Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.7% → 80.1%

Time: 19.9s

Alternatives: 16

Speedup: 1.4×

• 22.9% 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: 66.7% 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: 80.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\sqrt[3]{h}}{\ell}\\ t_1 := \sqrt{-d}\\ \mathbf{if}\;d \leq -58000000:\\ \;\;\;\;\left(\frac{t\_1}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D\_m}{d} \cdot \left(0.5 \cdot M\_m\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{elif}\;d \leq -6 \cdot 10^{-300}:\\ \;\;\;\;\frac{t\_1}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D\_m \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-194}:\\ \;\;\;\;\left(\left|t\_0\right| \cdot \sqrt{t\_0}\right) \cdot \left(-0.125 \cdot \left({D\_m}^{2} \cdot \frac{{M\_m}^{2}}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + h \cdot \frac{-0.125 \cdot {\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2}}{\ell}\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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 (/ (cbrt h) l)) (t_1 (sqrt (- d))))
   (if (<= d -58000000.0)
     (*
      (* (/ t_1 (sqrt (- h))) (sqrt (/ d l)))
      (- 1.0 (* 0.5 (pow (* (* (/ D_m d) (* 0.5 M_m)) (sqrt (/ h l))) 2.0))))
     (if (<= d -6e-300)
       (*
        (/ t_1 (sqrt (- l)))
        (*
         (sqrt (/ d h))
         (+ 1.0 (* (/ h l) (* (pow (* D_m (/ (/ M_m 2.0) d)) 2.0) -0.5)))))
       (if (<= d 1.05e-194)
         (*
          (* (fabs t_0) (sqrt t_0))
          (* -0.125 (* (pow D_m 2.0) (/ (pow M_m 2.0) d))))
         (*
          (+ 1.0 (* h (/ (* -0.125 (pow (* D_m (/ M_m d)) 2.0)) l)))
          (/ d (* (sqrt l) (sqrt h)))))))))

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 = cbrt(h) / l;
	double t_1 = sqrt(-d);
	double tmp;
	if (d <= -58000000.0) {
		tmp = ((t_1 / sqrt(-h)) * sqrt((d / l))) * (1.0 - (0.5 * pow((((D_m / d) * (0.5 * M_m)) * sqrt((h / l))), 2.0)));
	} else if (d <= -6e-300) {
		tmp = (t_1 / sqrt(-l)) * (sqrt((d / h)) * (1.0 + ((h / l) * (pow((D_m * ((M_m / 2.0) / d)), 2.0) * -0.5))));
	} else if (d <= 1.05e-194) {
		tmp = (fabs(t_0) * sqrt(t_0)) * (-0.125 * (pow(D_m, 2.0) * (pow(M_m, 2.0) / d)));
	} else {
		tmp = (1.0 + (h * ((-0.125 * pow((D_m * (M_m / d)), 2.0)) / l))) * (d / (sqrt(l) * sqrt(h)));
	}
	return tmp;
}

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.cbrt(h) / l;
	double t_1 = Math.sqrt(-d);
	double tmp;
	if (d <= -58000000.0) {
		tmp = ((t_1 / Math.sqrt(-h)) * Math.sqrt((d / l))) * (1.0 - (0.5 * Math.pow((((D_m / d) * (0.5 * M_m)) * Math.sqrt((h / l))), 2.0)));
	} else if (d <= -6e-300) {
		tmp = (t_1 / Math.sqrt(-l)) * (Math.sqrt((d / h)) * (1.0 + ((h / l) * (Math.pow((D_m * ((M_m / 2.0) / d)), 2.0) * -0.5))));
	} else if (d <= 1.05e-194) {
		tmp = (Math.abs(t_0) * Math.sqrt(t_0)) * (-0.125 * (Math.pow(D_m, 2.0) * (Math.pow(M_m, 2.0) / d)));
	} else {
		tmp = (1.0 + (h * ((-0.125 * Math.pow((D_m * (M_m / d)), 2.0)) / l))) * (d / (Math.sqrt(l) * Math.sqrt(h)));
	}
	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(cbrt(h) / l)
	t_1 = sqrt(Float64(-d))
	tmp = 0.0
	if (d <= -58000000.0)
		tmp = Float64(Float64(Float64(t_1 / sqrt(Float64(-h))) * sqrt(Float64(d / l))) * Float64(1.0 - Float64(0.5 * (Float64(Float64(Float64(D_m / d) * Float64(0.5 * M_m)) * sqrt(Float64(h / l))) ^ 2.0))));
	elseif (d <= -6e-300)
		tmp = Float64(Float64(t_1 / sqrt(Float64(-l))) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D_m * Float64(Float64(M_m / 2.0) / d)) ^ 2.0) * -0.5)))));
	elseif (d <= 1.05e-194)
		tmp = Float64(Float64(abs(t_0) * sqrt(t_0)) * Float64(-0.125 * Float64((D_m ^ 2.0) * Float64((M_m ^ 2.0) / d))));
	else
		tmp = Float64(Float64(1.0 + Float64(h * Float64(Float64(-0.125 * (Float64(D_m * Float64(M_m / d)) ^ 2.0)) / l))) * Float64(d / Float64(sqrt(l) * sqrt(h))));
	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[(N[Power[h, 1/3], $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[d, -58000000.0], 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[Power[N[(N[(N[(D$95$m / d), $MachinePrecision] * N[(0.5 * M$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -6e-300], N[(N[(t$95$1 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / 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[d, 1.05e-194], N[(N[(N[Abs[t$95$0], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(-0.125 * N[(N[Power[D$95$m, 2.0], $MachinePrecision] * N[(N[Power[M$95$m, 2.0], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(h * N[(N[(-0.125 * N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -5.8e7

   1. Initial program 77.8%

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

   3. Simplified 77.8%

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

      1. add-sqr-sqrt 77.7%

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

      2. pow2 77.7%

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

      3. sqrt-prod 77.7%

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

      4. sqrt-pow1 79.1%

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

      5. metadata-eval 79.1%

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

      6. pow1 79.1%

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

      7. *-commutative 79.1%

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

      8. div-inv 79.1%

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

      9. metadata-eval 79.1%

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

   6. Applied egg-rr 79.1%

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

      1. frac-2neg 79.1%

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

      2. sqrt-div 92.8%

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

   8. Applied egg-rr 92.8%

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

   if -5.8e7 < d < -6.00000000000000048e-300

   1. Initial program 63.7%

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

   3. Simplified 63.7%

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

      1. frac-2neg 63.7%

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

      2. sqrt-div 72.2%

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

   6. Applied egg-rr 72.2%

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

   if -6.00000000000000048e-300 < d < 1.05e-194

   1. Initial program 18.4%

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

   3. Simplified 18.4%

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

   5. Taylor expanded in d around 0 36.8%

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

      1. associate-*r* 36.8%

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

      2. *-commutative 36.8%

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

      3. associate-/l* 36.8%

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

   7. Simplified 36.8%

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

      1. pow1/2 36.8%

         \[\leadsto \color{blue}{{\left(\frac{h}{{\ell}^{3}}\right)}^{0.5}} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)\right) \]

      2. add-cube-cbrt 36.7%

         \[\leadsto {\color{blue}{\left(\left(\sqrt[3]{\frac{h}{{\ell}^{3}}} \cdot \sqrt[3]{\frac{h}{{\ell}^{3}}}\right) \cdot \sqrt[3]{\frac{h}{{\ell}^{3}}}\right)}}^{0.5} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)\right) \]

      3. unpow-prod-down 36.7%

         \[\leadsto \color{blue}{\left({\left(\sqrt[3]{\frac{h}{{\ell}^{3}}} \cdot \sqrt[3]{\frac{h}{{\ell}^{3}}}\right)}^{0.5} \cdot {\left(\sqrt[3]{\frac{h}{{\ell}^{3}}}\right)}^{0.5}\right)} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)\right) \]

      4. pow2 36.7%

         \[\leadsto \left({\color{blue}{\left({\left(\sqrt[3]{\frac{h}{{\ell}^{3}}}\right)}^{2}\right)}}^{0.5} \cdot {\left(\sqrt[3]{\frac{h}{{\ell}^{3}}}\right)}^{0.5}\right) \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)\right) \]

      5. cbrt-div 36.6%

         \[\leadsto \left({\left({\color{blue}{\left(\frac{\sqrt[3]{h}}{\sqrt[3]{{\ell}^{3}}}\right)}}^{2}\right)}^{0.5} \cdot {\left(\sqrt[3]{\frac{h}{{\ell}^{3}}}\right)}^{0.5}\right) \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)\right) \]

      6. unpow3 36.6%

         \[\leadsto \left({\left({\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\color{blue}{\left(\ell \cdot \ell\right) \cdot \ell}}}\right)}^{2}\right)}^{0.5} \cdot {\left(\sqrt[3]{\frac{h}{{\ell}^{3}}}\right)}^{0.5}\right) \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)\right) \]

      7. add-cbrt-cube 36.7%

         \[\leadsto \left({\left({\left(\frac{\sqrt[3]{h}}{\color{blue}{\ell}}\right)}^{2}\right)}^{0.5} \cdot {\left(\sqrt[3]{\frac{h}{{\ell}^{3}}}\right)}^{0.5}\right) \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)\right) \]

      8. cbrt-div 36.7%

         \[\leadsto \left({\left({\left(\frac{\sqrt[3]{h}}{\ell}\right)}^{2}\right)}^{0.5} \cdot {\color{blue}{\left(\frac{\sqrt[3]{h}}{\sqrt[3]{{\ell}^{3}}}\right)}}^{0.5}\right) \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)\right) \]

      9. unpow3 36.7%

         \[\leadsto \left({\left({\left(\frac{\sqrt[3]{h}}{\ell}\right)}^{2}\right)}^{0.5} \cdot {\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\color{blue}{\left(\ell \cdot \ell\right) \cdot \ell}}}\right)}^{0.5}\right) \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)\right) \]

      10. add-cbrt-cube 55.2%

          \[\leadsto \left({\left({\left(\frac{\sqrt[3]{h}}{\ell}\right)}^{2}\right)}^{0.5} \cdot {\left(\frac{\sqrt[3]{h}}{\color{blue}{\ell}}\right)}^{0.5}\right) \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)\right) \]

   9. Applied egg-rr 55.2%

      \[\leadsto \color{blue}{\left({\left({\left(\frac{\sqrt[3]{h}}{\ell}\right)}^{2}\right)}^{0.5} \cdot {\left(\frac{\sqrt[3]{h}}{\ell}\right)}^{0.5}\right)} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)\right) \]
   10. Step-by-step derivation

       1. unpow1/2 55.2%

          \[\leadsto \left(\color{blue}{\sqrt{{\left(\frac{\sqrt[3]{h}}{\ell}\right)}^{2}}} \cdot {\left(\frac{\sqrt[3]{h}}{\ell}\right)}^{0.5}\right) \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)\right) \]

       2. unpow2 55.2%

          \[\leadsto \left(\sqrt{\color{blue}{\frac{\sqrt[3]{h}}{\ell} \cdot \frac{\sqrt[3]{h}}{\ell}}} \cdot {\left(\frac{\sqrt[3]{h}}{\ell}\right)}^{0.5}\right) \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)\right) \]

       3. rem-sqrt-square 59.8%

          \[\leadsto \left(\color{blue}{\left|\frac{\sqrt[3]{h}}{\ell}\right|} \cdot {\left(\frac{\sqrt[3]{h}}{\ell}\right)}^{0.5}\right) \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)\right) \]

