(-0.125 * (Float64(M_m * Float64(D_m / d)) ^ 2.0)), Float64(h / l), 1.0) / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[l, -1.5e-179], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D$95$m * N[(N[(M$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -4e-310], N[(N[(1.0 - N[(0.5 * N[(N[(h * N[(0.25 * N[Power[N[(M$95$m / N[(d / D$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[(-0.125 * N[Power[N[(M$95$m * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.50000000000000003e-179

   1. Initial program 58.3%

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

   3. Simplified 58.3%

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

      1. frac-2neg 58.4%

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

      2. sqrt-div 69.4%

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

   6. Applied egg-rr 72.4%

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

   if -1.50000000000000003e-179 < l < -3.999999999999988e-310

   1. Initial program 52.9%

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

   3. Simplified 48.9%

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

      1. associate-*r/ 65.3%

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

      2. associate-*l/ 65.3%

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

      3. div-inv 65.3%

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

      4. metadata-eval 65.3%

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

      5. *-commutative 65.3%

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

      6. unpow-prod-down 65.3%

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

      7. metadata-eval 65.3%

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

      8. clear-num 65.3%

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

      9. un-div-inv 65.3%

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

   6. Applied egg-rr 65.3%

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

      1. frac-2neg 49.1%

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

      2. sqrt-div 52.6%

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

   8. Applied egg-rr 83.9%

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

   if -3.999999999999988e-310 < l

   1. Initial program 65.6%

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

   3. Simplified 67.1%

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

      1. pow1 67.1%

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

      2. associate-*r* 67.1%

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

      3. sqrt-div 79.6%

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

      4. sqrt-div 83.4%

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

      5. frac-times 83.3%

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

      6. add-sqr-sqrt 83.4%

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

      7. unpow-prod-down 83.4%

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

      8. metadata-eval 83.4%

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

      9. clear-num 82.9%

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

      10. un-div-inv 82.9%

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

   6. Applied egg-rr 82.9%

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

      1. unpow1 82.9%

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

      2. associate-*l/ 84.4%

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

      3. associate-/l* 85.1%

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

      4. fma-define 85.1%

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

      5. associate-*r* 85.1%

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

      6. fma-define 85.1%

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

      7. *-commutative 85.1%

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

      8. associate-*l* 85.1%

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

      9. metadata-eval 85.1%

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

      10. *-commutative 85.1%

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

      11. associate-/r/ 85.1%

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

      12. associate-*l/ 83.7%

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

      13. associate-/l* 85.7%

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

   8. Simplified 85.7%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.5 \cdot 10^{-179}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot \left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right)}{\ell}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := {\left(M\_m \cdot \frac{D\_m}{d}\right)}^{2}\\ \mathbf{if}\;\ell \leq -7.5 \cdot 10^{-187}:\\ \;\;\;\;t\_0 \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D\_m \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{-309}:\\ \;\;\;\;\left(t\_0 \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{t\_1}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot t\_1, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l))) (t_1 (pow (* M_m (/ D_m d)) 2.0)))
   (if (<= l -7.5e-187)
     (*
      t_0
      (*
       (/ (sqrt (- d)) (sqrt (- h)))
       (+ 1.0 (* (/ h l) (* (pow (* D_m (/ (/ M_m 2.0) d)) 2.0) -0.5)))))
     (if (<= l 8e-309)
       (* (* t_0 (sqrt (/ d h))) (- 1.0 (* 0.5 (* h (* 0.25 (/ t_1 l))))))
       (* d (/ (fma (* -0.125 t_1) (/ h l) 1.0) (* (sqrt h) (sqrt l))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d / l));
	double t_1 = pow((M_m * (D_m / d)), 2.0);
	double tmp;
	if (l <= -7.5e-187) {
		tmp = t_0 * ((sqrt(-d) / sqrt(-h)) * (1.0 + ((h / l) * (pow((D_m * ((M_m / 2.0) / d)), 2.0) * -0.5))));
	} else if (l <= 8e-309) {
		tmp = (t_0 * sqrt((d / h))) * (1.0 - (0.5 * (h * (0.25 * (t_1 / l)))));
	} else {
		tmp = d * (fma((-0.125 * t_1), (h / l), 1.0) / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(d / l))
	t_1 = Float64(M_m * Float64(D_m / d)) ^ 2.0
	tmp = 0.0
	if (l <= -7.5e-187)
		tmp = Float64(t_0 * Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D_m * Float64(Float64(M_m / 2.0) / d)) ^ 2.0) * -0.5)))));
	elseif (l <= 8e-309)
		tmp = Float64(Float64(t_0 * sqrt(Float64(d / h))) * Float64(1.0 - Float64(0.5 * Float64(h * Float64(0.25 * Float64(t_1 / l))))));
	else
		tmp = Float64(d * Float64(fma(Float64(-0.125 * t_1), Float64(h / l), 1.0) / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(M$95$m * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[l, -7.5e-187], N[(t$95$0 * N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D$95$m * N[(N[(M$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 8e-309], N[(N[(t$95$0 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(h * N[(0.25 * N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[(-0.125 * t$95$1), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -7.5000000000000004e-187

   1. Initial program 58.3%

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

   3. Simplified 58.3%

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

      1. frac-2neg 58.4%

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

      2. sqrt-div 69.4%

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

   6. Applied egg-rr 72.4%

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

   if -7.5000000000000004e-187 < l < 8.0000000000000003e-309

   1. Initial program 52.9%

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

   3. Simplified 48.9%

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

   5. Taylor expanded in M around 0 49.5%

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

      1. associate-*r* 49.8%

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

      2. times-frac 40.9%

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

      3. associate-/l* 40.9%

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

      4. *-commutative 40.9%

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

      5. unpow2 40.9%

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

      6. unpow2 40.9%

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

      7. times-frac 40.9%

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

      8. unpow2 40.9%

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

      9. swap-sqr 52.9%

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

      10. associate-/r/ 48.9%

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

      11. associate-/r/ 48.9%

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

      12. unpow2 48.9%

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

      13. associate-*l* 48.9%

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

      14. *-commutative 48.9%

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

      15. associate-*l/ 65.3%

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

      16. associate-/l* 65.3%

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

      17. associate-/l* 65.3%

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

   7. Simplified 65.3%

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

   if 8.0000000000000003e-309 < l

   1. Initial program 65.6%

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

   3. Simplified 67.1%

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

      1. pow1 67.1%

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

      2. associate-*r* 67.1%

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

      3. sqrt-div 79.6%

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

      4. sqrt-div 83.4%

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

      5. frac-times 83.3%

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

      6. add-sqr-sqrt 83.4%

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

      7. unpow-prod-down 83.4%

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

      8. metadata-eval 83.4%

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

      9. clear-num 82.9%

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

      10. un-div-inv 82.9%

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

   6. Applied egg-rr 82.9%

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

      1. unpow1 82.9%

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

      2. associate-*l/ 84.4%

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

      3. associate-/l* 85.1%

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

      4. fma-define 85.1%

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

      5. associate-*r* 85.1%

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

      6. fma-define 85.1%

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

      7. *-commutative 85.1%

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

      8. associate-*l* 85.1%

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

      9. metadata-eval 85.1%

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

      10. *-commutative 85.1%

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

      11. associate-/r/ 85.1%

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

      12. associate-*l/ 83.7%

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

      13. associate-/l* 85.7%

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

   8. Simplified 85.7%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7.5 \cdot 10^{-187}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{-309}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := {\left(M\_m \cdot \frac{D\_m}{d}\right)}^{2}\\ \mathbf{if}\;d \leq -1.5 \cdot 10^{-143}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{t\_0}{\ell}\right)\right)\right)\\ \mathbf{elif}\;d \leq -1.35 \cdot 10^{-303}:\\ \;\;\;\;\left({D\_m}^{2} \cdot \left({M\_m}^{2} \cdot 0.125\right)\right) \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot t\_0, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (pow (* M_m (/ D_m d)) 2.0)))
   (if (<= d -1.5e-143)
     (*
      (* (sqrt (/ d l)) (sqrt (/ d h)))
      (- 1.0 (* 0.5 (* h (* 0.25 (/ t_0 l))))))
     (if (<= d -1.35e-303)
       (*
        (* (pow D_m 2.0) (* (pow M_m 2.0) 0.125))
        (/ (sqrt (/ h (pow l 3.0))) d))
       (* d (/ (fma (* -0.125 t_0) (/ h l) 1.0) (* (sqrt h) (sqrt l))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = pow((M_m * (D_m / d)), 2.0);
	double tmp;
	if (d <= -1.5e-143) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.5 * (h * (0.25 * (t_0 / l)))));
	} else if (d <= -1.35e-303) {
		tmp = (pow(D_m, 2.0) * (pow(M_m, 2.0) * 0.125)) * (sqrt((h / pow(l, 3.0))) / d);
	} else {
		tmp = d * (fma((-0.125 * t_0), (h / l), 1.0) / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(M_m * Float64(D_m / d)) ^ 2.0
	tmp = 0.0
	if (d <= -1.5e-143)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(0.5 * Float64(h * Float64(0.25 * Float64(t_0 / l))))));
	elseif (d <= -1.35e-303)
		tmp = Float64(Float64((D_m ^ 2.0) * Float64((M_m ^ 2.0) * 0.125)) * Float64(sqrt(Float64(h / (l ^ 3.0))) / d));
	else
		tmp = Float64(d * Float64(fma(Float64(-0.125 * t_0), Float64(h / l), 1.0) / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Power[N[(M$95$m * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[d, -1.5e-143], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(h * N[(0.25 * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.35e-303], N[(N[(N[Power[D$95$m, 2.0], $MachinePrecision] * N[(N[Power[M$95$m, 2.0], $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[(-0.125 * t$95$0), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.49999999999999993e-143

   1. Initial program 71.3%

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

   3. Simplified 70.2%

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

   5. Taylor expanded in M around 0 55.6%

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

      1. associate-*r* 56.1%

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

      2. times-frac 51.5%

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

      3. associate-/l* 51.5%

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

      4. *-commutative 51.5%

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

      5. unpow2 51.5%

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

      6. unpow2 51.5%

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

      7. times-frac 58.3%

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

      8. unpow2 58.3%

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

      9. swap-sqr 71.4%

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

      10. associate-/r/ 70.2%

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

      11. associate-/r/ 70.2%

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

      12. unpow2 70.2%

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

      13. associate-*l* 70.2%

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

      14. *-commutative 70.2%

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

      15. associate-*l/ 74.8%

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

      16. associate-/l* 75.9%

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

      17. associate-/l* 75.9%

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

   7. Simplified 75.9%

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

   if -1.49999999999999993e-143 < d < -1.34999999999999993e-303

   1. Initial program 19.7%

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

   3. Simplified 19.7%

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

   5. Taylor expanded in M around 0 9.5%

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

      1. associate-*r* 9.5%

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

      2. times-frac 9.5%

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

      3. associate-/l* 9.5%

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

      4. *-commutative 9.5%

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

      5. unpow2 9.5%

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

      6. unpow2 9.5%

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

      7. times-frac 19.3%

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

      8. unpow2 19.3%

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

      9. swap-sqr 19.7%

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

      10. associate-/r/ 19.7%

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

      11. associate-/r/ 19.7%

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

      12. unpow2 19.7%

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

      13. associate-*l* 19.7%

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

      14. *-commutative 19.7%

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

      15. associate-*l/ 20.0%

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

      16. associate-/l* 20.0%

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

      17. associate-/l* 20.0%

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

   7. Simplified 20.0%

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

      1. add-sqr-sqrt 20.0%

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

      2. pow2 20.0%

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

      3. associate-*r* 20.0%

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

      4. associate-*r/ 20.0%

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

   9. Applied egg-rr 20.0%

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

   10. Taylor expanded in h around -inf 0.0%

       \[\leadsto \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot {\left(\sqrt{-0.125}\right)}^{2}\right)\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]
   11. Step-by-step derivation

       1. associate-*l/ 0.0%

          \[\leadsto \color{blue}{\frac{\left({D}^{2} \cdot \left({M}^{2} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot {\left(\sqrt{-0.125}\right)}^{2}\right)\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d}} \]

       2. associate-/l* 0.0%

          \[\leadsto \color{blue}{\left({D}^{2} \cdot \left({M}^{2} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot {\left(\sqrt{-0.125}\right)}^{2}\right)\right)\right) \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d}} \]

       3. unpow2 0.0%

          \[\leadsto \left({D}^{2} \cdot \left({M}^{2} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot {\left(\sqrt{-0.125}\right)}^{2}\right)\right)\right) \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d} \]

       4. rem-square-sqrt 0.0%

          \[\leadsto \left({D}^{2} \cdot \left({M}^{2} \cdot \left(\color{blue}{-1} \cdot {\left(\sqrt{-0.125}\right)}^{2}\right)\right)\right) \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d} \]

       5. unpow2 0.0%

          \[\leadsto \left({D}^{2} \cdot \left({M}^{2} \cdot \left(-1 \cdot \color{blue}{\left(\sqrt{-0.125} \cdot \sqrt{-0.125}\right)}\right)\right)\right) \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d} \]

       6. rem-square-sqrt 35.3%

          \[\leadsto \left({D}^{2} \cdot \left({M}^{2} \cdot \left(-1 \cdot \color{blue}{-0.125}\right)\right)\right) \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d} \]

       7. metadata-eval 35.3%

          \[\leadsto \left({D}^{2} \cdot \left({M}^{2} \cdot \color{blue}{0.125}\right)\right) \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d} \]

