Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.6% → 81.3%

Time: 1.8min

Alternatives: 17

Speedup: 1.0×

• could not determine a ground truth

  1. d = 1.0729845265480665e+222
  2. h = 2.145552694129482e-108
  3. l = -3.2941842806862084e-305
  4. M = 9.53363375566394e+87
  5. D = 5.389320583061149e+306

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: 66.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (d h l M D)
 :precision binary64
 (*
  (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
  (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))

C

double code(double d, double h, double l, double M, double D) {
	return (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
}

Python

def code(d, h, l, M, D):
	return (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))

Julia

function code(d, h, l, M, D)
	return Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 81.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = sqrt(-d);
	double tmp;
	if (l <= -5e-310) {
		tmp = (t_0 / sqrt(-l)) * ((t_0 / sqrt(-h)) * (1.0 + ((h / l) * (pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5))));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - ((h / l) * (0.5 * pow(((M_m / d) * (D / 2.0)), 2.0))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(-d)
    if (l <= (-5d-310)) then
        tmp = (t_0 / sqrt(-l)) * ((t_0 / sqrt(-h)) * (1.0d0 + ((h / l) * (((d_1 * ((m_m / 2.0d0) / d)) ** 2.0d0) * (-0.5d0)))))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 - ((h / l) * (0.5d0 * (((m_m / d) * (d_1 / 2.0d0)) ** 2.0d0))))
    end if
    code = tmp
end function

Java

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

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = math.sqrt(-d)
	tmp = 0
	if l <= -5e-310:
		tmp = (t_0 / math.sqrt(-l)) * ((t_0 / math.sqrt(-h)) * (1.0 + ((h / l) * (math.pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5))))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 - ((h / l) * (0.5 * math.pow(((M_m / d) * (D / 2.0)), 2.0))))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = sqrt(Float64(-d))
	tmp = 0.0
	if (l <= -5e-310)
		tmp = Float64(Float64(t_0 / sqrt(Float64(-l))) * Float64(Float64(t_0 / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D * Float64(Float64(M_m / 2.0) / d)) ^ 2.0) * -0.5)))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(M_m / d) * Float64(D / 2.0)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 69.5%

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

   3. Simplified 67.5%

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

      1. frac-2neg 67.5%

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

      2. sqrt-div 78.5%

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

   6. Applied egg-rr 78.5%

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

      1. frac-2neg 78.5%

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

      2. sqrt-div 82.2%

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

   8. Applied egg-rr 82.2%

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

   if -4.999999999999985e-310 < l

   1. Initial program 64.4%

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

      1. pow1 64.4%

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

   4. Applied egg-rr 81.0%

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

4. Final simplification 81.6%

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

Alternative 2: 75.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1.46 \cdot 10^{+251}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;d \leq -4.2 \cdot 10^{-233}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{M\_m}{d} \cdot \frac{D}{2}\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \left({M\_m}^{2} \cdot \frac{-1}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left({\left(M\_m \cdot \frac{D}{d \cdot 2}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= d -1.46e+251)
   (* (- d) (sqrt (/ (/ 1.0 h) l)))
   (if (<= d -4.2e-233)
     (*
      (* (sqrt (/ d l)) (sqrt (/ d h)))
      (- 1.0 (* 0.5 (/ (* h (pow (* (/ M_m d) (/ D 2.0)) 2.0)) l))))
     (if (<= d -5e-310)
       (*
        (sqrt (/ h (pow l 3.0)))
        (* -0.125 (* (pow D 2.0) (* (pow M_m 2.0) (/ -1.0 d)))))
       (*
        d
        (/
         (fma (pow (* M_m (/ D (* d 2.0))) 2.0) (* (/ h l) -0.5) 1.0)
         (* (sqrt l) (sqrt h))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= -1.46e+251) {
		tmp = -d * sqrt(((1.0 / h) / l));
	} else if (d <= -4.2e-233) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.5 * ((h * pow(((M_m / d) * (D / 2.0)), 2.0)) / l)));
	} else if (d <= -5e-310) {
		tmp = sqrt((h / pow(l, 3.0))) * (-0.125 * (pow(D, 2.0) * (pow(M_m, 2.0) * (-1.0 / d))));
	} else {
		tmp = d * (fma(pow((M_m * (D / (d * 2.0))), 2.0), ((h / l) * -0.5), 1.0) / (sqrt(l) * sqrt(h)));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (d <= -1.46e+251)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / h) / l)));
	elseif (d <= -4.2e-233)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(Float64(M_m / d) * Float64(D / 2.0)) ^ 2.0)) / l))));
	elseif (d <= -5e-310)
		tmp = Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(-0.125 * Float64((D ^ 2.0) * Float64((M_m ^ 2.0) * Float64(-1.0 / d)))));
	else
		tmp = Float64(d * Float64(fma((Float64(M_m * Float64(D / Float64(d * 2.0))) ^ 2.0), Float64(Float64(h / l) * -0.5), 1.0) / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[d, -1.46e+251], N[((-d) * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4.2e-233], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(N[(M$95$m / d), $MachinePrecision] * N[(D / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.125 * N[(N[Power[D, 2.0], $MachinePrecision] * N[(N[Power[M$95$m, 2.0], $MachinePrecision] * N[(-1.0 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[Power[N[(M$95$m * N[(D / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.4600000000000001e251