       4. unpow1/2 59.8%

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

   11. Simplified 59.8%

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

   if 1.05e-194 < d

   1. Initial program 68.1%

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

   3. Simplified 67.1%

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

      1. add-sqr-sqrt 67.1%

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

      2. pow2 67.1%

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

      3. sqrt-prod 67.1%

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

      4. sqrt-pow1 71.0%

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

      5. metadata-eval 71.0%

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

      6. pow1 71.0%

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

      7. *-commutative 71.0%

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

      8. div-inv 71.0%

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

      9. metadata-eval 71.0%

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

   6. Applied egg-rr 71.0%

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

   7. Taylor expanded in h around inf 52.3%

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

      1. sub-neg 52.3%

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

      2. distribute-lft-in 52.3%

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

      3. rgt-mult-inverse 52.4%

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

      4. distribute-lft-neg-in 52.4%

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

      5. metadata-eval 52.4%

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

      6. associate-/r* 53.5%

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

      7. associate-*r/ 53.5%

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

   9. Simplified 71.5%

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

       1. *-commutative 71.5%

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

       2. sqrt-div 75.4%

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

       3. sqrt-div 90.9%

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

       4. frac-times 90.9%

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

       5. add-sqr-sqrt 91.0%

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

   11. Applied egg-rr 91.0%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 + h \cdot \frac{-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -58000000:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{elif}\;d \leq -6 \cdot 10^{-300}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-194}:\\ \;\;\;\;\left(\left|\frac{\sqrt[3]{h}}{\ell}\right| \cdot \sqrt{\frac{\sqrt[3]{h}}{\ell}}\right) \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + h \cdot \frac{-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{-d}\\ \mathbf{if}\;d \leq -27000000:\\ \;\;\;\;\left(\frac{t\_0}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D\_m}{d} \cdot \left(0.5 \cdot M\_m\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{elif}\;d \leq -6 \cdot 10^{-300}:\\ \;\;\;\;\frac{t\_0}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D\_m \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + h \cdot \frac{-0.125 \cdot {\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2}}{\ell}\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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 (<= d -27000000.0)
     (*
      (* (/ t_0 (sqrt (- h))) (sqrt (/ d l)))
      (- 1.0 (* 0.5 (pow (* (* (/ D_m d) (* 0.5 M_m)) (sqrt (/ h l))) 2.0))))
     (if (<= d -6e-300)
       (*
        (/ t_0 (sqrt (- l)))
        (*
         (sqrt (/ d h))
         (+ 1.0 (* (/ h l) (* (pow (* D_m (/ (/ M_m 2.0) d)) 2.0) -0.5)))))
       (*
        (+ 1.0 (* h (/ (* -0.125 (pow (* D_m (/ M_m d)) 2.0)) l)))
        (/ d (* (sqrt l) (sqrt h))))))))

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 (d <= -27000000.0) {
		tmp = ((t_0 / sqrt(-h)) * sqrt((d / l))) * (1.0 - (0.5 * pow((((D_m / d) * (0.5 * M_m)) * sqrt((h / l))), 2.0)));
	} else if (d <= -6e-300) {
		tmp = (t_0 / sqrt(-l)) * (sqrt((d / h)) * (1.0 + ((h / l) * (pow((D_m * ((M_m / 2.0) / d)), 2.0) * -0.5))));
	} else {
		tmp = (1.0 + (h * ((-0.125 * pow((D_m * (M_m / d)), 2.0)) / l))) * (d / (sqrt(l) * sqrt(h)));
	}
	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(-d)
    if (d <= (-27000000.0d0)) then
        tmp = ((t_0 / sqrt(-h)) * sqrt((d / l))) * (1.0d0 - (0.5d0 * ((((d_m / d) * (0.5d0 * m_m)) * sqrt((h / l))) ** 2.0d0)))
    else if (d <= (-6d-300)) then
        tmp = (t_0 / sqrt(-l)) * (sqrt((d / h)) * (1.0d0 + ((h / l) * (((d_m * ((m_m / 2.0d0) / d)) ** 2.0d0) * (-0.5d0)))))
    else
        tmp = (1.0d0 + (h * (((-0.125d0) * ((d_m * (m_m / d)) ** 2.0d0)) / l))) * (d / (sqrt(l) * sqrt(h)))
    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(-d);
	double tmp;
	if (d <= -27000000.0) {
		tmp = ((t_0 / Math.sqrt(-h)) * Math.sqrt((d / l))) * (1.0 - (0.5 * Math.pow((((D_m / d) * (0.5 * M_m)) * Math.sqrt((h / l))), 2.0)));
	} else if (d <= -6e-300) {
		tmp = (t_0 / Math.sqrt(-l)) * (Math.sqrt((d / h)) * (1.0 + ((h / l) * (Math.pow((D_m * ((M_m / 2.0) / d)), 2.0) * -0.5))));
	} else {
		tmp = (1.0 + (h * ((-0.125 * Math.pow((D_m * (M_m / d)), 2.0)) / l))) * (d / (Math.sqrt(l) * Math.sqrt(h)));
	}
	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(-d)
	tmp = 0
	if d <= -27000000.0:
		tmp = ((t_0 / math.sqrt(-h)) * math.sqrt((d / l))) * (1.0 - (0.5 * math.pow((((D_m / d) * (0.5 * M_m)) * math.sqrt((h / l))), 2.0)))
	elif d <= -6e-300:
		tmp = (t_0 / math.sqrt(-l)) * (math.sqrt((d / h)) * (1.0 + ((h / l) * (math.pow((D_m * ((M_m / 2.0) / d)), 2.0) * -0.5))))
	else:
		tmp = (1.0 + (h * ((-0.125 * math.pow((D_m * (M_m / d)), 2.0)) / l))) * (d / (math.sqrt(l) * math.sqrt(h)))
	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 (d <= -27000000.0)
		tmp = Float64(Float64(Float64(t_0 / sqrt(Float64(-h))) * sqrt(Float64(d / l))) * Float64(1.0 - Float64(0.5 * (Float64(Float64(Float64(D_m / d) * Float64(0.5 * M_m)) * sqrt(Float64(h / l))) ^ 2.0))));
	elseif (d <= -6e-300)
		tmp = Float64(Float64(t_0 / sqrt(Float64(-l))) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D_m * Float64(Float64(M_m / 2.0) / d)) ^ 2.0) * -0.5)))));
	else
		tmp = Float64(Float64(1.0 + Float64(h * Float64(Float64(-0.125 * (Float64(D_m * Float64(M_m / d)) ^ 2.0)) / l))) * Float64(d / Float64(sqrt(l) * sqrt(h))));
	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(-d);
	tmp = 0.0;
	if (d <= -27000000.0)
		tmp = ((t_0 / sqrt(-h)) * sqrt((d / l))) * (1.0 - (0.5 * ((((D_m / d) * (0.5 * M_m)) * sqrt((h / l))) ^ 2.0)));
	elseif (d <= -6e-300)
		tmp = (t_0 / sqrt(-l)) * (sqrt((d / h)) * (1.0 + ((h / l) * (((D_m * ((M_m / 2.0) / d)) ^ 2.0) * -0.5))));
	else
		tmp = (1.0 + (h * ((-0.125 * ((D_m * (M_m / d)) ^ 2.0)) / l))) * (d / (sqrt(l) * sqrt(h)));
	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[(-d)], $MachinePrecision]}, If[LessEqual[d, -27000000.0], N[(N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[Power[N[(N[(N[(D$95$m / d), $MachinePrecision] * N[(0.5 * M$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -6e-300], N[(N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / 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], N[(N[(1.0 + N[(h * N[(N[(-0.125 * N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.7e7

   1. Initial program 77.8%

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

   3. Simplified 77.8%

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

      1. add-sqr-sqrt 77.7%

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

      2. pow2 77.7%

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

      3. sqrt-prod 77.7%

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

      4. sqrt-pow1 79.1%

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

      5. metadata-eval 79.1%

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

      6. pow1 79.1%

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

      7. *-commutative 79.1%

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

      8. div-inv 79.1%

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

      9. metadata-eval 79.1%

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

   6. Applied egg-rr 79.1%

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

      1. frac-2neg 79.1%

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

      2. sqrt-div 92.8%

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

   8. Applied egg-rr 92.8%

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

   if -2.7e7 < d < -6.00000000000000048e-300

   1. Initial program 63.7%

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

   3. Simplified 63.7%

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

      1. frac-2neg 63.7%

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

      2. sqrt-div 72.2%

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

   6. Applied egg-rr 72.2%

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

   if -6.00000000000000048e-300 < d

   1. Initial program 58.9%

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

   3. Simplified 58.1%

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

      1. add-sqr-sqrt 58.1%

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

      2. pow2 58.1%

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

      3. sqrt-prod 58.1%

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

      4. sqrt-pow1 61.3%

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

      5. metadata-eval 61.3%

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

      6. pow1 61.3%

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

      7. *-commutative 61.3%

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

      8. div-inv 61.3%

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

      9. metadata-eval 61.3%

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

   6. Applied egg-rr 61.3%

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

   7. Taylor expanded in h around inf 43.6%

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

      1. sub-neg 43.6%

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

      2. distribute-lft-in 43.6%

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

      3. rgt-mult-inverse 43.6%

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

      4. distribute-lft-neg-in 43.6%

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

      5. metadata-eval 43.6%

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

      6. associate-/r* 44.5%

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

      7. associate-*r/ 44.5%

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

   9. Simplified 61.7%

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

       1. *-commutative 61.7%

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

       2. sqrt-div 64.9%

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

       3. sqrt-div 81.7%

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

       4. frac-times 81.7%

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

       5. add-sqr-sqrt 81.8%

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

   11. Applied egg-rr 81.8%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -27000000:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{elif}\;d \leq -6 \cdot 10^{-300}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + h \cdot \frac{-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{-d}\\ \mathbf{if}\;d \leq -5.5 \cdot 10^{+33}:\\ \;\;\;\;\left(\frac{t\_2}{\sqrt{-h}} \cdot t\_0\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \frac{M\_m}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -7.2 \cdot 10^{-144}:\\ \;\;\;\;\left(t\_0 \cdot t\_1\right) \cdot \left(1 - \left({\left(D\_m \cdot M\_m\right)}^{2} \cdot \frac{h}{{d}^{2}}\right) \cdot \frac{0.125}{\ell}\right)\\ \mathbf{elif}\;d \leq -6 \cdot 10^{-300}:\\ \;\;\;\;\frac{t\_2}{\sqrt{-\ell}} \cdot \left(t\_1 \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D\_m \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + h \cdot \frac{-0.125 \cdot {\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2}}{\ell}\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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 (sqrt (/ d h))) (t_2 (sqrt (- d))))
   (if (<= d -5.5e+33)
     (*
      (* (/ t_2 (sqrt (- h))) t_0)
      (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D_m d) (/ M_m 2.0)) 2.0)))))
     (if (<= d -7.2e-144)
       (*
        (* t_0 t_1)
        (- 1.0 (* (* (pow (* D_m M_m) 2.0) (/ h (pow d 2.0))) (/ 0.125 l))))
       (if (<= d -6e-300)
         (*
          (/ t_2 (sqrt (- l)))
          (*
           t_1
           (+ 1.0 (* (/ h l) (* (pow (* D_m (/ (/ M_m 2.0) d)) 2.0) -0.5)))))
         (*
          (+ 1.0 (* h (/ (* -0.125 (pow (* D_m (/ M_m d)) 2.0)) l)))
          (/ d (* (sqrt l) (sqrt h)))))))))