   12. Simplified 35.3%

       \[\leadsto \color{blue}{\left({D}^{2} \cdot \left({M}^{2} \cdot 0.125\right)\right) \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d}} \]

   if -1.34999999999999993e-303 < d

   1. Initial program 65.1%

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

   3. Simplified 66.6%

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

      1. pow1 66.6%

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

      2. associate-*r* 66.6%

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

      3. sqrt-div 79.0%

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

      4. sqrt-div 82.8%

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

      5. frac-times 82.7%

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

      6. add-sqr-sqrt 82.8%

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

      7. unpow-prod-down 82.8%

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

      8. metadata-eval 82.8%

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

      9. clear-num 82.3%

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

      10. un-div-inv 82.3%

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

   6. Applied egg-rr 82.3%

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

      1. unpow1 82.3%

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

      2. associate-*l/ 83.8%

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

      3. associate-/l* 84.5%

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

      4. fma-define 84.5%

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

      5. associate-*r* 84.5%

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

      6. fma-define 84.5%

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

      7. *-commutative 84.5%

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

      8. associate-*l* 84.5%

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

      9. metadata-eval 84.5%

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

      10. *-commutative 84.5%

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

      11. associate-/r/ 84.4%

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

      12. associate-*l/ 83.0%

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

      13. associate-/l* 85.0%

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

   8. Simplified 85.0%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.5 \cdot 10^{-143}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{elif}\;d \leq -1.35 \cdot 10^{-303}:\\ \;\;\;\;\left({D}^{2} \cdot \left({M}^{2} \cdot 0.125\right)\right) \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1.55 \cdot 10^{-142}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{{\left(M\_m \cdot \frac{D\_m}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{elif}\;d \leq -1.35 \cdot 10^{-303}:\\ \;\;\;\;\left({D\_m}^{2} \cdot \left({M\_m}^{2} \cdot 0.125\right)\right) \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \frac{M\_m}{2}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -1.55e-142)
   (*
    (* (sqrt (/ d l)) (sqrt (/ d h)))
    (- 1.0 (* 0.5 (* h (* 0.25 (/ (pow (* M_m (/ D_m d)) 2.0) l))))))
   (if (<= d -1.35e-303)
     (*
      (* (pow D_m 2.0) (* (pow M_m 2.0) 0.125))
      (/ (sqrt (/ h (pow l 3.0))) d))
     (*
      (/ d (* (sqrt h) (sqrt l)))
      (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D_m d) (/ M_m 2.0)) 2.0))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -1.55e-142) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.5 * (h * (0.25 * (pow((M_m * (D_m / d)), 2.0) / l)))));
	} else if (d <= -1.35e-303) {
		tmp = (pow(D_m, 2.0) * (pow(M_m, 2.0) * 0.125)) * (sqrt((h / pow(l, 3.0))) / d);
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - (0.5 * ((h / l) * pow(((D_m / d) * (M_m / 2.0)), 2.0))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-1.55d-142)) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - (0.5d0 * (h * (0.25d0 * (((m_m * (d_m / d)) ** 2.0d0) / l)))))
    else if (d <= (-1.35d-303)) then
        tmp = ((d_m ** 2.0d0) * ((m_m ** 2.0d0) * 0.125d0)) * (sqrt((h / (l ** 3.0d0))) / d)
    else
        tmp = (d / (sqrt(h) * sqrt(l))) * (1.0d0 - (0.5d0 * ((h / l) * (((d_m / d) * (m_m / 2.0d0)) ** 2.0d0))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -1.55e-142) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (0.5 * (h * (0.25 * (Math.pow((M_m * (D_m / d)), 2.0) / l)))));
	} else if (d <= -1.35e-303) {
		tmp = (Math.pow(D_m, 2.0) * (Math.pow(M_m, 2.0) * 0.125)) * (Math.sqrt((h / Math.pow(l, 3.0))) / d);
	} else {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * (1.0 - (0.5 * ((h / l) * Math.pow(((D_m / d) * (M_m / 2.0)), 2.0))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= -1.55e-142:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (0.5 * (h * (0.25 * (math.pow((M_m * (D_m / d)), 2.0) / l)))))
	elif d <= -1.35e-303:
		tmp = (math.pow(D_m, 2.0) * (math.pow(M_m, 2.0) * 0.125)) * (math.sqrt((h / math.pow(l, 3.0))) / d)
	else:
		tmp = (d / (math.sqrt(h) * math.sqrt(l))) * (1.0 - (0.5 * ((h / l) * math.pow(((D_m / d) * (M_m / 2.0)), 2.0))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -1.55e-142)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(0.5 * Float64(h * Float64(0.25 * Float64((Float64(M_m * Float64(D_m / d)) ^ 2.0) / l))))));
	elseif (d <= -1.35e-303)
		tmp = Float64(Float64((D_m ^ 2.0) * Float64((M_m ^ 2.0) * 0.125)) * Float64(sqrt(Float64(h / (l ^ 3.0))) / d));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(M_m / 2.0)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= -1.55e-142)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.5 * (h * (0.25 * (((M_m * (D_m / d)) ^ 2.0) / l)))));
	elseif (d <= -1.35e-303)
		tmp = ((D_m ^ 2.0) * ((M_m ^ 2.0) * 0.125)) * (sqrt((h / (l ^ 3.0))) / d);
	else
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - (0.5 * ((h / l) * (((D_m / d) * (M_m / 2.0)) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -1.55e-142], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(h * N[(0.25 * N[(N[Power[N[(M$95$m * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.35e-303], N[(N[(N[Power[D$95$m, 2.0], $MachinePrecision] * N[(N[Power[M$95$m, 2.0], $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(M$95$m / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.55e-142

   1. Initial program 71.3%

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

   3. Simplified 70.2%

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

   5. Taylor expanded in M around 0 55.6%

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

      1. associate-*r* 56.1%

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

      2. times-frac 51.5%

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

      3. associate-/l* 51.5%

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

      4. *-commutative 51.5%

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

      5. unpow2 51.5%

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

      6. unpow2 51.5%

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

      7. times-frac 58.3%

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

      8. unpow2 58.3%

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

      9. swap-sqr 71.4%

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

      10. associate-/r/ 70.2%

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

      11. associate-/r/ 70.2%

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

      12. unpow2 70.2%

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

      13. associate-*l* 70.2%

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

      14. *-commutative 70.2%

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

      15. associate-*l/ 74.8%

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

      16. associate-/l* 75.9%

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

      17. associate-/l* 75.9%

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

   7. Simplified 75.9%

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

   if -1.55e-142 < d < -1.34999999999999993e-303

   1. Initial program 19.7%

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

   3. Simplified 19.7%

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

   5. Taylor expanded in M around 0 9.5%

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

      1. associate-*r* 9.5%

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

      2. times-frac 9.5%

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

      3. associate-/l* 9.5%

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

      4. *-commutative 9.5%

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

      5. unpow2 9.5%

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

      6. unpow2 9.5%

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

      7. times-frac 19.3%

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

      8. unpow2 19.3%

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

      9. swap-sqr 19.7%

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

      10. associate-/r/ 19.7%

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

      11. associate-/r/ 19.7%

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

      12. unpow2 19.7%

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

      13. associate-*l* 19.7%

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

      14. *-commutative 19.7%

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

      15. associate-*l/ 20.0%

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

      16. associate-/l* 20.0%

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

      17. associate-/l* 20.0%

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

   7. Simplified 20.0%

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

      1. add-sqr-sqrt 20.0%

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

      2. pow2 20.0%

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

      3. associate-*r* 20.0%

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

      4. associate-*r/ 20.0%

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

   9. Applied egg-rr 20.0%

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

   10. Taylor expanded in h around -inf 0.0%

       \[\leadsto \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot {\left(\sqrt{-0.125}\right)}^{2}\right)\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]
   11. Step-by-step derivation

       1. associate-*l/ 0.0%

          \[\leadsto \color{blue}{\frac{\left({D}^{2} \cdot \left({M}^{2} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot {\left(\sqrt{-0.125}\right)}^{2}\right)\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d}} \]

       2. associate-/l* 0.0%

          \[\leadsto \color{blue}{\left({D}^{2} \cdot \left({M}^{2} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot {\left(\sqrt{-0.125}\right)}^{2}\right)\right)\right) \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d}} \]

       3. unpow2 0.0%

          \[\leadsto \left({D}^{2} \cdot \left({M}^{2} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot {\left(\sqrt{-0.125}\right)}^{2}\right)\right)\right) \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d} \]

       4. rem-square-sqrt 0.0%

          \[\leadsto \left({D}^{2} \cdot \left({M}^{2} \cdot \left(\color{blue}{-1} \cdot {\left(\sqrt{-0.125}\right)}^{2}\right)\right)\right) \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d} \]

       5. unpow2 0.0%

          \[\leadsto \left({D}^{2} \cdot \left({M}^{2} \cdot \left(-1 \cdot \color{blue}{\left(\sqrt{-0.125} \cdot \sqrt{-0.125}\right)}\right)\right)\right) \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d} \]

       6. rem-square-sqrt 35.3%

          \[\leadsto \left({D}^{2} \cdot \left({M}^{2} \cdot \left(-1 \cdot \color{blue}{-0.125}\right)\right)\right) \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d} \]

       7. metadata-eval 35.3%

          \[\leadsto \left({D}^{2} \cdot \left({M}^{2} \cdot \color{blue}{0.125}\right)\right) \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d} \]

   12. Simplified 35.3%

       \[\leadsto \color{blue}{\left({D}^{2} \cdot \left({M}^{2} \cdot 0.125\right)\right) \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d}} \]

   if -1.34999999999999993e-303 < d

   1. Initial program 65.1%

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

   3. Simplified 66.6%

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

      1. *-commutative 66.6%

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

      2. sqrt-div 70.5%

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

      3. sqrt-div 82.8%

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

      4. frac-times 82.7%

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

      5. add-sqr-sqrt 82.8%

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

   6. Applied egg-rr 82.8%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.55 \cdot 10^{-142}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{elif}\;d \leq -1.35 \cdot 10^{-303}:\\ \;\;\;\;\left({D}^{2} \cdot \left({M}^{2} \cdot 0.125\right)\right) \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -9.8 \cdot 10^{-209}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{{\left(M\_m \cdot \frac{D\_m}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{elif}\;h \leq 2.25 \cdot 10^{-305}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \frac{M\_m}{2}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= h -9.8e-209)
   (*
    (* (sqrt (/ d l)) (sqrt (/ d h)))
    (- 1.0 (* 0.5 (* h (* 0.25 (/ (pow (* M_m (/ D_m d)) 2.0) l))))))
   (if (<= h 2.25e-305)
     (* d (- (pow (* h l) -0.5)))
     (*
      (/ d (* (sqrt h) (sqrt l)))
      (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D_m d) (/ M_m 2.0)) 2.0))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= -9.8e-209) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.5 * (h * (0.25 * (pow((M_m * (D_m / d)), 2.0) / l)))));
	} else if (h <= 2.25e-305) {
		tmp = d * -pow((h * l), -0.5);
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - (0.5 * ((h / l) * pow(((D_m / d) * (M_m / 2.0)), 2.0))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (h <= (-9.8d-209)) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - (0.5d0 * (h * (0.25d0 * (((m_m * (d_m / d)) ** 2.0d0) / l)))))
    else if (h <= 2.25d-305) then
        tmp = d * -((h * l) ** (-0.5d0))
    else
        tmp = (d / (sqrt(h) * sqrt(l))) * (1.0d0 - (0.5d0 * ((h / l) * (((d_m / d) * (m_m / 2.0d0)) ** 2.0d0))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= -9.8e-209) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (0.5 * (h * (0.25 * (Math.pow((M_m * (D_m / d)), 2.0) / l)))));
	} else if (h <= 2.25e-305) {
		tmp = d * -Math.pow((h * l), -0.5);
	} else {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * (1.0 - (0.5 * ((h / l) * Math.pow(((D_m / d) * (M_m / 2.0)), 2.0))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if h <= -9.8e-209:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (0.5 * (h * (0.25 * (math.pow((M_m * (D_m / d)), 2.0) / l)))))
	elif h <= 2.25e-305:
		tmp = d * -math.pow((h * l), -0.5)
	else:
		tmp = (d / (math.sqrt(h) * math.sqrt(l))) * (1.0 - (0.5 * ((h / l) * math.pow(((D_m / d) * (M_m / 2.0)), 2.0))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (h <= -9.8e-209)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(0.5 * Float64(h * Float64(0.25 * Float64((Float64(M_m * Float64(D_m / d)) ^ 2.0) / l))))));
	elseif (h <= 2.25e-305)
		tmp = Float64(d * Float64(-(Float64(h * l) ^ -0.5)));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(M_m / 2.0)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (h <= -9.8e-209)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.5 * (h * (0.25 * (((M_m * (D_m / d)) ^ 2.0) / l)))));
	elseif (h <= 2.25e-305)
		tmp = d * -((h * l) ^ -0.5);
	else
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - (0.5 * ((h / l) * (((D_m / d) * (M_m / 2.0)) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[h, -9.8e-209], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(h * N[(0.25 * N[(N[Power[N[(M$95$m * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 2.25e-305], N[(d * (-N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(M$95$m / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -9.8000000000000007e-209