   1. Initial program 53.2%

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

   3. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-/r* 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 92.0%

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

      6. neg-mul-1 92.0%

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

   5. Simplified 92.0%

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

   if -1.4600000000000001e251 < d < -4.1999999999999997e-233

   1. Initial program 78.0%

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

   3. Simplified 78.0%

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

      1. associate-*r/ 82.2%

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

      2. frac-times 82.3%

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

      3. *-commutative 82.3%

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

      4. times-frac 80.2%

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

   6. Applied egg-rr 80.2%

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

   if -4.1999999999999997e-233 < d < -4.999999999999985e-310

   1. Initial program 40.5%

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

   3. Taylor expanded in h around -inf 0.0%

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

      1. associate-*r* 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-/l* 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 63.2%

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

      6. associate-/l* 63.2%

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

   5. Simplified 63.2%

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

   if -4.999999999999985e-310 < d

   1. Initial program 64.4%

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

      1. pow1 64.4%

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

   4. Applied egg-rr 81.0%

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

      1. unpow1 81.0%

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

      2. associate-*l/ 81.9%

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

      3. associate-/l* 81.7%

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

   6. Simplified 82.5%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.46 \cdot 10^{+251}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;d \leq -4.2 \cdot 10^{-233}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot \frac{-1}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := {\left(\frac{M\_m}{d} \cdot \frac{D}{2}\right)}^{2}\\ \mathbf{if}\;d \leq -3.1 \cdot 10^{+251}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-236}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot t\_0}{\ell}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \left({M\_m}^{2} \cdot \frac{-1}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot t\_0\right)\right) \cdot \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (pow (* (/ M_m d) (/ D 2.0)) 2.0)))
   (if (<= d -3.1e+251)
     (* (- d) (sqrt (/ (/ 1.0 h) l)))
     (if (<= d -1e-236)
       (* (* (sqrt (/ d l)) (sqrt (/ d h))) (- 1.0 (* 0.5 (/ (* h t_0) l))))
       (if (<= d -5e-310)
         (*
          (sqrt (/ h (pow l 3.0)))
          (* -0.125 (* (pow D 2.0) (* (pow M_m 2.0) (/ -1.0 d)))))
         (* (- 1.0 (* (/ h l) (* 0.5 t_0))) (/ (/ d (sqrt l)) (sqrt h))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = pow(((M_m / d) * (D / 2.0)), 2.0);
	double tmp;
	if (d <= -3.1e+251) {
		tmp = -d * sqrt(((1.0 / h) / l));
	} else if (d <= -1e-236) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.5 * ((h * t_0) / l)));
	} else if (d <= -5e-310) {
		tmp = sqrt((h / pow(l, 3.0))) * (-0.125 * (pow(D, 2.0) * (pow(M_m, 2.0) * (-1.0 / d))));
	} else {
		tmp = (1.0 - ((h / l) * (0.5 * t_0))) * ((d / sqrt(l)) / sqrt(h));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((m_m / d) * (d_1 / 2.0d0)) ** 2.0d0
    if (d <= (-3.1d+251)) then
        tmp = -d * sqrt(((1.0d0 / h) / l))
    else if (d <= (-1d-236)) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - (0.5d0 * ((h * t_0) / l)))
    else if (d <= (-5d-310)) then