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 = sqrt((d / h));
	double t_2 = sqrt(-d);
	double tmp;
	if (d <= -5.5e+33) {
		tmp = ((t_2 / sqrt(-h)) * t_0) * (1.0 - (0.5 * ((h / l) * pow(((D_m / d) * (M_m / 2.0)), 2.0))));
	} else if (d <= -7.2e-144) {
		tmp = (t_0 * t_1) * (1.0 - ((pow((D_m * M_m), 2.0) * (h / pow(d, 2.0))) * (0.125 / l)));
	} else if (d <= -6e-300) {
		tmp = (t_2 / sqrt(-l)) * (t_1 * (1.0 + ((h / l) * (pow((D_m * ((M_m / 2.0) / d)), 2.0) * -0.5))));
	} else {
		tmp = (1.0 + (h * ((-0.125 * pow((D_m * (M_m / d)), 2.0)) / l))) * (d / (sqrt(l) * sqrt(h)));
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt((d / l))
    t_1 = sqrt((d / h))
    t_2 = sqrt(-d)
    if (d <= (-5.5d+33)) then
        tmp = ((t_2 / sqrt(-h)) * t_0) * (1.0d0 - (0.5d0 * ((h / l) * (((d_m / d) * (m_m / 2.0d0)) ** 2.0d0))))
    else if (d <= (-7.2d-144)) then
        tmp = (t_0 * t_1) * (1.0d0 - ((((d_m * m_m) ** 2.0d0) * (h / (d ** 2.0d0))) * (0.125d0 / l)))
    else if (d <= (-6d-300)) then
        tmp = (t_2 / sqrt(-l)) * (t_1 * (1.0d0 + ((h / l) * (((d_m * ((m_m / 2.0d0) / d)) ** 2.0d0) * (-0.5d0)))))
    else
        tmp = (1.0d0 + (h * (((-0.125d0) * ((d_m * (m_m / d)) ** 2.0d0)) / l))) * (d / (sqrt(l) * sqrt(h)))
    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((d / l));
	double t_1 = Math.sqrt((d / h));
	double t_2 = Math.sqrt(-d);
	double tmp;
	if (d <= -5.5e+33) {
		tmp = ((t_2 / Math.sqrt(-h)) * t_0) * (1.0 - (0.5 * ((h / l) * Math.pow(((D_m / d) * (M_m / 2.0)), 2.0))));
	} else if (d <= -7.2e-144) {
		tmp = (t_0 * t_1) * (1.0 - ((Math.pow((D_m * M_m), 2.0) * (h / Math.pow(d, 2.0))) * (0.125 / l)));
	} else if (d <= -6e-300) {
		tmp = (t_2 / Math.sqrt(-l)) * (t_1 * (1.0 + ((h / l) * (Math.pow((D_m * ((M_m / 2.0) / d)), 2.0) * -0.5))));
	} else {
		tmp = (1.0 + (h * ((-0.125 * Math.pow((D_m * (M_m / d)), 2.0)) / l))) * (d / (Math.sqrt(l) * Math.sqrt(h)));
	}
	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((d / l))
	t_1 = math.sqrt((d / h))
	t_2 = math.sqrt(-d)
	tmp = 0
	if d <= -5.5e+33:
		tmp = ((t_2 / math.sqrt(-h)) * t_0) * (1.0 - (0.5 * ((h / l) * math.pow(((D_m / d) * (M_m / 2.0)), 2.0))))
	elif d <= -7.2e-144:
		tmp = (t_0 * t_1) * (1.0 - ((math.pow((D_m * M_m), 2.0) * (h / math.pow(d, 2.0))) * (0.125 / l)))
	elif d <= -6e-300:
		tmp = (t_2 / math.sqrt(-l)) * (t_1 * (1.0 + ((h / l) * (math.pow((D_m * ((M_m / 2.0) / d)), 2.0) * -0.5))))
	else:
		tmp = (1.0 + (h * ((-0.125 * math.pow((D_m * (M_m / d)), 2.0)) / l))) * (d / (math.sqrt(l) * math.sqrt(h)))
	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 = sqrt(Float64(d / h))
	t_2 = sqrt(Float64(-d))
	tmp = 0.0
	if (d <= -5.5e+33)
		tmp = Float64(Float64(Float64(t_2 / sqrt(Float64(-h))) * t_0) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(M_m / 2.0)) ^ 2.0)))));
	elseif (d <= -7.2e-144)
		tmp = Float64(Float64(t_0 * t_1) * Float64(1.0 - Float64(Float64((Float64(D_m * M_m) ^ 2.0) * Float64(h / (d ^ 2.0))) * Float64(0.125 / l))));
	elseif (d <= -6e-300)
		tmp = Float64(Float64(t_2 / sqrt(Float64(-l))) * Float64(t_1 * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D_m * Float64(Float64(M_m / 2.0) / d)) ^ 2.0) * -0.5)))));
	else
		tmp = Float64(Float64(1.0 + Float64(h * Float64(Float64(-0.125 * (Float64(D_m * Float64(M_m / d)) ^ 2.0)) / l))) * Float64(d / Float64(sqrt(l) * sqrt(h))));
	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((d / l));
	t_1 = sqrt((d / h));
	t_2 = sqrt(-d);
	tmp = 0.0;
	if (d <= -5.5e+33)
		tmp = ((t_2 / sqrt(-h)) * t_0) * (1.0 - (0.5 * ((h / l) * (((D_m / d) * (M_m / 2.0)) ^ 2.0))));
	elseif (d <= -7.2e-144)
		tmp = (t_0 * t_1) * (1.0 - ((((D_m * M_m) ^ 2.0) * (h / (d ^ 2.0))) * (0.125 / l)));
	elseif (d <= -6e-300)
		tmp = (t_2 / sqrt(-l)) * (t_1 * (1.0 + ((h / l) * (((D_m * ((M_m / 2.0) / d)) ^ 2.0) * -0.5))));
	else
		tmp = (1.0 + (h * ((-0.125 * ((D_m * (M_m / d)) ^ 2.0)) / l))) * (d / (sqrt(l) * sqrt(h)));
	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[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[d, -5.5e+33], N[(N[(N[(t$95$2 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$0), $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], If[LessEqual[d, -7.2e-144], N[(N[(t$95$0 * t$95$1), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[N[(D$95$m * M$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.125 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -6e-300], N[(N[(t$95$2 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * 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], N[(N[(1.0 + N[(h * N[(N[(-0.125 * N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -5.5000000000000006e33

   1. Initial program 79.5%

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

   3. Simplified 79.5%

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

      1. frac-2neg 79.5%

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

      2. sqrt-div 95.0%

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

   6. Applied egg-rr 93.6%

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

   if -5.5000000000000006e33 < d < -7.2000000000000001e-144

   1. Initial program 79.6%

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

   3. Simplified 79.6%

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

      1. expm1-log1p-u 78.5%

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

      2. expm1-undefine 78.5%

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

      3. *-commutative 78.5%

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

      4. *-commutative 78.5%

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

      5. div-inv 78.5%

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

      6. metadata-eval 78.5%

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

   6. Applied egg-rr 78.5%

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

      1. expm1-define 78.5%

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

      2. associate-*l/ 91.4%

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

      3. associate-/l* 91.4%

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

   8. Simplified 91.4%

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

   9. Taylor expanded in h around 0 59.6%

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

       1. associate-/r* 68.0%

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

       2. associate-*r/ 68.0%

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

       3. *-commutative 68.0%

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

       4. associate-/l* 68.0%

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

       5. associate-*r* 71.1%

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

       6. associate-/l* 71.1%

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

       7. unpow2 71.1%

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

       8. unpow2 71.1%

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

       9. swap-sqr 93.9%

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

       10. unpow2 93.9%

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

   11. Simplified 93.9%

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

   if -7.2000000000000001e-144 < d < -6.00000000000000048e-300

   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%

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

      1. frac-2neg 49.5%

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

      2. sqrt-div 63.9%

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

   6. Applied egg-rr 63.9%

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

   if -6.00000000000000048e-300 < d

   1. Initial program 58.9%

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

   3. Simplified 58.1%

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

      1. add-sqr-sqrt 58.1%

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

      2. pow2 58.1%

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

      3. sqrt-prod 58.1%

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

      4. sqrt-pow1 61.3%

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

      5. metadata-eval 61.3%

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

      6. pow1 61.3%

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

      7. *-commutative 61.3%

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

      8. div-inv 61.3%

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

      9. metadata-eval 61.3%

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

   6. Applied egg-rr 61.3%

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

   7. Taylor expanded in h around inf 43.6%

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

      1. sub-neg 43.6%

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

      2. distribute-lft-in 43.6%

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

      3. rgt-mult-inverse 43.6%

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

      4. distribute-lft-neg-in 43.6%

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

      5. metadata-eval 43.6%

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

      6. associate-/r* 44.5%

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

      7. associate-*r/ 44.5%

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

   9. Simplified 61.7%

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

       1. *-commutative 61.7%

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

       2. sqrt-div 64.9%

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

       3. sqrt-div 81.7%

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

       4. frac-times 81.7%

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

       5. add-sqr-sqrt 81.8%

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

   11. Applied egg-rr 81.8%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 + h \cdot \frac{-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.5 \cdot 10^{+33}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -7.2 \cdot 10^{-144}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left({\left(D \cdot M\right)}^{2} \cdot \frac{h}{{d}^{2}}\right) \cdot \frac{0.125}{\ell}\right)\\ \mathbf{elif}\;d \leq -6 \cdot 10^{-300}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + h \cdot \frac{-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \frac{M\_m}{2}\right)}^{2}\right)\\ \mathbf{if}\;d \leq -1.8 \cdot 10^{+33}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t\_0\right) \cdot \left(1 - t\_1\right)\\ \mathbf{elif}\;d \leq -7.8 \cdot 10^{-144}:\\ \;\;\;\;\left(t\_0 \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left({\left(D\_m \cdot M\_m\right)}^{2} \cdot \frac{h}{{d}^{2}}\right) \cdot \frac{0.125}{\ell}\right)\\ \mathbf{elif}\;d \leq -6 \cdot 10^{-300}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(t\_1 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + h \cdot \frac{-0.125 \cdot {\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2}}{\ell}\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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 (* 0.5 (* (/ h l) (pow (* (/ D_m d) (/ M_m 2.0)) 2.0)))))
   (if (<= d -1.8e+33)
     (* (* (/ (sqrt (- d)) (sqrt (- h))) t_0) (- 1.0 t_1))
     (if (<= d -7.8e-144)
       (*
        (* t_0 (sqrt (/ d h)))
        (- 1.0 (* (* (pow (* D_m M_m) 2.0) (/ h (pow d 2.0))) (/ 0.125 l))))
       (if (<= d -6e-300)
         (* (* d (sqrt (/ 1.0 (* h l)))) (+ t_1 -1.0))
         (*
          (+ 1.0 (* h (/ (* -0.125 (pow (* D_m (/ M_m d)) 2.0)) l)))
          (/ d (* (sqrt l) (sqrt h)))))))))