   1. Initial program 57.8%

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

   3. Simplified 56.8%

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

   5. Taylor expanded in M around 0 47.8%

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

      1. associate-*r* 48.2%

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

      2. times-frac 44.2%

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

      3. associate-/l* 44.2%

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

      4. *-commutative 44.2%

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

      5. unpow2 44.2%

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

      6. unpow2 44.2%

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

      7. times-frac 50.3%

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

      8. unpow2 50.3%

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

      9. swap-sqr 57.8%

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

      10. associate-/r/ 56.8%

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

      11. associate-/r/ 56.7%

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

      12. unpow2 56.7%

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

      13. associate-*l* 56.7%

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

      14. *-commutative 56.7%

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

      15. associate-*l/ 61.0%

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

      16. associate-/l* 61.9%

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

      17. associate-/l* 61.9%

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

   7. Simplified 61.9%

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

   if -9.8000000000000007e-209 < h < 2.2500000000000001e-305

   1. Initial program 51.9%

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

   3. Simplified 51.9%

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

   5. Taylor expanded in d around inf 6.0%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. associate-*l* 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 73.2%

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

      4. mul-1-neg 73.2%

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

      5. unpow-1 73.2%

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

      6. metadata-eval 73.2%

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

      7. pow-sqr 73.3%

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

      8. rem-sqrt-square 73.3%

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

      9. rem-square-sqrt 73.1%

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

      10. fabs-sqr 73.1%

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

      11. rem-square-sqrt 73.3%

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

   8. Simplified 73.3%

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

   if 2.2500000000000001e-305 < h

   1. Initial program 66.1%

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

   3. Simplified 67.6%

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

      1. *-commutative 67.6%

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

      2. sqrt-div 71.5%

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

      3. sqrt-div 84.0%

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

      4. frac-times 83.9%

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

      5. add-sqr-sqrt 84.1%

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

   6. Applied egg-rr 84.1%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -9.8 \cdot 10^{-209}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{elif}\;h \leq 2.25 \cdot 10^{-305}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -4.45 \cdot 10^{-209}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;h \leq 2.25 \cdot 10^{-305}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \frac{M\_m}{2}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= h -4.45e-209)
   (*
    (* (sqrt (/ d l)) (sqrt (/ d h)))
    (- 1.0 (* 0.125 (* (/ h l) (pow (* D_m (/ M_m d)) 2.0)))))
   (if (<= h 2.25e-305)
     (* d (- (pow (* h l) -0.5)))
     (*
      (/ d (* (sqrt h) (sqrt l)))
      (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D_m d) (/ M_m 2.0)) 2.0))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= -4.45e-209) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.125 * ((h / l) * pow((D_m * (M_m / d)), 2.0))));
	} else if (h <= 2.25e-305) {
		tmp = d * -pow((h * l), -0.5);
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - (0.5 * ((h / l) * pow(((D_m / d) * (M_m / 2.0)), 2.0))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (h <= (-4.45d-209)) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - (0.125d0 * ((h / l) * ((d_m * (m_m / d)) ** 2.0d0))))
    else if (h <= 2.25d-305) then
        tmp = d * -((h * l) ** (-0.5d0))
    else
        tmp = (d / (sqrt(h) * sqrt(l))) * (1.0d0 - (0.5d0 * ((h / l) * (((d_m / d) * (m_m / 2.0d0)) ** 2.0d0))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= -4.45e-209) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (0.125 * ((h / l) * Math.pow((D_m * (M_m / d)), 2.0))));
	} else if (h <= 2.25e-305) {
		tmp = d * -Math.pow((h * l), -0.5);
	} else {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * (1.0 - (0.5 * ((h / l) * Math.pow(((D_m / d) * (M_m / 2.0)), 2.0))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if h <= -4.45e-209:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (0.125 * ((h / l) * math.pow((D_m * (M_m / d)), 2.0))))
	elif h <= 2.25e-305:
		tmp = d * -math.pow((h * l), -0.5)
	else:
		tmp = (d / (math.sqrt(h) * math.sqrt(l))) * (1.0 - (0.5 * ((h / l) * math.pow(((D_m / d) * (M_m / 2.0)), 2.0))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (h <= -4.45e-209)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(0.125 * Float64(Float64(h / l) * (Float64(D_m * Float64(M_m / d)) ^ 2.0)))));
	elseif (h <= 2.25e-305)
		tmp = Float64(d * Float64(-(Float64(h * l) ^ -0.5)));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(M_m / 2.0)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (h <= -4.45e-209)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.125 * ((h / l) * ((D_m * (M_m / d)) ^ 2.0))));
	elseif (h <= 2.25e-305)
		tmp = d * -((h * l) ^ -0.5);
	else
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - (0.5 * ((h / l) * (((D_m / d) * (M_m / 2.0)) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[h, -4.45e-209], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.125 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 2.25e-305], N[(d * (-N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(M$95$m / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -4.4500000000000003e-209

   1. Initial program 57.8%

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

   3. Simplified 56.8%

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

   5. Taylor expanded in M around 0 47.8%

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

      1. associate-*r* 48.2%

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

      2. times-frac 44.2%

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

      3. associate-/l* 44.2%

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

      4. *-commutative 44.2%

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

      5. unpow2 44.2%

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

      6. unpow2 44.2%

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

      7. times-frac 50.3%

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

      8. unpow2 50.3%

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

      9. swap-sqr 57.8%

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

      10. associate-/r/ 56.8%

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

      11. associate-/r/ 56.7%

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

      12. unpow2 56.7%

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

      13. associate-*l* 56.7%

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

      14. *-commutative 56.7%

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

      15. associate-*l/ 61.0%

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

      16. associate-/l* 61.9%

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

      17. associate-/l* 61.9%

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

   7. Simplified 61.9%

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

   8. Taylor expanded in h around 0 47.8%

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

      1. associate-*r* 48.2%

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

      2. times-frac 44.2%

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

      3. *-commutative 44.2%

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

      4. associate-/l* 42.1%

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

      5. unpow2 42.1%

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

      6. unpow2 42.1%

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

      7. unpow2 42.1%

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

      8. times-frac 48.1%

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

      9. swap-sqr 56.8%

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

      10. unpow2 56.8%

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

      11. *-commutative 56.8%

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

      12. associate-*r/ 57.8%

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

      13. *-commutative 57.8%

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

      14. associate-/l* 57.8%

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

   10. Simplified 57.8%

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

   if -4.4500000000000003e-209 < h < 2.2500000000000001e-305

   1. Initial program 51.9%

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

   3. Simplified 51.9%

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

   5. Taylor expanded in d around inf 6.0%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. associate-*l* 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 73.2%

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

      4. mul-1-neg 73.2%

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

      5. unpow-1 73.2%

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

      6. metadata-eval 73.2%

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

      7. pow-sqr 73.3%

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

      8. rem-sqrt-square 73.3%

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

      9. rem-square-sqrt 73.1%

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

      10. fabs-sqr 73.1%

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

      11. rem-square-sqrt 73.3%

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

   8. Simplified 73.3%

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

   if 2.2500000000000001e-305 < h

   1. Initial program 66.1%

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

   3. Simplified 67.6%

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

      1. *-commutative 67.6%

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

      2. sqrt-div 71.5%

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

      3. sqrt-div 84.0%

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

      4. frac-times 83.9%

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

      5. add-sqr-sqrt 84.1%

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

   6. Applied egg-rr 84.1%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -4.45 \cdot 10^{-209}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;h \leq 2.25 \cdot 10^{-305}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{+131}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \frac{M\_m}{2}\right)}^{2}\right)\right) \cdot \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -4e-310)
   (*
    (* (sqrt (/ d l)) (sqrt (/ d h)))
    (- 1.0 (* 0.125 (* (/ h l) (pow (* D_m (/ M_m d)) 2.0)))))
   (if (<= l 2.2e+131)
     (*
      (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D_m d) (/ M_m 2.0)) 2.0))))
      (* d (pow (* h l) -0.5)))
     (/ d (* (sqrt h) (sqrt l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -4e-310) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.125 * ((h / l) * pow((D_m * (M_m / d)), 2.0))));
	} else if (l <= 2.2e+131) {
		tmp = (1.0 - (0.5 * ((h / l) * pow(((D_m / d) * (M_m / 2.0)), 2.0)))) * (d * pow((h * l), -0.5));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-4d-310)) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - (0.125d0 * ((h / l) * ((d_m * (m_m / d)) ** 2.0d0))))
    else if (l <= 2.2d+131) then
        tmp = (1.0d0 - (0.5d0 * ((h / l) * (((d_m / d) * (m_m / 2.0d0)) ** 2.0d0)))) * (d * ((h * l) ** (-0.5d0)))
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -4e-310) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (0.125 * ((h / l) * Math.pow((D_m * (M_m / d)), 2.0))));
	} else if (l <= 2.2e+131) {
		tmp = (1.0 - (0.5 * ((h / l) * Math.pow(((D_m / d) * (M_m / 2.0)), 2.0)))) * (d * Math.pow((h * l), -0.5));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -4e-310:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (0.125 * ((h / l) * math.pow((D_m * (M_m / d)), 2.0))))
	elif l <= 2.2e+131:
		tmp = (1.0 - (0.5 * ((h / l) * math.pow(((D_m / d) * (M_m / 2.0)), 2.0)))) * (d * math.pow((h * l), -0.5))
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -4e-310)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(0.125 * Float64(Float64(h / l) * (Float64(D_m * Float64(M_m / d)) ^ 2.0)))));
	elseif (l <= 2.2e+131)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(M_m / 2.0)) ^ 2.0)))) * Float64(d * (Float64(h * l) ^ -0.5)));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -4e-310)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.125 * ((h / l) * ((D_m * (M_m / d)) ^ 2.0))));
	elseif (l <= 2.2e+131)
		tmp = (1.0 - (0.5 * ((h / l) * (((D_m / d) * (M_m / 2.0)) ^ 2.0)))) * (d * ((h * l) ^ -0.5));
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -4e-310], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.125 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.2e+131], N[(N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(M$95$m / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.999999999999988e-310

   1. Initial program 57.2%

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

   3. Simplified 56.4%

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

   5. Taylor expanded in M around 0 43.1%

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

      1. associate-*r* 43.4%

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

      2. times-frac 40.1%

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

      3. associate-/l* 40.1%

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

      4. *-commutative 40.1%

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

      5. unpow2 40.1%

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

      6. unpow2 40.1%

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

      7. times-frac 47.6%

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

      8. unpow2 47.6%

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

      9. swap-sqr 57.2%

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

      10. associate-/r/ 56.4%

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

      11. associate-/r/ 56.3%

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

      12. unpow2 56.3%

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

      13. associate-*l* 56.3%

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

      14. *-commutative 56.3%

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

      15. associate-*l/ 59.8%

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

      16. associate-/l* 60.6%

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

      17. associate-/l* 60.6%

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

   7. Simplified 60.6%

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

   8. Taylor expanded in h around 0 43.1%

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

      1. associate-*r* 43.4%

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

      2. times-frac 40.1%

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

      3. *-commutative 40.1%

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

      4. associate-/l* 38.4%

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

      5. unpow2 38.4%

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

      6. unpow2 38.4%

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

      7. unpow2 38.4%

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

      8. times-frac 45.0%

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

      9. swap-sqr 56.4%

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

      10. unpow2 56.4%

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

      11. *-commutative 56.4%

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

      12. associate-*r/ 57.2%

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

      13. *-commutative 57.2%

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

      14. associate-/l* 57.2%

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

   10. Simplified 57.2%

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

   if -3.999999999999988e-310 < l < 2.1999999999999999e131

   1. Initial program 72.3%

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

   3. Simplified 73.4%

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

      1. add-cube-cbrt 73.0%

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

      2. pow3 73.1%

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

      3. sqrt-div 87.7%

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

      4. sqrt-div 90.3%

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

      5. frac-times 90.3%

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

      6. add-sqr-sqrt 90.4%

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

   6. Applied egg-rr 90.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}}\right)}^{3}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. rem-cube-cbrt 90.8%