        tmp = sqrt((h / (l ** 3.0d0))) * ((-0.125d0) * ((d_1 ** 2.0d0) * ((m_m ** 2.0d0) * ((-1.0d0) / d))))
    else
        tmp = (1.0d0 - ((h / l) * (0.5d0 * t_0))) * ((d / sqrt(l)) / sqrt(h))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = Math.pow(((M_m / d) * (D / 2.0)), 2.0);
	double tmp;
	if (d <= -3.1e+251) {
		tmp = -d * Math.sqrt(((1.0 / h) / l));
	} else if (d <= -1e-236) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (0.5 * ((h * t_0) / l)));
	} else if (d <= -5e-310) {
		tmp = Math.sqrt((h / Math.pow(l, 3.0))) * (-0.125 * (Math.pow(D, 2.0) * (Math.pow(M_m, 2.0) * (-1.0 / d))));
	} else {
		tmp = (1.0 - ((h / l) * (0.5 * t_0))) * ((d / Math.sqrt(l)) / Math.sqrt(h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = math.pow(((M_m / d) * (D / 2.0)), 2.0)
	tmp = 0
	if d <= -3.1e+251:
		tmp = -d * math.sqrt(((1.0 / h) / l))
	elif d <= -1e-236:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (0.5 * ((h * t_0) / l)))
	elif d <= -5e-310:
		tmp = math.sqrt((h / math.pow(l, 3.0))) * (-0.125 * (math.pow(D, 2.0) * (math.pow(M_m, 2.0) * (-1.0 / d))))
	else:
		tmp = (1.0 - ((h / l) * (0.5 * t_0))) * ((d / math.sqrt(l)) / math.sqrt(h))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(Float64(M_m / d) * Float64(D / 2.0)) ^ 2.0
	tmp = 0.0
	if (d <= -3.1e+251)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / h) / l)));
	elseif (d <= -1e-236)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * t_0) / l))));
	elseif (d <= -5e-310)
		tmp = Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(-0.125 * Float64((D ^ 2.0) * Float64((M_m ^ 2.0) * Float64(-1.0 / d)))));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * t_0))) * Float64(Float64(d / sqrt(l)) / sqrt(h)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = ((M_m / d) * (D / 2.0)) ^ 2.0;
	tmp = 0.0;
	if (d <= -3.1e+251)
		tmp = -d * sqrt(((1.0 / h) / l));
	elseif (d <= -1e-236)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.5 * ((h * t_0) / l)));
	elseif (d <= -5e-310)
		tmp = sqrt((h / (l ^ 3.0))) * (-0.125 * ((D ^ 2.0) * ((M_m ^ 2.0) * (-1.0 / d))));
	else
		tmp = (1.0 - ((h / l) * (0.5 * t_0))) * ((d / sqrt(l)) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[Power[N[(N[(M$95$m / d), $MachinePrecision] * N[(D / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[d, -3.1e+251], N[((-d) * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1e-236], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * t$95$0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.125 * N[(N[Power[D, 2.0], $MachinePrecision] * N[(N[Power[M$95$m, 2.0], $MachinePrecision] * N[(-1.0 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -3.0999999999999998e251

   1. Initial program 53.2%

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

   3. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-/r* 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 92.0%

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

      6. neg-mul-1 92.0%

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

   5. Simplified 92.0%

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

   if -3.0999999999999998e251 < d < -1e-236

   1. Initial program 78.0%

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

   3. Simplified 78.0%

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

      1. associate-*r/ 82.2%

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

      2. frac-times 82.3%

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

      3. *-commutative 82.3%

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

      4. times-frac 80.2%

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

   6. Applied egg-rr 80.2%

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

   if -1e-236 < d < -4.999999999999985e-310

   1. Initial program 40.5%

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

   3. Taylor expanded in h around -inf 0.0%

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

      1. associate-*r* 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-/l* 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 63.2%