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 = 0.5 * ((h / l) * pow(((D_m / d) * (M_m / 2.0)), 2.0));
	double tmp;
	if (d <= -1.8e+33) {
		tmp = ((sqrt(-d) / sqrt(-h)) * t_0) * (1.0 - t_1);
	} else if (d <= -7.8e-144) {
		tmp = (t_0 * sqrt((d / h))) * (1.0 - ((pow((D_m * M_m), 2.0) * (h / pow(d, 2.0))) * (0.125 / l)));
	} else if (d <= -6e-300) {
		tmp = (d * sqrt((1.0 / (h * l)))) * (t_1 + -1.0);
	} else {
		tmp = (1.0 + (h * ((-0.125 * pow((D_m * (M_m / d)), 2.0)) / l))) * (d / (sqrt(l) * sqrt(h)));
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = sqrt((d / l))
    t_1 = 0.5d0 * ((h / l) * (((d_m / d) * (m_m / 2.0d0)) ** 2.0d0))
    if (d <= (-1.8d+33)) then
        tmp = ((sqrt(-d) / sqrt(-h)) * t_0) * (1.0d0 - t_1)
    else if (d <= (-7.8d-144)) then
        tmp = (t_0 * sqrt((d / h))) * (1.0d0 - ((((d_m * m_m) ** 2.0d0) * (h / (d ** 2.0d0))) * (0.125d0 / l)))
    else if (d <= (-6d-300)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * (t_1 + (-1.0d0))
    else
        tmp = (1.0d0 + (h * (((-0.125d0) * ((d_m * (m_m / d)) ** 2.0d0)) / l))) * (d / (sqrt(l) * sqrt(h)))
    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((d / l));
	double t_1 = 0.5 * ((h / l) * Math.pow(((D_m / d) * (M_m / 2.0)), 2.0));
	double tmp;
	if (d <= -1.8e+33) {
		tmp = ((Math.sqrt(-d) / Math.sqrt(-h)) * t_0) * (1.0 - t_1);
	} else if (d <= -7.8e-144) {
		tmp = (t_0 * Math.sqrt((d / h))) * (1.0 - ((Math.pow((D_m * M_m), 2.0) * (h / Math.pow(d, 2.0))) * (0.125 / l)));
	} else if (d <= -6e-300) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * (t_1 + -1.0);
	} else {
		tmp = (1.0 + (h * ((-0.125 * Math.pow((D_m * (M_m / d)), 2.0)) / l))) * (d / (Math.sqrt(l) * Math.sqrt(h)));
	}
	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((d / l))
	t_1 = 0.5 * ((h / l) * math.pow(((D_m / d) * (M_m / 2.0)), 2.0))
	tmp = 0
	if d <= -1.8e+33:
		tmp = ((math.sqrt(-d) / math.sqrt(-h)) * t_0) * (1.0 - t_1)
	elif d <= -7.8e-144:
		tmp = (t_0 * math.sqrt((d / h))) * (1.0 - ((math.pow((D_m * M_m), 2.0) * (h / math.pow(d, 2.0))) * (0.125 / l)))
	elif d <= -6e-300:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * (t_1 + -1.0)
	else:
		tmp = (1.0 + (h * ((-0.125 * math.pow((D_m * (M_m / d)), 2.0)) / l))) * (d / (math.sqrt(l) * math.sqrt(h)))
	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(0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(M_m / 2.0)) ^ 2.0)))
	tmp = 0.0
	if (d <= -1.8e+33)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * t_0) * Float64(1.0 - t_1));
	elseif (d <= -7.8e-144)
		tmp = Float64(Float64(t_0 * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64((Float64(D_m * M_m) ^ 2.0) * Float64(h / (d ^ 2.0))) * Float64(0.125 / l))));
	elseif (d <= -6e-300)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(t_1 + -1.0));
	else
		tmp = Float64(Float64(1.0 + Float64(h * Float64(Float64(-0.125 * (Float64(D_m * Float64(M_m / d)) ^ 2.0)) / l))) * Float64(d / Float64(sqrt(l) * sqrt(h))));
	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((d / l));
	t_1 = 0.5 * ((h / l) * (((D_m / d) * (M_m / 2.0)) ^ 2.0));
	tmp = 0.0;
	if (d <= -1.8e+33)
		tmp = ((sqrt(-d) / sqrt(-h)) * t_0) * (1.0 - t_1);
	elseif (d <= -7.8e-144)
		tmp = (t_0 * sqrt((d / h))) * (1.0 - ((((D_m * M_m) ^ 2.0) * (h / (d ^ 2.0))) * (0.125 / l)));
	elseif (d <= -6e-300)
		tmp = (d * sqrt((1.0 / (h * l)))) * (t_1 + -1.0);
	else
		tmp = (1.0 + (h * ((-0.125 * ((D_m * (M_m / d)) ^ 2.0)) / l))) * (d / (sqrt(l) * sqrt(h)));
	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[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[d, -1.8e+33], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(1.0 - t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -7.8e-144], N[(N[(t$95$0 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[N[(D$95$m * M$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.125 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -6e-300], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(h * N[(N[(-0.125 * N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.8000000000000001e33

   1. Initial program 79.5%

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

   3. Simplified 79.5%

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

      1. frac-2neg 79.5%

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

      2. sqrt-div 95.0%

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

   6. Applied egg-rr 93.6%

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

   if -1.8000000000000001e33 < d < -7.8000000000000003e-144

   1. Initial program 79.6%

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

   3. Simplified 79.6%

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

      1. expm1-log1p-u 78.5%

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

      2. expm1-undefine 78.5%

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

      3. *-commutative 78.5%

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

      4. *-commutative 78.5%

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

      5. div-inv 78.5%

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

      6. metadata-eval 78.5%

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

   6. Applied egg-rr 78.5%

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

      1. expm1-define 78.5%

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

      2. associate-*l/ 91.4%

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

      3. associate-/l* 91.4%

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

   8. Simplified 91.4%

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

   9. Taylor expanded in h around 0 59.6%

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

       1. associate-/r* 68.0%

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

       2. associate-*r/ 68.0%

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

       3. *-commutative 68.0%

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

       4. associate-/l* 68.0%

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

       5. associate-*r* 71.1%

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

       6. associate-/l* 71.1%

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

       7. unpow2 71.1%

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

       8. unpow2 71.1%

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

       9. swap-sqr 93.9%

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

       10. unpow2 93.9%

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

   11. Simplified 93.9%

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

   if -7.8000000000000003e-144 < d < -6.00000000000000048e-300

   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 46.9%

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

      1. frac-2neg 47.3%

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

      2. sqrt-div 52.6%

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

   6. Applied egg-rr 52.2%

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

   7. Taylor expanded in d around -inf 61.4%

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

   if -6.00000000000000048e-300 < d

   1. Initial program 58.9%

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

   3. Simplified 58.1%

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

      1. add-sqr-sqrt 58.1%

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

      2. pow2 58.1%

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

      3. sqrt-prod 58.1%

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

      4. sqrt-pow1 61.3%

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

      5. metadata-eval 61.3%

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

      6. pow1 61.3%

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

      7. *-commutative 61.3%

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

      8. div-inv 61.3%

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

      9. metadata-eval 61.3%

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

   6. Applied egg-rr 61.3%

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

   7. Taylor expanded in h around inf 43.6%

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

      1. sub-neg 43.6%

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

      2. distribute-lft-in 43.6%

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

      3. rgt-mult-inverse 43.6%

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

      4. distribute-lft-neg-in 43.6%

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

      5. metadata-eval 43.6%

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

      6. associate-/r* 44.5%

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

      7. associate-*r/ 44.5%

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

   9. Simplified 61.7%

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

       1. *-commutative 61.7%

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

       2. sqrt-div 64.9%

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

       3. sqrt-div 81.7%

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

       4. frac-times 81.7%

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

       5. add-sqr-sqrt 81.8%

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

   11. Applied egg-rr 81.8%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 + h \cdot \frac{-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.8 \cdot 10^{+33}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -7.8 \cdot 10^{-144}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left({\left(D \cdot M\right)}^{2} \cdot \frac{h}{{d}^{2}}\right) \cdot \frac{0.125}{\ell}\right)\\ \mathbf{elif}\;d \leq -6 \cdot 10^{-300}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + h \cdot \frac{-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \frac{M\_m}{2}\right)}^{2}\right)\\ \mathbf{if}\;h \leq -2.8 \cdot 10^{+112}:\\ \;\;\;\;\left(1 - t\_0\right) \cdot \frac{\sqrt{d \cdot \frac{d}{-\ell}}}{\sqrt{-h}}\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(t\_0 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + h \cdot \frac{-0.125 \cdot {\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2}}{\ell}\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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 (* 0.5 (* (/ h l) (pow (* (/ D_m d) (/ M_m 2.0)) 2.0)))))
   (if (<= h -2.8e+112)
     (* (- 1.0 t_0) (/ (sqrt (* d (/ d (- l)))) (sqrt (- h))))
     (if (<= h -5e-311)
       (* (* d (sqrt (/ 1.0 (* h l)))) (+ t_0 -1.0))
       (*
        (+ 1.0 (* h (/ (* -0.125 (pow (* D_m (/ M_m d)) 2.0)) l)))
        (/ d (* (sqrt l) (sqrt h))))))))

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 = 0.5 * ((h / l) * pow(((D_m / d) * (M_m / 2.0)), 2.0));
	double tmp;
	if (h <= -2.8e+112) {
		tmp = (1.0 - t_0) * (sqrt((d * (d / -l))) / sqrt(-h));
	} else if (h <= -5e-311) {
		tmp = (d * sqrt((1.0 / (h * l)))) * (t_0 + -1.0);
	} else {
		tmp = (1.0 + (h * ((-0.125 * pow((D_m * (M_m / d)), 2.0)) / l))) * (d / (sqrt(l) * sqrt(h)));
	}
	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 = 0.5d0 * ((h / l) * (((d_m / d) * (m_m / 2.0d0)) ** 2.0d0))
    if (h <= (-2.8d+112)) then
        tmp = (1.0d0 - t_0) * (sqrt((d * (d / -l))) / sqrt(-h))
    else if (h <= (-5d-311)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * (t_0 + (-1.0d0))
    else
        tmp = (1.0d0 + (h * (((-0.125d0) * ((d_m * (m_m / d)) ** 2.0d0)) / l))) * (d / (sqrt(l) * sqrt(h)))
    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 = 0.5 * ((h / l) * Math.pow(((D_m / d) * (M_m / 2.0)), 2.0));
	double tmp;
	if (h <= -2.8e+112) {
		tmp = (1.0 - t_0) * (Math.sqrt((d * (d / -l))) / Math.sqrt(-h));
	} else if (h <= -5e-311) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * (t_0 + -1.0);
	} else {
		tmp = (1.0 + (h * ((-0.125 * Math.pow((D_m * (M_m / d)), 2.0)) / l))) * (d / (Math.sqrt(l) * Math.sqrt(h)));
	}
	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 = 0.5 * ((h / l) * math.pow(((D_m / d) * (M_m / 2.0)), 2.0))
	tmp = 0
	if h <= -2.8e+112:
		tmp = (1.0 - t_0) * (math.sqrt((d * (d / -l))) / math.sqrt(-h))
	elif h <= -5e-311:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * (t_0 + -1.0)
	else:
		tmp = (1.0 + (h * ((-0.125 * math.pow((D_m * (M_m / d)), 2.0)) / l))) * (d / (math.sqrt(l) * math.sqrt(h)))
	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(0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(M_m / 2.0)) ^ 2.0)))
	tmp = 0.0
	if (h <= -2.8e+112)
		tmp = Float64(Float64(1.0 - t_0) * Float64(sqrt(Float64(d * Float64(d / Float64(-l)))) / sqrt(Float64(-h))));
	elseif (h <= -5e-311)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(t_0 + -1.0));
	else
		tmp = Float64(Float64(1.0 + Float64(h * Float64(Float64(-0.125 * (Float64(D_m * Float64(M_m / d)) ^ 2.0)) / l))) * Float64(d / Float64(sqrt(l) * sqrt(h))));
	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 = 0.5 * ((h / l) * (((D_m / d) * (M_m / 2.0)) ^ 2.0));
	tmp = 0.0;
	if (h <= -2.8e+112)
		tmp = (1.0 - t_0) * (sqrt((d * (d / -l))) / sqrt(-h));
	elseif (h <= -5e-311)
		tmp = (d * sqrt((1.0 / (h * l)))) * (t_0 + -1.0);
	else
		tmp = (1.0 + (h * ((-0.125 * ((D_m * (M_m / d)) ^ 2.0)) / l))) * (d / (sqrt(l) * sqrt(h)));
	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[(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]}, If[LessEqual[h, -2.8e+112], N[(N[(1.0 - t$95$0), $MachinePrecision] * N[(N[Sqrt[N[(d * N[(d / (-l)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -5e-311], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(h * N[(N[(-0.125 * N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if h < -2.8000000000000001e112