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

      2. div-inv 90.7%

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

      3. metadata-eval 90.7%

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

      4. sqrt-unprod 80.1%

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

      5. sqrt-div 80.1%

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

      6. *-commutative 80.1%

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

      7. pow1/2 80.1%

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

      8. inv-pow 80.1%

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

      9. pow-pow 80.1%

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

      10. metadata-eval 80.1%

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

   8. Applied egg-rr 80.1%

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

   if 2.1999999999999999e131 < l

   1. Initial program 52.4%

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

   3. Simplified 54.6%

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

   5. Taylor expanded in d around inf 41.6%

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

      1. sqrt-div 41.5%

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

      2. metadata-eval 41.5%

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

      3. sqrt-unprod 62.6%

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

      4. div-inv 62.6%

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

      5. associate-/r* 56.5%

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

   7. Applied egg-rr 56.5%

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

      1. associate-/l/ 62.6%

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

   9. Simplified 62.6%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{+131}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.72 \cdot 10^{+130}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{+132}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \frac{M\_m}{2}\right)}^{2}\right)\right) \cdot \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -1.72e+130)
   (* (sqrt (/ d l)) (sqrt (/ d h)))
   (if (<= l -4e-310)
     (* (- d) (sqrt (/ (/ 1.0 h) l)))
     (if (<= l 9e+132)
       (*
        (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D_m d) (/ M_m 2.0)) 2.0))))
        (* d (pow (* h l) -0.5)))
       (/ d (* (sqrt h) (sqrt l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -1.72e+130) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (l <= -4e-310) {
		tmp = -d * sqrt(((1.0 / h) / l));
	} else if (l <= 9e+132) {
		tmp = (1.0 - (0.5 * ((h / l) * pow(((D_m / d) * (M_m / 2.0)), 2.0)))) * (d * pow((h * l), -0.5));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-1.72d+130)) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else if (l <= (-4d-310)) then
        tmp = -d * sqrt(((1.0d0 / h) / l))
    else if (l <= 9d+132) then
        tmp = (1.0d0 - (0.5d0 * ((h / l) * (((d_m / d) * (m_m / 2.0d0)) ** 2.0d0)))) * (d * ((h * l) ** (-0.5d0)))
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -1.72e+130) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else if (l <= -4e-310) {
		tmp = -d * Math.sqrt(((1.0 / h) / l));
	} else if (l <= 9e+132) {
		tmp = (1.0 - (0.5 * ((h / l) * Math.pow(((D_m / d) * (M_m / 2.0)), 2.0)))) * (d * Math.pow((h * l), -0.5));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -1.72e+130:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	elif l <= -4e-310:
		tmp = -d * math.sqrt(((1.0 / h) / l))
	elif l <= 9e+132:
		tmp = (1.0 - (0.5 * ((h / l) * math.pow(((D_m / d) * (M_m / 2.0)), 2.0)))) * (d * math.pow((h * l), -0.5))
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -1.72e+130)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (l <= -4e-310)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / h) / l)));
	elseif (l <= 9e+132)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(M_m / 2.0)) ^ 2.0)))) * Float64(d * (Float64(h * l) ^ -0.5)));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -1.72e+130)
		tmp = sqrt((d / l)) * sqrt((d / h));
	elseif (l <= -4e-310)
		tmp = -d * sqrt(((1.0 / h) / l));
	elseif (l <= 9e+132)
		tmp = (1.0 - (0.5 * ((h / l) * (((D_m / d) * (M_m / 2.0)) ^ 2.0)))) * (d * ((h * l) ^ -0.5));
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -1.72e+130], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -4e-310], N[((-d) * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 9e+132], N[(N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(M$95$m / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.72000000000000008e130

   1. Initial program 60.7%

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

   3. Simplified 60.7%

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

   5. Taylor expanded in d around inf 49.9%

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

   if -1.72000000000000008e130 < l < -3.999999999999988e-310

   1. Initial program 55.8%

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

   3. Simplified 54.6%

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

   5. Taylor expanded in M around 0 39.6%

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

      1. associate-*r* 40.1%

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

      2. times-frac 37.5%

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

      3. associate-/l* 37.5%

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

      4. *-commutative 37.5%

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

      5. unpow2 37.5%

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

      6. unpow2 37.5%

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

      7. times-frac 45.8%

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

      8. unpow2 45.8%

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

      9. swap-sqr 55.8%

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

      10. associate-/r/ 54.6%

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

      11. associate-/r/ 54.6%

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

      12. unpow2 54.6%

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

      13. associate-*l* 54.6%

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

      14. *-commutative 54.6%

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

      15. associate-*l/ 59.4%

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

      16. associate-/l* 59.4%

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

      17. associate-/l* 59.4%

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

   7. Simplified 59.4%

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

      1. add-sqr-sqrt 59.4%

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

      2. pow2 59.4%

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

      3. associate-*r* 59.4%

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

      4. associate-*r/ 59.4%

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

   9. Applied egg-rr 59.4%

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

   10. Taylor expanded in l around -inf 0.0%

       \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   11. 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. associate-/r* 0.0%

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

       3. *-commutative 0.0%

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

       4. unpow2 0.0%

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

       5. rem-square-sqrt 44.8%

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

       6. neg-mul-1 44.8%

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

   12. Simplified 44.8%

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

   if -3.999999999999988e-310 < l < 8.99999999999999944e132

   1. Initial program 72.3%

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

   3. Simplified 73.4%

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

      1. add-cube-cbrt 73.0%

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

      2. pow3 73.1%

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

      3. sqrt-div 87.7%

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

      4. sqrt-div 90.3%

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

      5. frac-times 90.3%

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

      6. add-sqr-sqrt 90.4%

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

   6. Applied egg-rr 90.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}}\right)}^{3}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. rem-cube-cbrt 90.8%

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

      2. div-inv 90.7%

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

      3. metadata-eval 90.7%

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

      4. sqrt-unprod 80.1%

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

      5. sqrt-div 80.1%

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

      6. *-commutative 80.1%

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

      7. pow1/2 80.1%

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

      8. inv-pow 80.1%

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

      9. pow-pow 80.1%

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

      10. metadata-eval 80.1%

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

   8. Applied egg-rr 80.1%

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

   if 8.99999999999999944e132 < l

   1. Initial program 52.4%

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

   3. Simplified 54.6%

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

   5. Taylor expanded in d around inf 41.6%

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

      1. sqrt-div 41.5%

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

      2. metadata-eval 41.5%

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

      3. sqrt-unprod 62.6%

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

      4. div-inv 62.6%

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

      5. associate-/r* 56.5%

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

   7. Applied egg-rr 56.5%

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

      1. associate-/l/ 62.6%

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

   9. Simplified 62.6%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.72 \cdot 10^{+130}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{+132}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.5 \cdot 10^{+127}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{+133}:\\ \;\;\;\;d \cdot \frac{1 + \frac{h}{\ell} \cdot \left(-0.125 \cdot {\left(M\_m \cdot \frac{D\_m}{d}\right)}^{2}\right)}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -6.5e+127)
   (* (sqrt (/ d l)) (sqrt (/ d h)))
   (if (<= l -4e-310)
     (* (- d) (sqrt (/ (/ 1.0 h) l)))
     (if (<= l 6.5e+133)
       (*
        d
        (/
         (+ 1.0 (* (/ h l) (* -0.125 (pow (* M_m (/ D_m d)) 2.0))))
         (sqrt (* h l))))
       (/ d (* (sqrt h) (sqrt l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -6.5e+127) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (l <= -4e-310) {
		tmp = -d * sqrt(((1.0 / h) / l));
	} else if (l <= 6.5e+133) {
		tmp = d * ((1.0 + ((h / l) * (-0.125 * pow((M_m * (D_m / d)), 2.0)))) / sqrt((h * l)));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-6.5d+127)) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else if (l <= (-4d-310)) then
        tmp = -d * sqrt(((1.0d0 / h) / l))
    else if (l <= 6.5d+133) then
        tmp = d * ((1.0d0 + ((h / l) * ((-0.125d0) * ((m_m * (d_m / d)) ** 2.0d0)))) / sqrt((h * l)))
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -6.5e+127) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else if (l <= -4e-310) {
		tmp = -d * Math.sqrt(((1.0 / h) / l));
	} else if (l <= 6.5e+133) {
		tmp = d * ((1.0 + ((h / l) * (-0.125 * Math.pow((M_m * (D_m / d)), 2.0)))) / Math.sqrt((h * l)));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -6.5e+127:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	elif l <= -4e-310:
		tmp = -d * math.sqrt(((1.0 / h) / l))
	elif l <= 6.5e+133:
		tmp = d * ((1.0 + ((h / l) * (-0.125 * math.pow((M_m * (D_m / d)), 2.0)))) / math.sqrt((h * l)))
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -6.5e+127)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (l <= -4e-310)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / h) / l)));
	elseif (l <= 6.5e+133)
		tmp = Float64(d * Float64(Float64(1.0 + Float64(Float64(h / l) * Float64(-0.125 * (Float64(M_m * Float64(D_m / d)) ^ 2.0)))) / sqrt(Float64(h * l))));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -6.5e+127)
		tmp = sqrt((d / l)) * sqrt((d / h));
	elseif (l <= -4e-310)
		tmp = -d * sqrt(((1.0 / h) / l));
	elseif (l <= 6.5e+133)
		tmp = d * ((1.0 + ((h / l) * (-0.125 * ((M_m * (D_m / d)) ^ 2.0)))) / sqrt((h * l)));
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -6.5e+127], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -4e-310], N[((-d) * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.5e+133], N[(d * N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.125 * N[Power[N[(M$95$m * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -6.5e127

   1. Initial program 60.7%

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

   3. Simplified 60.7%

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

   5. Taylor expanded in d around inf 49.9%

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

   if -6.5e127 < l < -3.999999999999988e-310

   1. Initial program 55.8%

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

   3. Simplified 54.6%

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

   5. Taylor expanded in M around 0 39.6%

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

      1. associate-*r* 40.1%

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

      2. times-frac 37.5%

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

      3. associate-/l* 37.5%

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

      4. *-commutative 37.5%

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

      5. unpow2 37.5%

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

      6. unpow2 37.5%

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

      7. times-frac 45.8%

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

      8. unpow2 45.8%

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

      9. swap-sqr 55.8%

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

      10. associate-/r/ 54.6%

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

      11. associate-/r/ 54.6%

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

      12. unpow2 54.6%

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

      13. associate-*l* 54.6%

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

      14. *-commutative 54.6%

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

      15. associate-*l/ 59.4%

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

      16. associate-/l* 59.4%

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

      17. associate-/l* 59.4%

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

   7. Simplified 59.4%

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

      1. add-sqr-sqrt 59.4%

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

      2. pow2 59.4%

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

      3. associate-*r* 59.4%

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

      4. associate-*r/ 59.4%

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

   9. Applied egg-rr 59.4%

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

   10. Taylor expanded in l around -inf 0.0%

       \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   11. 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. associate-/r* 0.0%

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

       3. *-commutative 0.0%

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

       4. unpow2 0.0%

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

       5. rem-square-sqrt 44.8%

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

       6. neg-mul-1 44.8%

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

   12. Simplified 44.8%

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

   if -3.999999999999988e-310 < l < 6.5000000000000004e133

   1. Initial program 72.3%

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

   3. Simplified 73.4%

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

      1. add-cube-cbrt 73.0%

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

      2. pow3 73.1%

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

      3. sqrt-div 87.7%

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

      4. sqrt-div 90.3%

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

      5. frac-times 90.3%

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

      6. add-sqr-sqrt 90.4%

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

   6. Applied egg-rr 90.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}}\right)}^{3}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. pow1 90.4%

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

      2. rem-cube-cbrt 90.8%

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

      3. sqrt-unprod 80.2%

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

      4. cancel-sign-sub-inv 80.2%

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

      5. metadata-eval 80.2%

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

      6. *-commutative 80.2%

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

      7. div-inv 80.2%

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

      8. metadata-eval 80.2%

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

   8. Applied egg-rr 80.2%

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

      1. unpow1 80.2%

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

      2. associate-*l/ 80.2%

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

      3. associate-/l* 80.1%

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

   10. Simplified 79.2%

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

       1. fma-undefine 79.2%

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

       2. *-commutative 79.2%

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

       3. associate-/r/ 79.3%

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

       4. div-inv 79.3%

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

       5. clear-num 80.1%

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

   12. Applied egg-rr 80.1%

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

   if 6.5000000000000004e133 < l

   1. Initial program 52.4%

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

   3. Simplified 54.6%

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

   5. Taylor expanded in d around inf 41.6%

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

      1. sqrt-div 41.5%

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

      2. metadata-eval 41.5%

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

      3. sqrt-unprod 62.6%

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

      4. div-inv 62.6%

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

      5. associate-/r* 56.5%

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

   7. Applied egg-rr 56.5%

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

      1. associate-/l/ 62.6%

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

   9. Simplified 62.6%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.5 \cdot 10^{+127}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{+133}:\\ \;\;\;\;d \cdot \frac{1 + \frac{h}{\ell} \cdot \left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right)}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 50.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -4.8 \cdot 10^{+203}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq 3.3 \cdot 10^{-289}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq 8 \cdot 10^{-45}:\\ \;\;\;\;d \cdot \frac{h \cdot \left({\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2} \cdot \frac{-0.125}{\ell}\right)}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -4.8e+203)
   (* (sqrt (/ d l)) (sqrt (/ d h)))
   (if (<= d 3.3e-289)
     (* d (- (pow (* h l) -0.5)))
     (if (<= d 8e-45)
       (*
        d
        (/ (* h (* (pow (* D_m (/ M_m d)) 2.0) (/ -0.125 l))) (sqrt (* h l))))
       (/ d (* (sqrt h) (sqrt l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -4.8e+203) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (d <= 3.3e-289) {
		tmp = d * -pow((h * l), -0.5);
	} else if (d <= 8e-45) {
		tmp = d * ((h * (pow((D_m * (M_m / d)), 2.0) * (-0.125 / l))) / sqrt((h * l)));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-4.8d+203)) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else if (d <= 3.3d-289) then
        tmp = d * -((h * l) ** (-0.5d0))
    else if (d <= 8d-45) then
        tmp = d * ((h * (((d_m * (m_m / d)) ** 2.0d0) * ((-0.125d0) / l))) / sqrt((h * l)))
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -4.8e+203) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else if (d <= 3.3e-289) {
		tmp = d * -Math.pow((h * l), -0.5);
	} else if (d <= 8e-45) {
		tmp = d * ((h * (Math.pow((D_m * (M_m / d)), 2.0) * (-0.125 / l))) / Math.sqrt((h * l)));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= -4.8e+203:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	elif d <= 3.3e-289:
		tmp = d * -math.pow((h * l), -0.5)
	elif d <= 8e-45:
		tmp = d * ((h * (math.pow((D_m * (M_m / d)), 2.0) * (-0.125 / l))) / math.sqrt((h * l)))
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -4.8e+203)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (d <= 3.3e-289)
		tmp = Float64(d * Float64(-(Float64(h * l) ^ -0.5)));
	elseif (d <= 8e-45)
		tmp = Float64(d * Float64(Float64(h * Float64((Float64(D_m * Float64(M_m / d)) ^ 2.0) * Float64(-0.125 / l))) / sqrt(Float64(h * l))));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= -4.8e+203)
		tmp = sqrt((d / l)) * sqrt((d / h));
	elseif (d <= 3.3e-289)
		tmp = d * -((h * l) ^ -0.5);
	elseif (d <= 8e-45)
		tmp = d * ((h * (((D_m * (M_m / d)) ^ 2.0) * (-0.125 / l))) / sqrt((h * l)));
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -4.8e+203], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.3e-289], N[(d * (-N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], If[LessEqual[d, 8e-45], N[(d * N[(N[(h * N[(N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.125 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -4.8000000000000002e203