   1. Initial program 71.5%

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

   3. Simplified 71.5%

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

      1. frac-2neg 71.5%

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

      2. sqrt-div 77.1%

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

   6. Applied egg-rr 77.0%

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

      1. associate-*l/ 77.1%

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

      2. pow1/2 77.1%

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

      3. pow1/2 77.1%

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

      4. pow-prod-down 77.3%

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

   8. Applied egg-rr 77.3%

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

      1. unpow1/2 77.3%

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

      2. *-commutative 77.3%

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

   10. Simplified 77.3%

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

   if -2.8000000000000001e112 < h < -5.00000000000023e-311

   1. Initial program 70.4%

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

   3. Simplified 69.5%

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

      1. frac-2neg 70.5%

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

      2. sqrt-div 80.2%

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

   6. Applied egg-rr 78.3%

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

   7. Taylor expanded in d around -inf 78.2%

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

   if -5.00000000000023e-311 < h

   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 58.6%

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

      1. add-sqr-sqrt 58.6%

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

      2. pow2 58.6%

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

      3. sqrt-prod 58.6%

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

      4. sqrt-pow1 61.8%

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

      5. metadata-eval 61.8%

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

      6. pow1 61.8%

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

      7. *-commutative 61.8%

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

      8. div-inv 61.8%

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

      9. metadata-eval 61.8%

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

   6. Applied egg-rr 61.8%

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

   7. Taylor expanded in h around inf 43.9%

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

      1. sub-neg 43.9%

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

      2. distribute-lft-in 43.9%

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

      3. rgt-mult-inverse 44.0%

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

      4. distribute-lft-neg-in 44.0%

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

      5. metadata-eval 44.0%

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

      6. associate-/r* 44.9%

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

      7. associate-*r/ 44.9%

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

   9. Simplified 62.2%

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

       1. *-commutative 62.2%

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

       2. sqrt-div 65.4%

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

       3. sqrt-div 82.4%

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

       4. frac-times 82.4%

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

       5. add-sqr-sqrt 82.5%

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

   11. Applied egg-rr 82.5%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2.8 \cdot 10^{+112}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot \frac{\sqrt{d \cdot \frac{d}{-\ell}}}{\sqrt{-h}}\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + h \cdot \frac{-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := 1 + h \cdot \frac{-0.125 \cdot {\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2}}{\ell}\\ \mathbf{if}\;h \leq -1.22 \cdot 10^{-17}:\\ \;\;\;\;t\_0 \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \frac{M\_m}{2}\right)}^{2}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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 (+ 1.0 (* h (/ (* -0.125 (pow (* D_m (/ M_m d)) 2.0)) l)))))
   (if (<= h -1.22e-17)
     (* t_0 (* (sqrt (/ d l)) (sqrt (/ d h))))
     (if (<= h -5e-311)
       (*
        (* d (sqrt (/ 1.0 (* h l))))
        (+ (* 0.5 (* (/ h l) (pow (* (/ D_m d) (/ M_m 2.0)) 2.0))) -1.0))
       (* t_0 (/ d (* (sqrt l) (sqrt h))))))))

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 = 1.0 + (h * ((-0.125 * pow((D_m * (M_m / d)), 2.0)) / l));
	double tmp;
	if (h <= -1.22e-17) {
		tmp = t_0 * (sqrt((d / l)) * sqrt((d / h)));
	} else if (h <= -5e-311) {
		tmp = (d * sqrt((1.0 / (h * l)))) * ((0.5 * ((h / l) * pow(((D_m / d) * (M_m / 2.0)), 2.0))) + -1.0);
	} else {
		tmp = t_0 * (d / (sqrt(l) * sqrt(h)));
	}
	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 = 1.0d0 + (h * (((-0.125d0) * ((d_m * (m_m / d)) ** 2.0d0)) / l))
    if (h <= (-1.22d-17)) then
        tmp = t_0 * (sqrt((d / l)) * sqrt((d / h)))
    else if (h <= (-5d-311)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((0.5d0 * ((h / l) * (((d_m / d) * (m_m / 2.0d0)) ** 2.0d0))) + (-1.0d0))
    else
        tmp = t_0 * (d / (sqrt(l) * sqrt(h)))
    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 = 1.0 + (h * ((-0.125 * Math.pow((D_m * (M_m / d)), 2.0)) / l));
	double tmp;
	if (h <= -1.22e-17) {
		tmp = t_0 * (Math.sqrt((d / l)) * Math.sqrt((d / h)));
	} else if (h <= -5e-311) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * ((0.5 * ((h / l) * Math.pow(((D_m / d) * (M_m / 2.0)), 2.0))) + -1.0);
	} else {
		tmp = t_0 * (d / (Math.sqrt(l) * Math.sqrt(h)));
	}
	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 = 1.0 + (h * ((-0.125 * math.pow((D_m * (M_m / d)), 2.0)) / l))
	tmp = 0
	if h <= -1.22e-17:
		tmp = t_0 * (math.sqrt((d / l)) * math.sqrt((d / h)))
	elif h <= -5e-311:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * ((0.5 * ((h / l) * math.pow(((D_m / d) * (M_m / 2.0)), 2.0))) + -1.0)
	else:
		tmp = t_0 * (d / (math.sqrt(l) * math.sqrt(h)))
	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(1.0 + Float64(h * Float64(Float64(-0.125 * (Float64(D_m * Float64(M_m / d)) ^ 2.0)) / l)))
	tmp = 0.0
	if (h <= -1.22e-17)
		tmp = Float64(t_0 * Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))));
	elseif (h <= -5e-311)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(M_m / 2.0)) ^ 2.0))) + -1.0));
	else
		tmp = Float64(t_0 * Float64(d / Float64(sqrt(l) * sqrt(h))));
	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 = 1.0 + (h * ((-0.125 * ((D_m * (M_m / d)) ^ 2.0)) / l));
	tmp = 0.0;
	if (h <= -1.22e-17)
		tmp = t_0 * (sqrt((d / l)) * sqrt((d / h)));
	elseif (h <= -5e-311)
		tmp = (d * sqrt((1.0 / (h * l)))) * ((0.5 * ((h / l) * (((D_m / d) * (M_m / 2.0)) ^ 2.0))) + -1.0);
	else
		tmp = t_0 * (d / (sqrt(l) * sqrt(h)));
	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[(1.0 + N[(h * N[(N[(-0.125 * N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -1.22e-17], N[(t$95$0 * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -5e-311], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(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] + -1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.22e-17

   1. Initial program 75.4%

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

   3. Simplified 75.4%

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

      1. add-sqr-sqrt 75.4%

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

      2. pow2 75.4%

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

      3. sqrt-prod 75.4%

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

      4. sqrt-pow1 76.7%

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

      5. metadata-eval 76.7%

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

      6. pow1 76.7%

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

      7. *-commutative 76.7%

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

      8. div-inv 76.7%

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

      9. metadata-eval 76.7%

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

   6. Applied egg-rr 76.7%

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

   7. Taylor expanded in h around inf 47.4%

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

      1. sub-neg 47.4%

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

      2. distribute-lft-in 47.4%

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

      3. rgt-mult-inverse 47.5%

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

      4. distribute-lft-neg-in 47.5%

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

      5. metadata-eval 47.5%

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

      6. associate-/r* 50.3%

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

      7. associate-*r/ 50.3%

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

   9. Simplified 79.0%

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

   if -1.22e-17 < h < -5.00000000000023e-311

   1. Initial program 65.7%

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

   3. Simplified 64.3%

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

      1. frac-2neg 64.4%

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

      2. sqrt-div 79.5%

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

   6. Applied egg-rr 78.0%

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

   7. Taylor expanded in d around -inf 76.7%

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

   if -5.00000000000023e-311 < h

   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 58.6%

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

      1. add-sqr-sqrt 58.6%

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

      2. pow2 58.6%

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

      3. sqrt-prod 58.6%

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

      4. sqrt-pow1 61.8%

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

      5. metadata-eval 61.8%

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

      6. pow1 61.8%

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

      7. *-commutative 61.8%

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

      8. div-inv 61.8%

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

      9. metadata-eval 61.8%

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

   6. Applied egg-rr 61.8%

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

   7. Taylor expanded in h around inf 43.9%

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

      1. sub-neg 43.9%

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

      2. distribute-lft-in 43.9%

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

      3. rgt-mult-inverse 44.0%

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

      4. distribute-lft-neg-in 44.0%

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

      5. metadata-eval 44.0%

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

      6. associate-/r* 44.9%

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

      7. associate-*r/ 44.9%

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

   9. Simplified 62.2%

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

       1. *-commutative 62.2%

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

       2. sqrt-div 65.4%

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

       3. sqrt-div 82.4%

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

       4. frac-times 82.4%

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

       5. add-sqr-sqrt 82.5%

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

   11. Applied egg-rr 82.5%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1.22 \cdot 10^{-17}:\\ \;\;\;\;\left(1 + h \cdot \frac{-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + h \cdot \frac{-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -3 \cdot 10^{+98}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \frac{M\_m}{2}\right)}^{2}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + h \cdot \frac{-0.125 \cdot {\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2}}{\ell}\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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 -3e+98)
   (*
    (sqrt (* (/ d l) (/ d h)))
    (+ 1.0 (* -0.5 (* h (/ (pow (* M_m (* 0.5 (/ D_m d))) 2.0) l)))))
   (if (<= h -5e-311)
     (*
      (* d (sqrt (/ 1.0 (* h l))))
      (+ (* 0.5 (* (/ h l) (pow (* (/ D_m d) (/ M_m 2.0)) 2.0))) -1.0))
     (*
      (+ 1.0 (* h (/ (* -0.125 (pow (* D_m (/ M_m d)) 2.0)) l)))
      (/ d (* (sqrt l) (sqrt h)))))))

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 <= -3e+98) {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * (h * (pow((M_m * (0.5 * (D_m / d))), 2.0) / l))));
	} else if (h <= -5e-311) {
		tmp = (d * sqrt((1.0 / (h * l)))) * ((0.5 * ((h / l) * pow(((D_m / d) * (M_m / 2.0)), 2.0))) + -1.0);
	} else {
		tmp = (1.0 + (h * ((-0.125 * pow((D_m * (M_m / d)), 2.0)) / l))) * (d / (sqrt(l) * sqrt(h)));
	}
	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 <= (-3d+98)) then
        tmp = sqrt(((d / l) * (d / h))) * (1.0d0 + ((-0.5d0) * (h * (((m_m * (0.5d0 * (d_m / d))) ** 2.0d0) / l))))
    else if (h <= (-5d-311)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((0.5d0 * ((h / l) * (((d_m / d) * (m_m / 2.0d0)) ** 2.0d0))) + (-1.0d0))
    else
        tmp = (1.0d0 + (h * (((-0.125d0) * ((d_m * (m_m / d)) ** 2.0d0)) / l))) * (d / (sqrt(l) * sqrt(h)))
    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 <= -3e+98) {
		tmp = Math.sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * (h * (Math.pow((M_m * (0.5 * (D_m / d))), 2.0) / l))));
	} else if (h <= -5e-311) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * ((0.5 * ((h / l) * Math.pow(((D_m / d) * (M_m / 2.0)), 2.0))) + -1.0);
	} else {
		tmp = (1.0 + (h * ((-0.125 * Math.pow((D_m * (M_m / d)), 2.0)) / l))) * (d / (Math.sqrt(l) * Math.sqrt(h)));
	}
	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 <= -3e+98:
		tmp = math.sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * (h * (math.pow((M_m * (0.5 * (D_m / d))), 2.0) / l))))
	elif h <= -5e-311:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * ((0.5 * ((h / l) * math.pow(((D_m / d) * (M_m / 2.0)), 2.0))) + -1.0)
	else:
		tmp = (1.0 + (h * ((-0.125 * math.pow((D_m * (M_m / d)), 2.0)) / l))) * (d / (math.sqrt(l) * math.sqrt(h)))
	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 <= -3e+98)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 + Float64(-0.5 * Float64(h * Float64((Float64(M_m * Float64(0.5 * Float64(D_m / d))) ^ 2.0) / l)))));
	elseif (h <= -5e-311)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(M_m / 2.0)) ^ 2.0))) + -1.0));
	else
		tmp = Float64(Float64(1.0 + Float64(h * Float64(Float64(-0.125 * (Float64(D_m * Float64(M_m / d)) ^ 2.0)) / l))) * Float64(d / Float64(sqrt(l) * sqrt(h))));
	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 <= -3e+98)
		tmp = sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * (h * (((M_m * (0.5 * (D_m / d))) ^ 2.0) / l))));
	elseif (h <= -5e-311)
		tmp = (d * sqrt((1.0 / (h * l)))) * ((0.5 * ((h / l) * (((D_m / d) * (M_m / 2.0)) ^ 2.0))) + -1.0);
	else
		tmp = (1.0 + (h * ((-0.125 * ((D_m * (M_m / d)) ^ 2.0)) / l))) * (d / (sqrt(l) * sqrt(h)));
	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, -3e+98], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(h * N[(N[Power[N[(M$95$m * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -5e-311], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(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] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(h * N[(N[(-0.125 * N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -3.0000000000000001e98

   1. Initial program 71.2%

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

   3. Simplified 71.2%

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

      1. expm1-log1p-u 71.0%

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

      2. expm1-undefine 71.0%

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

      3. *-commutative 71.0%

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

      4. *-commutative 71.0%

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

      5. div-inv 71.0%

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

      6. metadata-eval 71.0%

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

   6. Applied egg-rr 71.0%

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

      1. expm1-define 71.0%

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

      2. associate-*l/ 71.0%

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

      3. associate-/l* 73.3%

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

   8. Simplified 73.3%

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

      1. pow1 73.3%

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

   10. Applied egg-rr 61.1%

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

       1. unpow1 61.1%

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

       2. *-commutative 61.1%

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

       3. associate-*r/ 63.6%

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

       4. associate-*l* 63.6%

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

   12. Simplified 63.6%

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

   if -3.0000000000000001e98 < h < -5.00000000000023e-311

   1. Initial program 70.5%

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

   3. Simplified 69.5%

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

      1. frac-2neg 70.6%

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

      2. sqrt-div 80.6%

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

   6. Applied egg-rr 78.6%

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

   7. Taylor expanded in d around -inf 78.5%

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

   if -5.00000000000023e-311 < h

   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 58.6%

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

      1. add-sqr-sqrt 58.6%

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

      2. pow2 58.6%

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

      3. sqrt-prod 58.6%

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

      4. sqrt-pow1 61.8%

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

      5. metadata-eval 61.8%

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

      6. pow1 61.8%

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

      7. *-commutative 61.8%

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

      8. div-inv 61.8%

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

      9. metadata-eval 61.8%

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

   6. Applied egg-rr 61.8%

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

   7. Taylor expanded in h around inf 43.9%

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

      1. sub-neg 43.9%

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

      2. distribute-lft-in 43.9%

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

      3. rgt-mult-inverse 44.0%

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

      4. distribute-lft-neg-in 44.0%

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

      5. metadata-eval 44.0%

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

      6. associate-/r* 44.9%

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

      7. associate-*r/ 44.9%

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

   9. Simplified 62.2%

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

       1. *-commutative 62.2%

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

       2. sqrt-div 65.4%

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

       3. sqrt-div 82.4%

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

       4. frac-times 82.4%

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

       5. add-sqr-sqrt 82.5%

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

   11. Applied egg-rr 82.5%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -3 \cdot 10^{+98}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + h \cdot \frac{-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -1.4 \cdot 10^{+98}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \frac{M\_m}{2}\right)}^{2}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{h}{\ell} \cdot {\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2}, 1\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \end{array} \]

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 (<= h -1.4e+98)
   (*
    (sqrt (* (/ d l) (/ d h)))
    (+ 1.0 (* -0.5 (* h (/ (pow (* M_m (* 0.5 (/ D_m d))) 2.0) l)))))
   (if (<= h -5e-311)
     (*
      (* d (sqrt (/ 1.0 (* h l))))
      (+ (* 0.5 (* (/ h l) (pow (* (/ D_m d) (/ M_m 2.0)) 2.0))) -1.0))
     (*
      (fma -0.125 (* (/ h l) (pow (* D_m (/ M_m d)) 2.0)) 1.0)
      (/ 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 (h <= -1.4e+98) {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * (h * (pow((M_m * (0.5 * (D_m / d))), 2.0) / l))));
	} else if (h <= -5e-311) {
		tmp = (d * sqrt((1.0 / (h * l)))) * ((0.5 * ((h / l) * pow(((D_m / d) * (M_m / 2.0)), 2.0))) + -1.0);
	} else {
		tmp = fma(-0.125, ((h / l) * pow((D_m * (M_m / d)), 2.0)), 1.0) * (d / 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 (h <= -1.4e+98)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 + Float64(-0.5 * Float64(h * Float64((Float64(M_m * Float64(0.5 * Float64(D_m / d))) ^ 2.0) / l)))));
	elseif (h <= -5e-311)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(M_m / 2.0)) ^ 2.0))) + -1.0));
	else
		tmp = Float64(fma(-0.125, Float64(Float64(h / l) * (Float64(D_m * Float64(M_m / d)) ^ 2.0)), 1.0) * Float64(d / sqrt(Float64(h * 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_] := If[LessEqual[h, -1.4e+98], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(h * N[(N[Power[N[(M$95$m * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -5e-311], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(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] + -1.0), $MachinePrecision]), $MachinePrecision], 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] + 1.0), $MachinePrecision] * N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.4e98

   1. Initial program 71.2%

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

   3. Simplified 71.2%

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

      1. expm1-log1p-u 71.0%

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

      2. expm1-undefine 71.0%

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

      3. *-commutative 71.0%

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

      4. *-commutative 71.0%

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

      5. div-inv 71.0%

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

      6. metadata-eval 71.0%

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

   6. Applied egg-rr 71.0%

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

      1. expm1-define 71.0%

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

      2. associate-*l/ 71.0%

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

      3. associate-/l* 73.3%

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

   8. Simplified 73.3%

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

      1. pow1 73.3%

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

   10. Applied egg-rr 61.1%

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

       1. unpow1 61.1%

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

       2. *-commutative 61.1%

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

       3. associate-*r/ 63.6%

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

       4. associate-*l* 63.6%

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

   12. Simplified 63.6%

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

   if -1.4e98 < h < -5.00000000000023e-311

   1. Initial program 70.5%

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

   3. Simplified 69.5%

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

      1. frac-2neg 70.6%

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

      2. sqrt-div 80.6%

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

   6. Applied egg-rr 78.6%

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

   7. Taylor expanded in d around -inf 78.5%

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

   if -5.00000000000023e-311 < h

   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 58.6%

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

      1. add-sqr-sqrt 58.6%

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

      2. pow2 58.6%

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

      3. sqrt-prod 58.6%

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

      4. sqrt-pow1 61.8%

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

      5. metadata-eval 61.8%

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

      6. pow1 61.8%

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

      7. *-commutative 61.8%

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

      8. div-inv 61.8%

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

      9. metadata-eval 61.8%

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

   6. Applied egg-rr 61.8%

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

      1. pow1 61.8%

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

   8. Applied egg-rr 73.2%

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

   10. Simplified 74.0%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1.4 \cdot 10^{+98}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}, 1\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \frac{M\_m}{2}\right)}^{2}\right)\\ \mathbf{if}\;h \leq -9 \cdot 10^{+95}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(t\_0 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - t\_0\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* 0.5 (* (/ h l) (pow (* (/ D_m d) (/ M_m 2.0)) 2.0)))))
   (if (<= h -9e+95)
     (*
      (sqrt (* (/ d l) (/ d h)))
      (+ 1.0 (* -0.5 (* h (/ (pow (* M_m (* 0.5 (/ D_m d))) 2.0) l)))))
     (if (<= h -5e-311)
       (* (* d (sqrt (/ 1.0 (* h l)))) (+ t_0 -1.0))
       (* (- 1.0 t_0) (/ 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 t_0 = 0.5 * ((h / l) * pow(((D_m / d) * (M_m / 2.0)), 2.0));
	double tmp;
	if (h <= -9e+95) {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * (h * (pow((M_m * (0.5 * (D_m / d))), 2.0) / l))));
	} else if (h <= -5e-311) {
		tmp = (d * sqrt((1.0 / (h * l)))) * (t_0 + -1.0);
	} else {
		tmp = (1.0 - t_0) * (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) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * ((h / l) * (((d_m / d) * (m_m / 2.0d0)) ** 2.0d0))
    if (h <= (-9d+95)) then
        tmp = sqrt(((d / l) * (d / h))) * (1.0d0 + ((-0.5d0) * (h * (((m_m * (0.5d0 * (d_m / d))) ** 2.0d0) / l))))
    else if (h <= (-5d-311)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * (t_0 + (-1.0d0))
    else
        tmp = (1.0d0 - t_0) * (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 t_0 = 0.5 * ((h / l) * Math.pow(((D_m / d) * (M_m / 2.0)), 2.0));
	double tmp;
	if (h <= -9e+95) {
		tmp = Math.sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * (h * (Math.pow((M_m * (0.5 * (D_m / d))), 2.0) / l))));
	} else if (h <= -5e-311) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * (t_0 + -1.0);
	} else {
		tmp = (1.0 - t_0) * (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):
	t_0 = 0.5 * ((h / l) * math.pow(((D_m / d) * (M_m / 2.0)), 2.0))
	tmp = 0
	if h <= -9e+95:
		tmp = math.sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * (h * (math.pow((M_m * (0.5 * (D_m / d))), 2.0) / l))))
	elif h <= -5e-311:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * (t_0 + -1.0)
	else:
		tmp = (1.0 - t_0) * (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)
	t_0 = Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(M_m / 2.0)) ^ 2.0)))
	tmp = 0.0
	if (h <= -9e+95)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 + Float64(-0.5 * Float64(h * Float64((Float64(M_m * Float64(0.5 * Float64(D_m / d))) ^ 2.0) / l)))));
	elseif (h <= -5e-311)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(t_0 + -1.0));
	else
		tmp = Float64(Float64(1.0 - t_0) * 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)
	t_0 = 0.5 * ((h / l) * (((D_m / d) * (M_m / 2.0)) ^ 2.0));
	tmp = 0.0;
	if (h <= -9e+95)
		tmp = sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * (h * (((M_m * (0.5 * (D_m / d))) ^ 2.0) / l))));
	elseif (h <= -5e-311)
		tmp = (d * sqrt((1.0 / (h * l)))) * (t_0 + -1.0);
	else
		tmp = (1.0 - t_0) * (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_] := Block[{t$95$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]}, If[LessEqual[h, -9e+95], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(h * N[(N[Power[N[(M$95$m * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -5e-311], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] * N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if h < -9.00000000000000033e95

   1. Initial program 71.2%

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

   3. Simplified 71.2%

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

      1. expm1-log1p-u 71.0%

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

      2. expm1-undefine 71.0%

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

      3. *-commutative 71.0%

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

      4. *-commutative 71.0%

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

      5. div-inv 71.0%

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

      6. metadata-eval 71.0%

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

   6. Applied egg-rr 71.0%

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

      1. expm1-define 71.0%

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

      2. associate-*l/ 71.0%

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

      3. associate-/l* 73.3%

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

   8. Simplified 73.3%

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

      1. pow1 73.3%

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

   10. Applied egg-rr 61.1%

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

       1. unpow1 61.1%

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

       2. *-commutative 61.1%

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

       3. associate-*r/ 63.6%

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

       4. associate-*l* 63.6%

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

   12. Simplified 63.6%

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

   if -9.00000000000000033e95 < h < -5.00000000000023e-311

   1. Initial program 70.5%

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

   3. Simplified 69.5%

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

      1. frac-2neg 70.6%

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

      2. sqrt-div 80.6%

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

   6. Applied egg-rr 78.6%

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

   7. Taylor expanded in d around -inf 78.5%

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

   if -5.00000000000023e-311 < h

   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 58.6%

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

      1. *-commutative 62.2%

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

      2. sqrt-div 65.4%

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

      3. sqrt-div 82.4%

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

      4. frac-times 82.4%

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

      5. add-sqr-sqrt 82.5%

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

   6. Applied egg-rr 77.2%

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

   7. Taylor expanded in l around 0 73.2%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -9 \cdot 10^{+95}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -9 \cdot 10^{+130}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq 2.1 \cdot 10^{-265}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \frac{M\_m}{2}\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \end{array} \]