   1. Initial program 86.4%

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

   3. Simplified 86.2%

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

   5. Taylor expanded in d around inf 82.0%

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

   if -4.8000000000000002e203 < d < 3.29999999999999997e-289

   1. Initial program 49.8%

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

   3. Simplified 48.9%

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

   5. Taylor expanded in d around inf 6.1%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. associate-*l* 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 35.3%

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

      4. mul-1-neg 35.3%

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

      5. unpow-1 35.3%

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

      6. metadata-eval 35.3%

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

      7. pow-sqr 35.3%

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

      8. rem-sqrt-square 35.3%

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

      9. rem-square-sqrt 35.1%

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

      10. fabs-sqr 35.1%

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

      11. rem-square-sqrt 35.3%

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

   8. Simplified 35.3%

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

   if 3.29999999999999997e-289 < d < 7.99999999999999987e-45

   1. Initial program 49.5%

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

   3. Simplified 49.5%

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

      1. add-cube-cbrt 49.1%

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

      2. pow3 49.1%

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

      3. sqrt-div 67.3%

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

      4. sqrt-div 73.6%

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

      5. frac-times 73.6%

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

      6. add-sqr-sqrt 73.7%

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

   6. Applied egg-rr 73.7%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}}\right)}^{3}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. pow1 73.7%

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

      2. rem-cube-cbrt 73.9%

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

      3. sqrt-unprod 66.9%

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

      4. cancel-sign-sub-inv 66.9%

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

      5. metadata-eval 66.9%

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

      6. *-commutative 66.9%

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

      7. div-inv 66.9%

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

      8. metadata-eval 66.9%

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

   8. Applied egg-rr 66.9%

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

      1. unpow1 66.9%

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

      2. associate-*l/ 68.6%

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

      3. associate-/l* 68.5%

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

   10. Simplified 68.5%

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

   11. Taylor expanded in D around inf 42.0%

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

   13. Simplified 54.1%

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

   if 7.99999999999999987e-45 < d

   1. Initial program 80.9%

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

   3. Simplified 83.7%

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

   5. Taylor expanded in d around inf 52.3%

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

      1. sqrt-div 52.3%

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

      2. metadata-eval 52.3%

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

      3. sqrt-unprod 73.3%

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

      4. div-inv 73.3%

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

      5. associate-/r* 69.4%

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

   7. Applied egg-rr 69.4%

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

      1. associate-/l/ 73.3%

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

   9. Simplified 73.3%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.8 \cdot 10^{+203}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq 3.3 \cdot 10^{-289}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq 8 \cdot 10^{-45}:\\ \;\;\;\;d \cdot \frac{h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{-0.125}{\ell}\right)}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 49.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -8.5 \cdot 10^{+203}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq 3.3 \cdot 10^{-289}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq 3.4 \cdot 10^{-44}:\\ \;\;\;\;d \cdot \frac{-0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2}\right)}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -8.5e+203)
   (* (sqrt (/ d l)) (sqrt (/ d h)))
   (if (<= d 3.3e-289)
     (* d (- (pow (* h l) -0.5)))
     (if (<= d 3.4e-44)
       (*
        d
        (/ (* -0.125 (* (/ h l) (pow (* D_m (/ M_m d)) 2.0))) (sqrt (* h l))))
       (/ d (* (sqrt h) (sqrt l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -8.5e+203) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (d <= 3.3e-289) {
		tmp = d * -pow((h * l), -0.5);
	} else if (d <= 3.4e-44) {
		tmp = d * ((-0.125 * ((h / l) * pow((D_m * (M_m / d)), 2.0))) / sqrt((h * l)));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-8.5d+203)) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else if (d <= 3.3d-289) then
        tmp = d * -((h * l) ** (-0.5d0))
    else if (d <= 3.4d-44) then
        tmp = d * (((-0.125d0) * ((h / l) * ((d_m * (m_m / d)) ** 2.0d0))) / sqrt((h * l)))
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -8.5e+203) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else if (d <= 3.3e-289) {
		tmp = d * -Math.pow((h * l), -0.5);
	} else if (d <= 3.4e-44) {
		tmp = d * ((-0.125 * ((h / l) * Math.pow((D_m * (M_m / d)), 2.0))) / Math.sqrt((h * l)));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= -8.5e+203:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	elif d <= 3.3e-289:
		tmp = d * -math.pow((h * l), -0.5)
	elif d <= 3.4e-44:
		tmp = d * ((-0.125 * ((h / l) * math.pow((D_m * (M_m / d)), 2.0))) / math.sqrt((h * l)))
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -8.5e+203)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (d <= 3.3e-289)
		tmp = Float64(d * Float64(-(Float64(h * l) ^ -0.5)));
	elseif (d <= 3.4e-44)
		tmp = Float64(d * Float64(Float64(-0.125 * Float64(Float64(h / l) * (Float64(D_m * Float64(M_m / d)) ^ 2.0))) / sqrt(Float64(h * l))));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= -8.5e+203)
		tmp = sqrt((d / l)) * sqrt((d / h));
	elseif (d <= 3.3e-289)
		tmp = d * -((h * l) ^ -0.5);
	elseif (d <= 3.4e-44)
		tmp = d * ((-0.125 * ((h / l) * ((D_m * (M_m / d)) ^ 2.0))) / sqrt((h * l)));
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -8.5e+203], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.3e-289], N[(d * (-N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], If[LessEqual[d, 3.4e-44], N[(d * N[(N[(-0.125 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -8.50000000000000025e203

   1. Initial program 86.4%

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

   3. Simplified 86.2%

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

   5. Taylor expanded in d around inf 82.0%

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

   if -8.50000000000000025e203 < d < 3.29999999999999997e-289

   1. Initial program 49.8%

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

   3. Simplified 48.9%

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

   5. Taylor expanded in d around inf 6.1%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. associate-*l* 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 35.3%

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

      4. mul-1-neg 35.3%

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

      5. unpow-1 35.3%

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

      6. metadata-eval 35.3%

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

      7. pow-sqr 35.3%

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

      8. rem-sqrt-square 35.3%

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

      9. rem-square-sqrt 35.1%

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

      10. fabs-sqr 35.1%

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

      11. rem-square-sqrt 35.3%

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

   8. Simplified 35.3%

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

   if 3.29999999999999997e-289 < d < 3.40000000000000016e-44

   1. Initial program 49.5%

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

   3. Simplified 49.5%

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

      1. add-cube-cbrt 49.1%

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

      2. pow3 49.1%

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

      3. sqrt-div 67.3%

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

      4. sqrt-div 73.6%

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

      5. frac-times 73.6%

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

      6. add-sqr-sqrt 73.7%

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

   6. Applied egg-rr 73.7%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}}\right)}^{3}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. pow1 73.7%

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

      2. rem-cube-cbrt 73.9%

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

      3. sqrt-unprod 66.9%

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

      4. cancel-sign-sub-inv 66.9%

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

      5. metadata-eval 66.9%

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

      6. *-commutative 66.9%

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

      7. div-inv 66.9%

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

      8. metadata-eval 66.9%

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

   8. Applied egg-rr 66.9%

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

      1. unpow1 66.9%

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

      2. associate-*l/ 68.6%

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

      3. associate-/l* 68.5%

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

   10. Simplified 68.5%

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

       1. *-commutative 68.5%

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

       2. associate-/r/ 68.6%

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

       3. div-inv 68.6%

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

       4. clear-num 68.5%

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

       5. associate-*r/ 68.5%

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

   12. Applied egg-rr 68.5%

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

   13. Taylor expanded in M around inf 42.0%

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

       1. associate-/l* 42.0%

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

       2. times-frac 42.1%

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

       3. associate-*r* 42.0%

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

       4. *-commutative 42.0%

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

       5. unpow2 42.0%

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

       6. unpow2 42.0%

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

       7. times-frac 48.6%

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

       8. unpow2 48.6%

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

       9. swap-sqr 52.2%

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

       10. unpow2 52.2%

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

       11. associate-*l/ 52.2%

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

       12. associate-*r/ 52.2%

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

       13. *-commutative 52.2%

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

       14. associate-*r/ 52.2%

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

       15. associate-*l/ 52.2%

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

       16. *-commutative 52.2%

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

   15. Simplified 52.2%

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

   if 3.40000000000000016e-44 < d

   1. Initial program 80.9%

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

   3. Simplified 83.7%

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

   5. Taylor expanded in d around inf 52.3%

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

      1. sqrt-div 52.3%

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

      2. metadata-eval 52.3%

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

      3. sqrt-unprod 73.3%

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

      4. div-inv 73.3%

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

      5. associate-/r* 69.4%

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

   7. Applied egg-rr 69.4%

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

      1. associate-/l/ 73.3%

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

   9. Simplified 73.3%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8.5 \cdot 10^{+203}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq 3.3 \cdot 10^{-289}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq 3.4 \cdot 10^{-44}:\\ \;\;\;\;d \cdot \frac{-0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 64.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{-308}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{{\left(M\_m \cdot \frac{D\_m}{d}\right)}^{2}}{\ell}\right)\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{+133}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \frac{M\_m}{2}\right)}^{2}\right)\right) \cdot \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l 2e-308)
   (*
    (- 1.0 (* 0.5 (* h (* 0.25 (/ (pow (* M_m (/ D_m d)) 2.0) l)))))
    (sqrt (* (/ d l) (/ d h))))
   (if (<= l 3.8e+133)
     (*
      (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D_m d) (/ M_m 2.0)) 2.0))))
      (* d (pow (* h l) -0.5)))
     (/ d (* (sqrt h) (sqrt l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 2e-308) {
		tmp = (1.0 - (0.5 * (h * (0.25 * (pow((M_m * (D_m / d)), 2.0) / l))))) * sqrt(((d / l) * (d / h)));
	} else if (l <= 3.8e+133) {
		tmp = (1.0 - (0.5 * ((h / l) * pow(((D_m / d) * (M_m / 2.0)), 2.0)))) * (d * pow((h * l), -0.5));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= 2d-308) then
        tmp = (1.0d0 - (0.5d0 * (h * (0.25d0 * (((m_m * (d_m / d)) ** 2.0d0) / l))))) * sqrt(((d / l) * (d / h)))
    else if (l <= 3.8d+133) then
        tmp = (1.0d0 - (0.5d0 * ((h / l) * (((d_m / d) * (m_m / 2.0d0)) ** 2.0d0)))) * (d * ((h * l) ** (-0.5d0)))
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 2e-308) {
		tmp = (1.0 - (0.5 * (h * (0.25 * (Math.pow((M_m * (D_m / d)), 2.0) / l))))) * Math.sqrt(((d / l) * (d / h)));
	} else if (l <= 3.8e+133) {
		tmp = (1.0 - (0.5 * ((h / l) * Math.pow(((D_m / d) * (M_m / 2.0)), 2.0)))) * (d * Math.pow((h * l), -0.5));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= 2e-308:
		tmp = (1.0 - (0.5 * (h * (0.25 * (math.pow((M_m * (D_m / d)), 2.0) / l))))) * math.sqrt(((d / l) * (d / h)))
	elif l <= 3.8e+133:
		tmp = (1.0 - (0.5 * ((h / l) * math.pow(((D_m / d) * (M_m / 2.0)), 2.0)))) * (d * math.pow((h * l), -0.5))
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= 2e-308)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(h * Float64(0.25 * Float64((Float64(M_m * Float64(D_m / d)) ^ 2.0) / l))))) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	elseif (l <= 3.8e+133)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(M_m / 2.0)) ^ 2.0)))) * Float64(d * (Float64(h * l) ^ -0.5)));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= 2e-308)
		tmp = (1.0 - (0.5 * (h * (0.25 * (((M_m * (D_m / d)) ^ 2.0) / l))))) * sqrt(((d / l) * (d / h)));
	elseif (l <= 3.8e+133)
		tmp = (1.0 - (0.5 * ((h / l) * (((D_m / d) * (M_m / 2.0)) ^ 2.0)))) * (d * ((h * l) ^ -0.5));
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, 2e-308], N[(N[(1.0 - N[(0.5 * N[(h * N[(0.25 * N[(N[Power[N[(M$95$m * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.8e+133], N[(N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(M$95$m / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.9999999999999998e-308

   1. Initial program 57.2%

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

   3. Simplified 56.4%

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

   5. Taylor expanded in M around 0 43.1%

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

      1. associate-*r* 43.4%

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

      2. times-frac 40.1%

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

      3. associate-/l* 40.1%

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

      4. *-commutative 40.1%

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

      5. unpow2 40.1%

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

      6. unpow2 40.1%

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

      7. times-frac 47.6%

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

      8. unpow2 47.6%

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

      9. swap-sqr 57.2%

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

      10. associate-/r/ 56.4%

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

      11. associate-/r/ 56.3%

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

      12. unpow2 56.3%

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

      13. associate-*l* 56.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      14. *-commutative 56.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot \left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right)\right)}\right) \]