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 (<= l -9e+130)
   (* d (- (sqrt (/ 1.0 (* h l)))))
   (if (<= l 2.1e-265)
     (*
      (sqrt (* (/ d l) (/ d h)))
      (+ 1.0 (* -0.5 (* h (/ (pow (* M_m (* 0.5 (/ D_m d))) 2.0) l)))))
     (*
      (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D_m d) (/ M_m 2.0)) 2.0))))
      (/ 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 (l <= -9e+130) {
		tmp = d * -sqrt((1.0 / (h * l)));
	} else if (l <= 2.1e-265) {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * (h * (pow((M_m * (0.5 * (D_m / d))), 2.0) / l))));
	} else {
		tmp = (1.0 - (0.5 * ((h / l) * pow(((D_m / d) * (M_m / 2.0)), 2.0)))) * (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 (l <= (-9d+130)) then
        tmp = d * -sqrt((1.0d0 / (h * l)))
    else if (l <= 2.1d-265) then
        tmp = sqrt(((d / l) * (d / h))) * (1.0d0 + ((-0.5d0) * (h * (((m_m * (0.5d0 * (d_m / d))) ** 2.0d0) / l))))
    else
        tmp = (1.0d0 - (0.5d0 * ((h / l) * (((d_m / d) * (m_m / 2.0d0)) ** 2.0d0)))) * (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 (l <= -9e+130) {
		tmp = d * -Math.sqrt((1.0 / (h * l)));
	} else if (l <= 2.1e-265) {
		tmp = Math.sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * (h * (Math.pow((M_m * (0.5 * (D_m / d))), 2.0) / l))));
	} else {
		tmp = (1.0 - (0.5 * ((h / l) * Math.pow(((D_m / d) * (M_m / 2.0)), 2.0)))) * (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 l <= -9e+130:
		tmp = d * -math.sqrt((1.0 / (h * l)))
	elif l <= 2.1e-265:
		tmp = math.sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * (h * (math.pow((M_m * (0.5 * (D_m / d))), 2.0) / l))))
	else:
		tmp = (1.0 - (0.5 * ((h / l) * math.pow(((D_m / d) * (M_m / 2.0)), 2.0)))) * (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 (l <= -9e+130)
		tmp = Float64(d * Float64(-sqrt(Float64(1.0 / Float64(h * l)))));
	elseif (l <= 2.1e-265)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 + Float64(-0.5 * Float64(h * Float64((Float64(M_m * Float64(0.5 * Float64(D_m / d))) ^ 2.0) / l)))));
	else
		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 / 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 (l <= -9e+130)
		tmp = d * -sqrt((1.0 / (h * l)));
	elseif (l <= 2.1e-265)
		tmp = sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * (h * (((M_m * (0.5 * (D_m / d))) ^ 2.0) / l))));
	else
		tmp = (1.0 - (0.5 * ((h / l) * (((D_m / d) * (M_m / 2.0)) ^ 2.0)))) * (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[l, -9e+130], N[(d * (-N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, 2.1e-265], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(h * N[(N[Power[N[(M$95$m * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -9.00000000000000078e130

   1. Initial program 56.1%

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

   3. Simplified 56.1%

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

   5. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 62.2%

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

      5. neg-mul-1 62.2%

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

   7. Simplified 62.2%

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

   if -9.00000000000000078e130 < l < 2.10000000000000004e-265

   1. Initial program 74.9%

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

   3. Simplified 74.0%

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

      1. expm1-log1p-u 73.7%

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

      2. expm1-undefine 73.7%

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

      3. *-commutative 73.7%

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

      4. *-commutative 73.7%

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

      5. div-inv 73.7%

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

      6. metadata-eval 73.7%

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

   6. Applied egg-rr 73.7%

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

      1. expm1-define 73.7%

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

      2. associate-*l/ 79.7%

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

      3. associate-/l* 79.7%

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

   8. Simplified 79.7%

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

      1. pow1 79.7%

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

   10. Applied egg-rr 73.6%

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

       1. unpow1 73.6%

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

       2. *-commutative 73.6%

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

       3. associate-*r/ 73.6%

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

       4. associate-*l* 73.6%

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

   12. Simplified 73.6%

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

   if 2.10000000000000004e-265 < l

   1. Initial program 59.3%

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

   3. Simplified 58.4%

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

      1. *-commutative 60.4%

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

      2. sqrt-div 63.2%

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

      3. sqrt-div 80.8%

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

      4. frac-times 80.8%

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

      5. add-sqr-sqrt 80.9%

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

   6. Applied egg-rr 77.9%

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

   7. Taylor expanded in l around 0 73.5%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9 \cdot 10^{+130}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq 2.1 \cdot 10^{-265}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.25 \cdot 10^{+131}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq 7.2 \cdot 10^{-266}:\\ \;\;\;\;\left(1 + h \cdot \frac{-0.125 \cdot {\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2}}{\ell}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \frac{M\_m}{2}\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \end{array} \]

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 (<= l -1.25e+131)
   (* d (- (sqrt (/ 1.0 (* h l)))))
   (if (<= l 7.2e-266)
     (*
      (+ 1.0 (* h (/ (* -0.125 (pow (* D_m (/ M_m d)) 2.0)) l)))
      (sqrt (* (/ d l) (/ d h))))
     (*
      (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D_m d) (/ M_m 2.0)) 2.0))))
      (/ 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 (l <= -1.25e+131) {
		tmp = d * -sqrt((1.0 / (h * l)));
	} else if (l <= 7.2e-266) {
		tmp = (1.0 + (h * ((-0.125 * pow((D_m * (M_m / d)), 2.0)) / l))) * sqrt(((d / l) * (d / h)));
	} else {
		tmp = (1.0 - (0.5 * ((h / l) * pow(((D_m / d) * (M_m / 2.0)), 2.0)))) * (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 (l <= (-1.25d+131)) then
        tmp = d * -sqrt((1.0d0 / (h * l)))
    else if (l <= 7.2d-266) then
        tmp = (1.0d0 + (h * (((-0.125d0) * ((d_m * (m_m / d)) ** 2.0d0)) / l))) * sqrt(((d / l) * (d / h)))
    else
        tmp = (1.0d0 - (0.5d0 * ((h / l) * (((d_m / d) * (m_m / 2.0d0)) ** 2.0d0)))) * (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 (l <= -1.25e+131) {
		tmp = d * -Math.sqrt((1.0 / (h * l)));
	} else if (l <= 7.2e-266) {
		tmp = (1.0 + (h * ((-0.125 * Math.pow((D_m * (M_m / d)), 2.0)) / l))) * Math.sqrt(((d / l) * (d / h)));
	} else {
		tmp = (1.0 - (0.5 * ((h / l) * Math.pow(((D_m / d) * (M_m / 2.0)), 2.0)))) * (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 l <= -1.25e+131:
		tmp = d * -math.sqrt((1.0 / (h * l)))
	elif l <= 7.2e-266:
		tmp = (1.0 + (h * ((-0.125 * math.pow((D_m * (M_m / d)), 2.0)) / l))) * math.sqrt(((d / l) * (d / h)))
	else:
		tmp = (1.0 - (0.5 * ((h / l) * math.pow(((D_m / d) * (M_m / 2.0)), 2.0)))) * (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 (l <= -1.25e+131)
		tmp = Float64(d * Float64(-sqrt(Float64(1.0 / Float64(h * l)))));
	elseif (l <= 7.2e-266)
		tmp = Float64(Float64(1.0 + Float64(h * Float64(Float64(-0.125 * (Float64(D_m * Float64(M_m / d)) ^ 2.0)) / l))) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	else
		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 / 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 (l <= -1.25e+131)
		tmp = d * -sqrt((1.0 / (h * l)));
	elseif (l <= 7.2e-266)
		tmp = (1.0 + (h * ((-0.125 * ((D_m * (M_m / d)) ^ 2.0)) / l))) * sqrt(((d / l) * (d / h)));
	else
		tmp = (1.0 - (0.5 * ((h / l) * (((D_m / d) * (M_m / 2.0)) ^ 2.0)))) * (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[l, -1.25e+131], N[(d * (-N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, 7.2e-266], N[(N[(1.0 + N[(h * N[(N[(-0.125 * N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 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[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.24999999999999999e131

   1. Initial program 56.1%

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

   3. Simplified 56.1%

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

   5. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 62.2%

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

      5. neg-mul-1 62.2%

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

   7. Simplified 62.2%

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

   if -1.24999999999999999e131 < l < 7.1999999999999999e-266

   1. Initial program 74.9%

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

   3. Simplified 74.0%

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

      1. add-sqr-sqrt 74.0%

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

      2. pow2 74.0%

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

      3. sqrt-prod 74.0%

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

      4. sqrt-pow1 74.0%

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

      5. metadata-eval 74.0%

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

      6. pow1 74.0%

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

      7. *-commutative 74.0%

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

      8. div-inv 74.0%

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

      9. metadata-eval 74.0%

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

   6. Applied egg-rr 74.0%

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

   7. Taylor expanded in h around inf 51.7%

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

      1. sub-neg 51.7%

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

      2. distribute-lft-in 51.7%

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

      3. rgt-mult-inverse 51.8%

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

      4. distribute-lft-neg-in 51.8%

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

      5. metadata-eval 51.8%

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

      6. associate-/r* 54.4%

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

      7. associate-*r/ 54.4%

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

   9. Simplified 79.3%

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

       1. *-commutative 79.3%

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

       2. sqrt-unprod 72.9%

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

   11. Applied egg-rr 72.9%

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

   if 7.1999999999999999e-266 < l

   1. Initial program 59.3%

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

   3. Simplified 58.4%

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

      1. *-commutative 60.4%

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

      2. sqrt-div 63.2%

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

      3. sqrt-div 80.8%

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

      4. frac-times 80.8%

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

      5. add-sqr-sqrt 80.9%

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

   6. Applied egg-rr 77.9%

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

   7. Taylor expanded in l around 0 73.5%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.25 \cdot 10^{+131}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq 7.2 \cdot 10^{-266}:\\ \;\;\;\;\left(1 + h \cdot \frac{-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -8.8 \cdot 10^{+130}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq 3.4 \cdot 10^{+99}:\\ \;\;\;\;\left(1 + h \cdot \frac{-0.125 \cdot {\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2}}{\ell}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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 -8.8e+130)
   (* d (- (sqrt (/ 1.0 (* h l)))))
   (if (<= l 3.4e+99)
     (*
      (+ 1.0 (* h (/ (* -0.125 (pow (* D_m (/ M_m d)) 2.0)) l)))
      (sqrt (* (/ d l) (/ d h))))
     (/ d (* (sqrt l) (sqrt h))))))

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 <= -8.8e+130) {
		tmp = d * -sqrt((1.0 / (h * l)));
	} else if (l <= 3.4e+99) {
		tmp = (1.0 + (h * ((-0.125 * pow((D_m * (M_m / d)), 2.0)) / l))) * sqrt(((d / l) * (d / h)));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	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 <= (-8.8d+130)) then
        tmp = d * -sqrt((1.0d0 / (h * l)))
    else if (l <= 3.4d+99) then
        tmp = (1.0d0 + (h * (((-0.125d0) * ((d_m * (m_m / d)) ** 2.0d0)) / l))) * sqrt(((d / l) * (d / h)))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    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 <= -8.8e+130) {
		tmp = d * -Math.sqrt((1.0 / (h * l)));
	} else if (l <= 3.4e+99) {
		tmp = (1.0 + (h * ((-0.125 * Math.pow((D_m * (M_m / d)), 2.0)) / l))) * Math.sqrt(((d / l) * (d / h)));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	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 <= -8.8e+130:
		tmp = d * -math.sqrt((1.0 / (h * l)))
	elif l <= 3.4e+99:
		tmp = (1.0 + (h * ((-0.125 * math.pow((D_m * (M_m / d)), 2.0)) / l))) * math.sqrt(((d / l) * (d / h)))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	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 <= -8.8e+130)
		tmp = Float64(d * Float64(-sqrt(Float64(1.0 / Float64(h * l)))));
	elseif (l <= 3.4e+99)
		tmp = Float64(Float64(1.0 + Float64(h * Float64(Float64(-0.125 * (Float64(D_m * Float64(M_m / d)) ^ 2.0)) / l))) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	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 <= -8.8e+130)
		tmp = d * -sqrt((1.0 / (h * l)));
	elseif (l <= 3.4e+99)
		tmp = (1.0 + (h * ((-0.125 * ((D_m * (M_m / d)) ^ 2.0)) / l))) * sqrt(((d / l) * (d / h)));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	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, -8.8e+130], N[(d * (-N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, 3.4e+99], N[(N[(1.0 + N[(h * N[(N[(-0.125 * N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -8.79999999999999974e130

   1. Initial program 56.1%

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

   3. Simplified 56.1%

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

   5. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 62.2%

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

      5. neg-mul-1 62.2%

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

   7. Simplified 62.2%

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

   if -8.79999999999999974e130 < l < 3.39999999999999984e99

   1. Initial program 75.5%

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

   3. Simplified 74.4%

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

      1. add-sqr-sqrt 74.4%

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

      2. pow2 74.4%

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

      3. sqrt-prod 74.4%

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

      4. sqrt-pow1 74.4%

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

      5. metadata-eval 74.4%

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

      6. pow1 74.4%

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

      7. *-commutative 74.4%

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

      8. div-inv 74.4%

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

      9. metadata-eval 74.4%

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

   6. Applied egg-rr 74.4%

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

   7. Taylor expanded in h around inf 53.6%

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

      1. sub-neg 53.6%

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

      2. distribute-lft-in 53.6%

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

      3. rgt-mult-inverse 53.6%

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

      4. distribute-lft-neg-in 53.6%

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

      5. metadata-eval 53.6%

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

      6. associate-/r* 55.8%

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

      7. associate-*r/ 55.8%

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

   9. Simplified 78.8%

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

       1. *-commutative 78.8%

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

       2. sqrt-unprod 73.2%

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

   11. Applied egg-rr 73.2%

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

   if 3.39999999999999984e99 < l

   1. Initial program 32.5%

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

   3. Simplified 32.4%

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

   5. Taylor expanded in d around inf 46.0%

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

      1. sqrt-div 46.0%

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

      2. metadata-eval 46.0%

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

      3. *-commutative 46.0%

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

      4. sqrt-unprod 48.3%

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

      5. div-inv 48.3%

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

      6. associate-/r* 48.5%

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

   7. Applied egg-rr 48.5%

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

      1. associate-/l/ 48.3%

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

   9. Simplified 48.3%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8.8 \cdot 10^{+130}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq 3.4 \cdot 10^{+99}:\\ \;\;\;\;\left(1 + h \cdot \frac{-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 49.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;D\_m \leq 1.45 \cdot 10^{+141}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(-0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot M\_m\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

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_m 1.45e+141)
   (* (sqrt (/ d l)) (sqrt (/ d h)))
   (*
    (sqrt (* (/ d l) (/ d h)))
    (* -0.125 (* (/ h l) (pow (* (/ D_m d) M_m) 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_m <= 1.45e+141) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = sqrt(((d / l) * (d / h))) * (-0.125 * ((h / l) * pow(((D_m / d) * M_m), 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_m <= 1.45d+141) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else
        tmp = sqrt(((d / l) * (d / h))) * ((-0.125d0) * ((h / l) * (((d_m / d) * m_m) ** 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_m <= 1.45e+141) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else {
		tmp = Math.sqrt(((d / l) * (d / h))) * (-0.125 * ((h / l) * Math.pow(((D_m / d) * M_m), 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_m <= 1.45e+141:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	else:
		tmp = math.sqrt(((d / l) * (d / h))) * (-0.125 * ((h / l) * math.pow(((D_m / d) * M_m), 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_m <= 1.45e+141)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(-0.125 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * M_m) ^ 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_m <= 1.45e+141)
		tmp = sqrt((d / l)) * sqrt((d / h));
	else
		tmp = sqrt(((d / l) * (d / h))) * (-0.125 * ((h / l) * (((D_m / d) * M_m) ^ 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$95$m, 1.45e+141], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.125 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 1.45000000000000003e141

   1. Initial program 65.4%

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

   3. Simplified 65.4%

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

   5. Taylor expanded in d around inf 41.0%

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

   if 1.45000000000000003e141 < D

   1. Initial program 65.9%

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

   3. Simplified 60.0%

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

   5. Taylor expanded in M around inf 22.2%

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

      1. associate-*r* 22.2%

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

      2. times-frac 22.3%

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

      3. *-commutative 22.3%

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

      4. associate-/l* 22.4%

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

      5. unpow2 22.4%

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

      6. unpow2 22.4%

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

      7. unpow2 22.4%

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

      8. times-frac 25.8%

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

      9. swap-sqr 38.1%

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

      10. unpow2 38.1%

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

   7. Simplified 38.1%

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

      1. pow1 38.1%

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

      2. associate-*r* 38.0%

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

      3. sqrt-unprod 37.9%

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

      4. associate-*r/ 41.1%

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

   9. Applied egg-rr 41.1%

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

       1. unpow1 41.1%

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

       2. *-commutative 41.1%

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

       3. associate-/l* 37.9%

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

   11. Simplified 37.9%

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

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 1.45 \cdot 10^{+141}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(-0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot M\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 45.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 2.3 \cdot 10^{-143}:\\ \;\;\;\;\frac{d}{-\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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 2.3e-143) (/ d (- (sqrt (* h l)))) (/ d (* (sqrt l) (sqrt h)))))

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 <= 2.3e-143) {
		tmp = d / -sqrt((h * l));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	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 <= 2.3d-143) then
        tmp = d / -sqrt((h * l))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    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 <= 2.3e-143) {
		tmp = d / -Math.sqrt((h * l));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	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 <= 2.3e-143:
		tmp = d / -math.sqrt((h * l))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	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 <= 2.3e-143)
		tmp = Float64(d / Float64(-sqrt(Float64(h * l))));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	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 <= 2.3e-143)
		tmp = d / -sqrt((h * l));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	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, 2.3e-143], N[(d / (-N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 2.30000000000000011e-143

   1. Initial program 64.0%

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

   3. Simplified 62.9%

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

      1. add-sqr-sqrt 62.9%

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

      2. pow2 62.9%

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

      3. sqrt-prod 62.9%

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

      4. sqrt-pow1 63.5%

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

      5. metadata-eval 63.5%

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

      6. pow1 63.5%

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

      7. *-commutative 63.5%

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

      8. div-inv 63.5%

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

      9. metadata-eval 63.5%

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

   6. Applied egg-rr 63.5%

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

   7. Taylor expanded in l around -inf 0.0%

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

      1. remove-double-neg 0.0%

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

      2. distribute-lft-neg-in 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 39.9%

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

      6. neg-mul-1 39.9%

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

      7. remove-double-neg 39.9%

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

      8. associate-/r* 39.9%

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

      9. unpow1/2 39.9%

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

      10. associate-/r* 39.9%

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

      11. rem-exp-log 38.0%

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

      12. exp-neg 38.0%

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

      13. exp-prod 38.5%

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

      14. distribute-lft-neg-out 38.5%

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

      15. rec-exp 38.5%

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

      16. exp-to-pow 40.4%

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

      17. unpow1/2 40.4%

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

      18. associate-/l* 40.5%

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

      19. *-rgt-identity 40.5%

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

   9. Simplified 40.5%

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

   if 2.30000000000000011e-143 < d

   1. Initial program 68.2%

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

   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 59.4%

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

      1. sqrt-div 59.5%

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

      2. metadata-eval 59.5%

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

      3. *-commutative 59.5%

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

      4. sqrt-unprod 62.7%

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

      5. div-inv 62.8%

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

      6. associate-/r* 61.7%

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

   7. Applied egg-rr 61.7%

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

      1. associate-/l/ 62.8%

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

   9. Simplified 62.8%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 2.3 \cdot 10^{-143}:\\ \;\;\;\;\frac{d}{-\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 43.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{h \cdot \ell}\\ \mathbf{if}\;\ell \leq -1.5 \cdot 10^{-258}:\\ \;\;\;\;\frac{d}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{t\_0}\\ \end{array} \end{array} \]

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 (<= l -1.5e-258) (/ 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 (l <= -1.5e-258) {
		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 (l <= (-1.5d-258)) 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 (l <= -1.5e-258) {
		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 l <= -1.5e-258:
		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 (l <= -1.5e-258)
		tmp = Float64(d / Float64(-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 (l <= -1.5e-258)
		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[l, -1.5e-258], N[(d / (-t$95$0)), $MachinePrecision], N[(d / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -1.5000000000000001e-258

   1. Initial program 71.6%

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

   3. Simplified 70.8%

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

      1. add-sqr-sqrt 70.8%

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

      2. pow2 70.8%

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

      3. sqrt-prod 70.8%

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

      4. sqrt-pow1 71.7%

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

      5. metadata-eval 71.7%

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

      6. pow1 71.7%

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

      7. *-commutative 71.7%

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

      8. div-inv 71.7%

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

      9. metadata-eval 71.7%

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

   6. Applied egg-rr 71.7%

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

   7. Taylor expanded in l around -inf 0.0%

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

      1. remove-double-neg 0.0%

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

      2. distribute-lft-neg-in 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 48.3%

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

      6. neg-mul-1 48.3%

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

      7. remove-double-neg 48.3%

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

      8. associate-/r* 48.3%

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

      9. unpow1/2 48.3%

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

      10. associate-/r* 48.3%

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

      11. rem-exp-log 45.8%

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

      12. exp-neg 45.8%

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

      13. exp-prod 46.5%

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

      14. distribute-lft-neg-out 46.5%

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

      15. rec-exp 46.5%

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

      16. exp-to-pow 48.9%

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

      17. unpow1/2 48.9%

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

      18. associate-/l* 49.0%

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

      19. *-rgt-identity 49.0%

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

   9. Simplified 49.0%

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

   if -1.5000000000000001e-258 < l

   1. Initial program 59.6%

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

   3. Simplified 58.9%

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

   5. Taylor expanded in d around inf 46.6%

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

      1. sqrt-div 46.7%

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

      2. metadata-eval 46.7%

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

      3. *-commutative 46.7%

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

      4. sqrt-unprod 43.3%

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

      5. div-inv 43.4%

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

      6. sqrt-unprod 46.7%

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

      7. *-commutative 46.7%

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

   7. Applied egg-rr 46.7%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.5 \cdot 10^{-258}:\\ \;\;\;\;\frac{d}{-\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 26.7% 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 65.5%

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

3. Simplified 64.7%

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

5. Taylor expanded in d around inf 27.5%

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

   1. sqrt-div 26.4%

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

   2. metadata-eval 26.4%

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

   3. *-commutative 26.4%

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

   4. sqrt-unprod 22.2%

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

   5. div-inv 22.2%

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

   6. sqrt-unprod 26.5%

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

   7. *-commutative 26.5%

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

7. Applied egg-rr 26.5%

   \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024177 
(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)))))