      15. associate-*l/ 59.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{h \cdot \left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right)}{\ell}}\right) \]

      16. associate-/l* 60.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}}{\ell}\right)}\right) \]

      17. associate-/l* 60.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \color{blue}{\left(0.25 \cdot \frac{{\left(\frac{M}{\frac{d}{D}}\right)}^{2}}{\ell}\right)}\right)\right) \]

   7. Simplified 60.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative 60.6%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\right) \]

      2. sqrt-unprod 48.1%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\right) \]

   9. Applied egg-rr 48.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\right) \]

   if 1.9999999999999998e-308 < l < 3.8000000000000002e133

   1. Initial program 72.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 73.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cube-cbrt 73.0%

         \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \cdot \sqrt[3]{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}\right) \cdot \sqrt[3]{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. pow3 73.1%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}\right)}^{3}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. sqrt-div 87.7%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \sqrt{\frac{d}{\ell}}}\right)}^{3} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. sqrt-div 90.3%

         \[\leadsto {\left(\sqrt[3]{\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}}\right)}^{3} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. frac-times 90.3%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h} \cdot \sqrt{\ell}}}}\right)}^{3} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      6. add-sqr-sqrt 90.4%

         \[\leadsto {\left(\sqrt[3]{\frac{\color{blue}{d}}{\sqrt{h} \cdot \sqrt{\ell}}}\right)}^{3} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Applied egg-rr 90.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}}\right)}^{3}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. rem-cube-cbrt 90.8%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. div-inv 90.7%

         \[\leadsto \color{blue}{\left(d \cdot \frac{1}{\sqrt{h} \cdot \sqrt{\ell}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. metadata-eval 90.7%

         \[\leadsto \left(d \cdot \frac{\color{blue}{\sqrt{1}}}{\sqrt{h} \cdot \sqrt{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. sqrt-unprod 80.1%

         \[\leadsto \left(d \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{h \cdot \ell}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. sqrt-div 80.1%

         \[\leadsto \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      6. *-commutative 80.1%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      7. pow1/2 80.1%

         \[\leadsto \left(\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}} \cdot d\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. inv-pow 80.1%

         \[\leadsto \left({\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5} \cdot d\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      9. pow-pow 80.1%

         \[\leadsto \left(\color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 0.5\right)}} \cdot d\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      10. metadata-eval 80.1%

          \[\leadsto \left({\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. Applied egg-rr 80.1%

      \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   if 3.8000000000000002e133 < l

   1. Initial program 52.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 54.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 41.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. sqrt-div 41.5%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      2. metadata-eval 41.5%

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

      3. sqrt-unprod 62.6%

         \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      4. div-inv 62.6%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      5. associate-/r* 56.5%

         \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}} \]

   7. Applied egg-rr 56.5%

      \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}} \]
   8. Step-by-step derivation

      1. associate-/l/ 62.6%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]

   9. Simplified 62.6%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{-308}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{+133}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 64.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 9 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{{\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{+131}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \frac{M\_m}{2}\right)}^{2}\right)\right) \cdot \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l 9e-309)
   (*
    (sqrt (* (/ d l) (/ d h)))
    (- 1.0 (* 0.5 (* h (* 0.25 (/ (pow (* D_m (/ M_m d)) 2.0) l))))))
   (if (<= l 1.1e+131)
     (*
      (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D_m d) (/ M_m 2.0)) 2.0))))
      (* d (pow (* h l) -0.5)))
     (/ d (* (sqrt h) (sqrt l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 9e-309) {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - (0.5 * (h * (0.25 * (pow((D_m * (M_m / d)), 2.0) / l)))));
	} else if (l <= 1.1e+131) {
		tmp = (1.0 - (0.5 * ((h / l) * pow(((D_m / d) * (M_m / 2.0)), 2.0)))) * (d * pow((h * l), -0.5));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= 9d-309) then
        tmp = sqrt(((d / l) * (d / h))) * (1.0d0 - (0.5d0 * (h * (0.25d0 * (((d_m * (m_m / d)) ** 2.0d0) / l)))))
    else if (l <= 1.1d+131) then
        tmp = (1.0d0 - (0.5d0 * ((h / l) * (((d_m / d) * (m_m / 2.0d0)) ** 2.0d0)))) * (d * ((h * l) ** (-0.5d0)))
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 9e-309) {
		tmp = Math.sqrt(((d / l) * (d / h))) * (1.0 - (0.5 * (h * (0.25 * (Math.pow((D_m * (M_m / d)), 2.0) / l)))));
	} else if (l <= 1.1e+131) {
		tmp = (1.0 - (0.5 * ((h / l) * Math.pow(((D_m / d) * (M_m / 2.0)), 2.0)))) * (d * Math.pow((h * l), -0.5));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= 9e-309:
		tmp = math.sqrt(((d / l) * (d / h))) * (1.0 - (0.5 * (h * (0.25 * (math.pow((D_m * (M_m / d)), 2.0) / l)))))
	elif l <= 1.1e+131:
		tmp = (1.0 - (0.5 * ((h / l) * math.pow(((D_m / d) * (M_m / 2.0)), 2.0)))) * (d * math.pow((h * l), -0.5))
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= 9e-309)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 - Float64(0.5 * Float64(h * Float64(0.25 * Float64((Float64(D_m * Float64(M_m / d)) ^ 2.0) / l))))));
	elseif (l <= 1.1e+131)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(M_m / 2.0)) ^ 2.0)))) * Float64(d * (Float64(h * l) ^ -0.5)));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= 9e-309)
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - (0.5 * (h * (0.25 * (((D_m * (M_m / d)) ^ 2.0) / l)))));
	elseif (l <= 1.1e+131)
		tmp = (1.0 - (0.5 * ((h / l) * (((D_m / d) * (M_m / 2.0)) ^ 2.0)))) * (d * ((h * l) ^ -0.5));
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, 9e-309], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(h * N[(0.25 * N[(N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.1e+131], N[(N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(M$95$m / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 9.0000000000000021e-309

   1. Initial program 57.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 56.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in M around 0 43.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r* 43.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell}\right)\right) \]

      2. times-frac 40.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \frac{h}{\ell}\right)}\right)\right) \]

      3. associate-/l* 40.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\color{blue}{\left({D}^{2} \cdot \frac{{M}^{2}}{{d}^{2}}\right)} \cdot \frac{h}{\ell}\right)\right)\right) \]

      4. *-commutative 40.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\color{blue}{\left(\frac{{M}^{2}}{{d}^{2}} \cdot {D}^{2}\right)} \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow2 40.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\frac{\color{blue}{M \cdot M}}{{d}^{2}} \cdot {D}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. unpow2 40.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\frac{M \cdot M}{\color{blue}{d \cdot d}} \cdot {D}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. times-frac 47.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\color{blue}{\left(\frac{M}{d} \cdot \frac{M}{d}\right)} \cdot {D}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. unpow2 47.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\left(\frac{M}{d} \cdot \frac{M}{d}\right) \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      9. swap-sqr 57.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\color{blue}{\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)\right)} \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. associate-/r/ 56.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\color{blue}{\frac{M}{\frac{d}{D}}} \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. associate-/r/ 56.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\frac{M}{\frac{d}{D}} \cdot \color{blue}{\frac{M}{\frac{d}{D}}}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. unpow2 56.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\color{blue}{{\left(\frac{M}{\frac{d}{D}}\right)}^{2}} \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. associate-*l* 56.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      14. *-commutative 56.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot \left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right)\right)}\right) \]

      15. associate-*l/ 59.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{h \cdot \left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right)}{\ell}}\right) \]

      16. associate-/l* 60.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}}{\ell}\right)}\right) \]

      17. associate-/l* 60.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \color{blue}{\left(0.25 \cdot \frac{{\left(\frac{M}{\frac{d}{D}}\right)}^{2}}{\ell}\right)}\right)\right) \]

   7. Simplified 60.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. pow1 60.6%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\right)\right)}^{1}} \]

      2. sqrt-unprod 48.1%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\right)\right)}^{1} \]

      3. associate-*r* 48.1%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\left(0.5 \cdot h\right) \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)}\right)\right)}^{1} \]

      4. associate-*r/ 48.1%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(0.5 \cdot h\right) \cdot \color{blue}{\frac{0.25 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}}\right)\right)}^{1} \]

   9. Applied egg-rr 48.1%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(0.5 \cdot h\right) \cdot \frac{0.25 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)}^{1}} \]
   10. Step-by-step derivation

       1. unpow1 48.1%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(0.5 \cdot h\right) \cdot \frac{0.25 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)} \]

       2. *-commutative 48.1%

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(0.5 \cdot h\right) \cdot \frac{0.25 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right) \]

       3. associate-*l* 48.1%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \color{blue}{0.5 \cdot \left(h \cdot \frac{0.25 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)}\right) \]

       4. associate-/l* 48.1%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \color{blue}{\left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)}\right)\right) \]

       5. associate-*r/ 49.7%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}}{\ell}\right)\right)\right) \]

       6. *-commutative 49.7%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{{\left(\frac{\color{blue}{D \cdot M}}{d}\right)}^{2}}{\ell}\right)\right)\right) \]

       7. associate-/l* 49.7%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{{\color{blue}{\left(D \cdot \frac{M}{d}\right)}}^{2}}{\ell}\right)\right)\right) \]

   11. Simplified 49.7%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right)\right)\right)} \]

   if 9.0000000000000021e-309 < l < 1.0999999999999999e131

   1. Initial program 72.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 73.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cube-cbrt 73.0%

         \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \cdot \sqrt[3]{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}\right) \cdot \sqrt[3]{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. pow3 73.1%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}\right)}^{3}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. sqrt-div 87.7%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \sqrt{\frac{d}{\ell}}}\right)}^{3} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. sqrt-div 90.3%

         \[\leadsto {\left(\sqrt[3]{\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}}\right)}^{3} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. frac-times 90.3%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h} \cdot \sqrt{\ell}}}}\right)}^{3} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      6. add-sqr-sqrt 90.4%

         \[\leadsto {\left(\sqrt[3]{\frac{\color{blue}{d}}{\sqrt{h} \cdot \sqrt{\ell}}}\right)}^{3} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Applied egg-rr 90.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}}\right)}^{3}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. rem-cube-cbrt 90.8%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. div-inv 90.7%

         \[\leadsto \color{blue}{\left(d \cdot \frac{1}{\sqrt{h} \cdot \sqrt{\ell}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. metadata-eval 90.7%

         \[\leadsto \left(d \cdot \frac{\color{blue}{\sqrt{1}}}{\sqrt{h} \cdot \sqrt{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. sqrt-unprod 80.1%

         \[\leadsto \left(d \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{h \cdot \ell}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. sqrt-div 80.1%

         \[\leadsto \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      6. *-commutative 80.1%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      7. pow1/2 80.1%

         \[\leadsto \left(\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}} \cdot d\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. inv-pow 80.1%

         \[\leadsto \left({\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5} \cdot d\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      9. pow-pow 80.1%

         \[\leadsto \left(\color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 0.5\right)}} \cdot d\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      10. metadata-eval 80.1%

          \[\leadsto \left({\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. Applied egg-rr 80.1%

      \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   if 1.0999999999999999e131 < l

   1. Initial program 52.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 54.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 41.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. sqrt-div 41.5%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      2. metadata-eval 41.5%

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

      3. sqrt-unprod 62.6%

         \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      4. div-inv 62.6%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      5. associate-/r* 56.5%

         \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}} \]

   7. Applied egg-rr 56.5%

      \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}} \]
   8. Step-by-step derivation

      1. associate-/l/ 62.6%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]

   9. Simplified 62.6%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 9 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{+131}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 45.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.05 \cdot 10^{+127}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{-294}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -1.05e+127)
   (* (sqrt (/ d l)) (sqrt (/ d h)))
   (if (<= l 6.5e-294)
     (* (- d) (sqrt (/ (/ 1.0 h) l)))
     (/ d (* (sqrt h) (sqrt l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -1.05e+127) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (l <= 6.5e-294) {
		tmp = -d * sqrt(((1.0 / h) / l));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-1.05d+127)) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else if (l <= 6.5d-294) then
        tmp = -d * sqrt(((1.0d0 / h) / l))
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -1.05e+127) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else if (l <= 6.5e-294) {
		tmp = -d * Math.sqrt(((1.0 / h) / l));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -1.05e+127:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	elif l <= 6.5e-294:
		tmp = -d * math.sqrt(((1.0 / h) / l))
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -1.05e+127)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (l <= 6.5e-294)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / h) / l)));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -1.05e+127)
		tmp = sqrt((d / l)) * sqrt((d / h));
	elseif (l <= 6.5e-294)
		tmp = -d * sqrt(((1.0 / h) / l));
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -1.05e+127], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.5e-294], N[((-d) * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.04999999999999996e127

   1. Initial program 60.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 60.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 49.9%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

   if -1.04999999999999996e127 < l < 6.4999999999999995e-294

   1. Initial program 55.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 53.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in M around 0 39.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r* 39.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell}\right)\right) \]

      2. times-frac 37.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \frac{h}{\ell}\right)}\right)\right) \]

      3. associate-/l* 37.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\color{blue}{\left({D}^{2} \cdot \frac{{M}^{2}}{{d}^{2}}\right)} \cdot \frac{h}{\ell}\right)\right)\right) \]

      4. *-commutative 37.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\color{blue}{\left(\frac{{M}^{2}}{{d}^{2}} \cdot {D}^{2}\right)} \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow2 37.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\frac{\color{blue}{M \cdot M}}{{d}^{2}} \cdot {D}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. unpow2 37.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\frac{M \cdot M}{\color{blue}{d \cdot d}} \cdot {D}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. times-frac 45.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\color{blue}{\left(\frac{M}{d} \cdot \frac{M}{d}\right)} \cdot {D}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. unpow2 45.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\left(\frac{M}{d} \cdot \frac{M}{d}\right) \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      9. swap-sqr 55.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\color{blue}{\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)\right)} \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. associate-/r/ 53.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\color{blue}{\frac{M}{\frac{d}{D}}} \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. associate-/r/ 53.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\frac{M}{\frac{d}{D}} \cdot \color{blue}{\frac{M}{\frac{d}{D}}}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. unpow2 53.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\color{blue}{{\left(\frac{M}{\frac{d}{D}}\right)}^{2}} \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. associate-*l* 53.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      14. *-commutative 53.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot \left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right)\right)}\right) \]

      15. associate-*l/ 58.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{h \cdot \left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right)}{\ell}}\right) \]

      16. associate-/l* 58.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}}{\ell}\right)}\right) \]

      17. associate-/l* 58.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \color{blue}{\left(0.25 \cdot \frac{{\left(\frac{M}{\frac{d}{D}}\right)}^{2}}{\ell}\right)}\right)\right) \]

   7. Simplified 58.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 58.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\sqrt{0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)} \cdot \sqrt{0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)}}\right) \]

      2. pow2 58.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\sqrt{0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)}\right)}^{2}}\right) \]

      3. associate-*r* 58.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\sqrt{\color{blue}{\left(0.5 \cdot h\right) \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)}}\right)}^{2}\right) \]

      4. associate-*r/ 58.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\sqrt{\left(0.5 \cdot h\right) \cdot \color{blue}{\frac{0.25 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}}}\right)}^{2}\right) \]

   9. Applied egg-rr 58.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\sqrt{\left(0.5 \cdot h\right) \cdot \frac{0.25 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}}\right)}^{2}}\right) \]

   10. Taylor expanded in l around -inf 0.0%

       \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   11. 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. associate-/r* 0.0%

          \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

       3. *-commutative 0.0%

          \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

       4. unpow2 0.0%

          \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

       5. rem-square-sqrt 44.5%

          \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(\color{blue}{-1} \cdot d\right) \]

       6. neg-mul-1 44.5%

          \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \color{blue}{\left(-d\right)} \]

   12. Simplified 44.5%

       \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-d\right)} \]

   if 6.4999999999999995e-294 < l

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 67.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 39.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. sqrt-div 39.9%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      2. metadata-eval 39.9%

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

      3. sqrt-unprod 53.6%

         \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      4. div-inv 53.6%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      5. associate-/r* 51.5%

         \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}} \]

   7. Applied egg-rr 51.5%

      \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}} \]
   8. Step-by-step derivation

      1. associate-/l/ 53.6%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]

   9. Simplified 53.6%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.05 \cdot 10^{+127}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{-294}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 46.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 1.3 \cdot 10^{-270}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d 1.3e-270)
   (* (- d) (sqrt (/ (/ 1.0 h) l)))
   (/ d (* (sqrt h) (sqrt l)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= 1.3e-270) {
		tmp = -d * sqrt(((1.0 / h) / l));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= 1.3d-270) then
        tmp = -d * sqrt(((1.0d0 / h) / l))
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= 1.3e-270) {
		tmp = -d * Math.sqrt(((1.0 / h) / l));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= 1.3e-270:
		tmp = -d * math.sqrt(((1.0 / h) / l))
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= 1.3e-270)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / h) / l)));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= 1.3e-270)
		tmp = -d * sqrt(((1.0 / h) / l));
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, 1.3e-270], N[((-d) * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.3000000000000001e-270

   1. Initial program 56.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 56.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in M around 0 41.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r* 41.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell}\right)\right) \]

      2. times-frac 38.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \frac{h}{\ell}\right)}\right)\right) \]

      3. associate-/l* 38.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\color{blue}{\left({D}^{2} \cdot \frac{{M}^{2}}{{d}^{2}}\right)} \cdot \frac{h}{\ell}\right)\right)\right) \]

      4. *-commutative 38.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\color{blue}{\left(\frac{{M}^{2}}{{d}^{2}} \cdot {D}^{2}\right)} \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow2 38.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\frac{\color{blue}{M \cdot M}}{{d}^{2}} \cdot {D}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. unpow2 38.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\frac{M \cdot M}{\color{blue}{d \cdot d}} \cdot {D}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. times-frac 45.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\color{blue}{\left(\frac{M}{d} \cdot \frac{M}{d}\right)} \cdot {D}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. unpow2 45.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\left(\frac{M}{d} \cdot \frac{M}{d}\right) \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      9. swap-sqr 56.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\color{blue}{\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)\right)} \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. associate-/r/ 55.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\color{blue}{\frac{M}{\frac{d}{D}}} \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. associate-/r/ 55.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\frac{M}{\frac{d}{D}} \cdot \color{blue}{\frac{M}{\frac{d}{D}}}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. unpow2 55.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\color{blue}{{\left(\frac{M}{\frac{d}{D}}\right)}^{2}} \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. associate-*l* 55.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      14. *-commutative 55.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot \left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right)\right)}\right) \]

      15. associate-*l/ 58.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{h \cdot \left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right)}{\ell}}\right) \]

      16. associate-/l* 59.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}}{\ell}\right)}\right) \]

      17. associate-/l* 59.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \color{blue}{\left(0.25 \cdot \frac{{\left(\frac{M}{\frac{d}{D}}\right)}^{2}}{\ell}\right)}\right)\right) \]

   7. Simplified 60.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 59.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\sqrt{0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)} \cdot \sqrt{0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)}}\right) \]

      2. pow2 59.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\sqrt{0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)}\right)}^{2}}\right) \]

      3. associate-*r* 59.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\sqrt{\color{blue}{\left(0.5 \cdot h\right) \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)}}\right)}^{2}\right) \]

      4. associate-*r/ 59.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\sqrt{\left(0.5 \cdot h\right) \cdot \color{blue}{\frac{0.25 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}}}\right)}^{2}\right) \]

   9. Applied egg-rr 59.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\sqrt{\left(0.5 \cdot h\right) \cdot \frac{0.25 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}}\right)}^{2}}\right) \]

   10. Taylor expanded in l around -inf 0.0%

       \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   11. 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. associate-/r* 0.0%

          \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

       3. *-commutative 0.0%

          \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

       4. unpow2 0.0%

          \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

       5. rem-square-sqrt 39.1%

          \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(\color{blue}{-1} \cdot d\right) \]

       6. neg-mul-1 39.1%

          \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \color{blue}{\left(-d\right)} \]

   12. Simplified 39.1%

       \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-d\right)} \]

   if 1.3000000000000001e-270 < d

   1. Initial program 66.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 68.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 40.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. sqrt-div 40.7%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      2. metadata-eval 40.7%

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

      3. sqrt-unprod 54.9%

         \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      4. div-inv 55.0%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      5. associate-/r* 52.8%

         \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}} \]

   7. Applied egg-rr 52.8%

      \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}} \]
   8. Step-by-step derivation

      1. associate-/l/ 55.0%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]

   9. Simplified 55.0%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.3 \cdot 10^{-270}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 43.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{if}\;d \leq 1.8 \cdot 10^{-270}:\\ \;\;\;\;\left(-d\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot t\_0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ (/ 1.0 h) l))))
   (if (<= d 1.8e-270) (* (- d) t_0) (* d t_0))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt(((1.0 / h) / l));
	double tmp;
	if (d <= 1.8e-270) {
		tmp = -d * t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((1.0d0 / h) / l))
    if (d <= 1.8d-270) then
        tmp = -d * t_0
    else
        tmp = d * t_0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt(((1.0 / h) / l));
	double tmp;
	if (d <= 1.8e-270) {
		tmp = -d * t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt(((1.0 / h) / l))
	tmp = 0
	if d <= 1.8e-270:
		tmp = -d * t_0
	else:
		tmp = d * t_0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(Float64(1.0 / h) / l))
	tmp = 0.0
	if (d <= 1.8e-270)
		tmp = Float64(Float64(-d) * t_0);
	else
		tmp = Float64(d * t_0);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt(((1.0 / h) / l));
	tmp = 0.0;
	if (d <= 1.8e-270)
		tmp = -d * t_0;
	else
		tmp = d * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, 1.8e-270], N[((-d) * t$95$0), $MachinePrecision], N[(d * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < 1.7999999999999999e-270

   1. Initial program 56.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 56.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in M around 0 41.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r* 41.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell}\right)\right) \]

      2. times-frac 38.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \frac{h}{\ell}\right)}\right)\right) \]

      3. associate-/l* 38.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\color{blue}{\left({D}^{2} \cdot \frac{{M}^{2}}{{d}^{2}}\right)} \cdot \frac{h}{\ell}\right)\right)\right) \]

      4. *-commutative 38.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\color{blue}{\left(\frac{{M}^{2}}{{d}^{2}} \cdot {D}^{2}\right)} \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow2 38.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\frac{\color{blue}{M \cdot M}}{{d}^{2}} \cdot {D}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. unpow2 38.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\frac{M \cdot M}{\color{blue}{d \cdot d}} \cdot {D}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. times-frac 45.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\color{blue}{\left(\frac{M}{d} \cdot \frac{M}{d}\right)} \cdot {D}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. unpow2 45.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\left(\frac{M}{d} \cdot \frac{M}{d}\right) \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      9. swap-sqr 56.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\color{blue}{\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)\right)} \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. associate-/r/ 55.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\color{blue}{\frac{M}{\frac{d}{D}}} \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. associate-/r/ 55.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\frac{M}{\frac{d}{D}} \cdot \color{blue}{\frac{M}{\frac{d}{D}}}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. unpow2 55.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\color{blue}{{\left(\frac{M}{\frac{d}{D}}\right)}^{2}} \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. associate-*l* 55.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      14. *-commutative 55.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot \left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right)\right)}\right) \]

      15. associate-*l/ 58.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{h \cdot \left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right)}{\ell}}\right) \]

      16. associate-/l* 59.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}}{\ell}\right)}\right) \]

      17. associate-/l* 59.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \color{blue}{\left(0.25 \cdot \frac{{\left(\frac{M}{\frac{d}{D}}\right)}^{2}}{\ell}\right)}\right)\right) \]

   7. Simplified 60.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 59.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\sqrt{0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)} \cdot \sqrt{0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)}}\right) \]

      2. pow2 59.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\sqrt{0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)}\right)}^{2}}\right) \]

      3. associate-*r* 59.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\sqrt{\color{blue}{\left(0.5 \cdot h\right) \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)}}\right)}^{2}\right) \]

      4. associate-*r/ 59.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\sqrt{\left(0.5 \cdot h\right) \cdot \color{blue}{\frac{0.25 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}}}\right)}^{2}\right) \]

   9. Applied egg-rr 59.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\sqrt{\left(0.5 \cdot h\right) \cdot \frac{0.25 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}}\right)}^{2}}\right) \]

   10. Taylor expanded in l around -inf 0.0%

       \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   11. 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. associate-/r* 0.0%

          \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

       3. *-commutative 0.0%

          \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

       4. unpow2 0.0%

          \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

       5. rem-square-sqrt 39.1%

          \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(\color{blue}{-1} \cdot d\right) \]

       6. neg-mul-1 39.1%

          \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \color{blue}{\left(-d\right)} \]

   12. Simplified 39.1%

       \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-d\right)} \]

   if 1.7999999999999999e-270 < d

   1. Initial program 66.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 68.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in M around 0 37.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r* 39.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell}\right)\right) \]

      2. times-frac 39.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \frac{h}{\ell}\right)}\right)\right) \]

      3. associate-/l* 41.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\color{blue}{\left({D}^{2} \cdot \frac{{M}^{2}}{{d}^{2}}\right)} \cdot \frac{h}{\ell}\right)\right)\right) \]

      4. *-commutative 41.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\color{blue}{\left(\frac{{M}^{2}}{{d}^{2}} \cdot {D}^{2}\right)} \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow2 41.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\frac{\color{blue}{M \cdot M}}{{d}^{2}} \cdot {D}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. unpow2 41.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\frac{M \cdot M}{\color{blue}{d \cdot d}} \cdot {D}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. times-frac 53.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\color{blue}{\left(\frac{M}{d} \cdot \frac{M}{d}\right)} \cdot {D}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. unpow2 53.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\left(\frac{M}{d} \cdot \frac{M}{d}\right) \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      9. swap-sqr 67.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\color{blue}{\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)\right)} \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. associate-/r/ 67.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\color{blue}{\frac{M}{\frac{d}{D}}} \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. associate-/r/ 68.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\frac{M}{\frac{d}{D}} \cdot \color{blue}{\frac{M}{\frac{d}{D}}}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. unpow2 68.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\color{blue}{{\left(\frac{M}{\frac{d}{D}}\right)}^{2}} \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. associate-*l* 68.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      14. *-commutative 68.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot \left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right)\right)}\right) \]

      15. associate-*l/ 66.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{h \cdot \left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right)}{\ell}}\right) \]

      16. associate-/l* 68.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}}{\ell}\right)}\right) \]

      17. associate-/l* 68.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \color{blue}{\left(0.25 \cdot \frac{{\left(\frac{M}{\frac{d}{D}}\right)}^{2}}{\ell}\right)}\right)\right) \]

   7. Simplified 68.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 68.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\sqrt{0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)} \cdot \sqrt{0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)}}\right) \]

      2. pow2 68.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\sqrt{0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)}\right)}^{2}}\right) \]

      3. associate-*r* 68.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\sqrt{\color{blue}{\left(0.5 \cdot h\right) \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)}}\right)}^{2}\right) \]

      4. associate-*r/ 68.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\sqrt{\left(0.5 \cdot h\right) \cdot \color{blue}{\frac{0.25 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}}}\right)}^{2}\right) \]

   9. Applied egg-rr 68.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\sqrt{\left(0.5 \cdot h\right) \cdot \frac{0.25 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}}\right)}^{2}}\right) \]

   10. Taylor expanded in d around inf 40.7%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. associate-/r* 41.7%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   12. Simplified 41.7%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.8 \cdot 10^{-270}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 43.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 2.8 \cdot 10^{-269}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d 2.8e-269)
   (* d (- (pow (* h l) -0.5)))
   (* d (sqrt (/ (/ 1.0 h) l)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= 2.8e-269) {
		tmp = d * -pow((h * l), -0.5);
	} else {
		tmp = d * sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= 2.8d-269) then
        tmp = d * -((h * l) ** (-0.5d0))
    else
        tmp = d * sqrt(((1.0d0 / h) / l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= 2.8e-269) {
		tmp = d * -Math.pow((h * l), -0.5);
	} else {
		tmp = d * Math.sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= 2.8e-269:
		tmp = d * -math.pow((h * l), -0.5)
	else:
		tmp = d * math.sqrt(((1.0 / h) / l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= 2.8e-269)
		tmp = Float64(d * Float64(-(Float64(h * l) ^ -0.5)));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / h) / l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= 2.8e-269)
		tmp = d * -((h * l) ^ -0.5);
	else
		tmp = d * sqrt(((1.0 / h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, 2.8e-269], N[(d * (-N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 2.79999999999999995e-269

   1. Initial program 56.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 55.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 6.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   6. Taylor expanded in h around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. associate-*l* 0.0%

         \[\leadsto \color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. unpow2 0.0%

         \[\leadsto d \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. rem-square-sqrt 37.4%

         \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. mul-1-neg 37.4%

         \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      5. unpow-1 37.4%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \]

      6. metadata-eval 37.4%

         \[\leadsto d \cdot \left(-\sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \]

      7. pow-sqr 37.5%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      8. rem-sqrt-square 37.5%

         \[\leadsto d \cdot \left(-\color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \]

      9. rem-square-sqrt 37.3%

         \[\leadsto d \cdot \left(-\left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right|\right) \]

      10. fabs-sqr 37.3%

          \[\leadsto d \cdot \left(-\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      11. rem-square-sqrt 37.5%

          \[\leadsto d \cdot \left(-\color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \]

   8. Simplified 37.5%

      \[\leadsto \color{blue}{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \]

   if 2.79999999999999995e-269 < d

   1. Initial program 66.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 68.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in M around 0 37.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r* 39.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell}\right)\right) \]

      2. times-frac 39.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \frac{h}{\ell}\right)}\right)\right) \]

      3. associate-/l* 41.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\color{blue}{\left({D}^{2} \cdot \frac{{M}^{2}}{{d}^{2}}\right)} \cdot \frac{h}{\ell}\right)\right)\right) \]

      4. *-commutative 41.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\color{blue}{\left(\frac{{M}^{2}}{{d}^{2}} \cdot {D}^{2}\right)} \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow2 41.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\frac{\color{blue}{M \cdot M}}{{d}^{2}} \cdot {D}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. unpow2 41.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\frac{M \cdot M}{\color{blue}{d \cdot d}} \cdot {D}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. times-frac 53.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\color{blue}{\left(\frac{M}{d} \cdot \frac{M}{d}\right)} \cdot {D}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. unpow2 53.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\left(\frac{M}{d} \cdot \frac{M}{d}\right) \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      9. swap-sqr 67.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\color{blue}{\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)\right)} \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. associate-/r/ 67.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\color{blue}{\frac{M}{\frac{d}{D}}} \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. associate-/r/ 68.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\frac{M}{\frac{d}{D}} \cdot \color{blue}{\frac{M}{\frac{d}{D}}}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. unpow2 68.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\color{blue}{{\left(\frac{M}{\frac{d}{D}}\right)}^{2}} \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. associate-*l* 68.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      14. *-commutative 68.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot \left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right)\right)}\right) \]

      15. associate-*l/ 66.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{h \cdot \left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right)}{\ell}}\right) \]

      16. associate-/l* 68.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}}{\ell}\right)}\right) \]

      17. associate-/l* 68.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \color{blue}{\left(0.25 \cdot \frac{{\left(\frac{M}{\frac{d}{D}}\right)}^{2}}{\ell}\right)}\right)\right) \]

   7. Simplified 68.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 68.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\sqrt{0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)} \cdot \sqrt{0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)}}\right) \]

      2. pow2 68.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\sqrt{0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)}\right)}^{2}}\right) \]

      3. associate-*r* 68.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\sqrt{\color{blue}{\left(0.5 \cdot h\right) \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)}}\right)}^{2}\right) \]

      4. associate-*r/ 68.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\sqrt{\left(0.5 \cdot h\right) \cdot \color{blue}{\frac{0.25 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}}}\right)}^{2}\right) \]

   9. Applied egg-rr 68.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\sqrt{\left(0.5 \cdot h\right) \cdot \frac{0.25 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}}\right)}^{2}}\right) \]

   10. Taylor expanded in d around inf 40.7%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. associate-/r* 41.7%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   12. Simplified 41.7%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 43.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 2 \cdot 10^{-269}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d 2e-269) (* d (- (pow (* h l) -0.5))) (/ d (sqrt (* h l)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= 2e-269) {
		tmp = d * -pow((h * l), -0.5);
	} else {
		tmp = d / sqrt((h * l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= 2d-269) then
        tmp = d * -((h * l) ** (-0.5d0))
    else
        tmp = d / sqrt((h * l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= 2e-269) {
		tmp = d * -Math.pow((h * l), -0.5);
	} else {
		tmp = d / Math.sqrt((h * l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= 2e-269:
		tmp = d * -math.pow((h * l), -0.5)
	else:
		tmp = d / math.sqrt((h * l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= 2e-269)
		tmp = Float64(d * Float64(-(Float64(h * l) ^ -0.5)));
	else
		tmp = Float64(d / sqrt(Float64(h * l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= 2e-269)
		tmp = d * -((h * l) ^ -0.5);
	else
		tmp = d / sqrt((h * l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, 2e-269], N[(d * (-N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.9999999999999999e-269

   1. Initial program 56.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 55.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 6.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   6. Taylor expanded in h around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. associate-*l* 0.0%

         \[\leadsto \color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. unpow2 0.0%

         \[\leadsto d \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. rem-square-sqrt 37.4%

         \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. mul-1-neg 37.4%

         \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      5. unpow-1 37.4%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \]

      6. metadata-eval 37.4%

         \[\leadsto d \cdot \left(-\sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \]

      7. pow-sqr 37.5%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      8. rem-sqrt-square 37.5%

         \[\leadsto d \cdot \left(-\color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \]

      9. rem-square-sqrt 37.3%

         \[\leadsto d \cdot \left(-\left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right|\right) \]

      10. fabs-sqr 37.3%

          \[\leadsto d \cdot \left(-\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      11. rem-square-sqrt 37.5%

          \[\leadsto d \cdot \left(-\color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \]

   8. Simplified 37.5%

      \[\leadsto \color{blue}{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \]

   if 1.9999999999999999e-269 < d

   1. Initial program 66.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 68.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 40.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. sqrt-div 40.7%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      2. metadata-eval 40.7%

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

      3. sqrt-unprod 54.9%

         \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      4. div-inv 55.0%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      5. sqrt-unprod 40.7%

         \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

   7. Applied egg-rr 40.7%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 43.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{h \cdot \ell}\\ \mathbf{if}\;d \leq 2.6 \cdot 10^{-269}:\\ \;\;\;\;\frac{-d}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{t\_0}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (* h l)))) (if (<= d 2.6e-269) (/ (- d) t_0) (/ d t_0))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((h * l));
	double tmp;
	if (d <= 2.6e-269) {
		tmp = -d / t_0;
	} else {
		tmp = d / t_0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((h * l))
    if (d <= 2.6d-269) then
        tmp = -d / t_0
    else
        tmp = d / t_0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((h * l));
	double tmp;
	if (d <= 2.6e-269) {
		tmp = -d / t_0;
	} else {
		tmp = d / t_0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((h * l))
	tmp = 0
	if d <= 2.6e-269:
		tmp = -d / t_0
	else:
		tmp = d / t_0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(h * l))
	tmp = 0.0
	if (d <= 2.6e-269)
		tmp = Float64(Float64(-d) / t_0);
	else
		tmp = Float64(d / t_0);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((h * l));
	tmp = 0.0;
	if (d <= 2.6e-269)
		tmp = -d / t_0;
	else
		tmp = d / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, 2.6e-269], N[((-d) / t$95$0), $MachinePrecision], N[(d / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < 2.6e-269

   1. Initial program 56.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 56.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in M around 0 41.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r* 41.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell}\right)\right) \]

      2. times-frac 38.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \frac{h}{\ell}\right)}\right)\right) \]

      3. associate-/l* 38.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\color{blue}{\left({D}^{2} \cdot \frac{{M}^{2}}{{d}^{2}}\right)} \cdot \frac{h}{\ell}\right)\right)\right) \]

      4. *-commutative 38.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\color{blue}{\left(\frac{{M}^{2}}{{d}^{2}} \cdot {D}^{2}\right)} \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow2 38.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\frac{\color{blue}{M \cdot M}}{{d}^{2}} \cdot {D}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. unpow2 38.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\frac{M \cdot M}{\color{blue}{d \cdot d}} \cdot {D}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. times-frac 45.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\color{blue}{\left(\frac{M}{d} \cdot \frac{M}{d}\right)} \cdot {D}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. unpow2 45.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\left(\frac{M}{d} \cdot \frac{M}{d}\right) \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      9. swap-sqr 56.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\color{blue}{\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)\right)} \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. associate-/r/ 55.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\color{blue}{\frac{M}{\frac{d}{D}}} \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. associate-/r/ 55.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\frac{M}{\frac{d}{D}} \cdot \color{blue}{\frac{M}{\frac{d}{D}}}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. unpow2 55.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\color{blue}{{\left(\frac{M}{\frac{d}{D}}\right)}^{2}} \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. associate-*l* 55.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

      14. *-commutative 55.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot \left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right)\right)}\right) \]

      15. associate-*l/ 58.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{h \cdot \left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right)}{\ell}}\right) \]

      16. associate-/l* 59.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}}{\ell}\right)}\right) \]

      17. associate-/l* 59.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \color{blue}{\left(0.25 \cdot \frac{{\left(\frac{M}{\frac{d}{D}}\right)}^{2}}{\ell}\right)}\right)\right) \]

   7. Simplified 60.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 59.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\sqrt{0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)} \cdot \sqrt{0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)}}\right) \]

      2. pow2 59.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\sqrt{0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)}\right)}^{2}}\right) \]

      3. associate-*r* 59.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\sqrt{\color{blue}{\left(0.5 \cdot h\right) \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)}}\right)}^{2}\right) \]

      4. associate-*r/ 59.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\sqrt{\left(0.5 \cdot h\right) \cdot \color{blue}{\frac{0.25 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}}}\right)}^{2}\right) \]

   9. Applied egg-rr 59.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\sqrt{\left(0.5 \cdot h\right) \cdot \frac{0.25 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}}\right)}^{2}}\right) \]

   10. Taylor expanded in l around -inf 0.0%

       \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. associate-*l* 0.0%

          \[\leadsto \color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

       2. unpow2 0.0%

          \[\leadsto d \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

       3. rem-square-sqrt 37.4%

          \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

       4. mul-1-neg 37.4%

          \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

       5. distribute-rgt-neg-out 37.4%

          \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

       6. *-commutative 37.4%

          \[\leadsto -\color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   12. Simplified 37.4%

       \[\leadsto \color{blue}{-\frac{d}{\sqrt{h \cdot \ell}}} \]

   if 2.6e-269 < d

   1. Initial program 66.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 68.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 40.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. sqrt-div 40.7%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      2. metadata-eval 40.7%

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

      3. sqrt-unprod 54.9%

         \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      4. div-inv 55.0%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      5. sqrt-unprod 40.7%

         \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

   7. Applied egg-rr 40.7%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 2.6 \cdot 10^{-269}:\\ \;\;\;\;\frac{-d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 25.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \frac{d}{\sqrt{h \cdot \ell}} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m) :precision binary64 (/ d (sqrt (* h l))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	return d / sqrt((h * l));
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    code = d / sqrt((h * l))
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	return d / Math.sqrt((h * l));
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	return d / math.sqrt((h * l))

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	return Float64(d / sqrt(Float64(h * l)))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp = code(d, h, l, M_m, D_m)
	tmp = d / sqrt((h * l));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.6%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 62.0%

   \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in d around inf 23.1%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
6. Step-by-step derivation

   1. sqrt-div 23.1%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

   2. metadata-eval 23.1%

      \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

   3. sqrt-unprod 27.8%

      \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

   4. div-inv 27.8%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]

   5. sqrt-unprod 23.1%

      \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

7. Applied egg-rr 23.1%

   \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024181 
(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)))))