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

      6. associate-/l* 63.2%

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

   5. Simplified 63.2%

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

   if -4.999999999999985e-310 < d

   1. Initial program 64.4%

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

      1. pow1 64.4%

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

   4. Applied egg-rr 81.0%

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

      1. unpow1 81.0%

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

      2. associate-/r* 79.9%

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

   6. Simplified 79.9%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.1 \cdot 10^{+251}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-236}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot \frac{-1}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}\right)\right) \cdot \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := {\left(\frac{M\_m}{d} \cdot \frac{D}{2}\right)}^{2}\\ \mathbf{if}\;d \leq -1.35 \cdot 10^{+251}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;d \leq -5.8 \cdot 10^{-235}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot t\_0}{\ell}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \left({M\_m}^{2} \cdot \frac{-1}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (pow (* (/ M_m d) (/ D 2.0)) 2.0)))
   (if (<= d -1.35e+251)
     (* (- d) (sqrt (/ (/ 1.0 h) l)))
     (if (<= d -5.8e-235)
       (* (* (sqrt (/ d l)) (sqrt (/ d h))) (- 1.0 (* 0.5 (/ (* h t_0) l))))
       (if (<= d -5e-310)
         (*
          (sqrt (/ h (pow l 3.0)))
          (* -0.125 (* (pow D 2.0) (* (pow M_m 2.0) (/ -1.0 d)))))
         (* (/ d (* (sqrt l) (sqrt h))) (- 1.0 (* (/ h l) (* 0.5 t_0)))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = pow(((M_m / d) * (D / 2.0)), 2.0);
	double tmp;
	if (d <= -1.35e+251) {
		tmp = -d * sqrt(((1.0 / h) / l));
	} else if (d <= -5.8e-235) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.5 * ((h * t_0) / l)));
	} else if (d <= -5e-310) {
		tmp = sqrt((h / pow(l, 3.0))) * (-0.125 * (pow(D, 2.0) * (pow(M_m, 2.0) * (-1.0 / d))));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - ((h / l) * (0.5 * t_0)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((m_m / d) * (d_1 / 2.0d0)) ** 2.0d0
    if (d <= (-1.35d+251)) then
        tmp = -d * sqrt(((1.0d0 / h) / l))
    else if (d <= (-5.8d-235)) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - (0.5d0 * ((h * t_0) / l)))
    else if (d <= (-5d-310)) then
        tmp = sqrt((h / (l ** 3.0d0))) * ((-0.125d0) * ((d_1 ** 2.0d0) * ((m_m ** 2.0d0) * ((-1.0d0) / d))))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 - ((h / l) * (0.5d0 * t_0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = Math.pow(((M_m / d) * (D / 2.0)), 2.0);
	double tmp;
	if (d <= -1.35e+251) {
		tmp = -d * Math.sqrt(((1.0 / h) / l));
	} else if (d <= -5.8e-235) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (0.5 * ((h * t_0) / l)));
	} else if (d <= -5e-310) {
		tmp = Math.sqrt((h / Math.pow(l, 3.0))) * (-0.125 * (Math.pow(D, 2.0) * (Math.pow(M_m, 2.0) * (-1.0 / d))));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 - ((h / l) * (0.5 * t_0)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = math.pow(((M_m / d) * (D / 2.0)), 2.0)
	tmp = 0
	if d <= -1.35e+251:
		tmp = -d * math.sqrt(((1.0 / h) / l))
	elif d <= -5.8e-235:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (0.5 * ((h * t_0) / l)))
	elif d <= -5e-310:
		tmp = math.sqrt((h / math.pow(l, 3.0))) * (-0.125 * (math.pow(D, 2.0) * (math.pow(M_m, 2.0) * (-1.0 / d))))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 - ((h / l) * (0.5 * t_0)))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(Float64(M_m / d) * Float64(D / 2.0)) ^ 2.0
	tmp = 0.0
	if (d <= -1.35e+251)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / h) / l)));
	elseif (d <= -5.8e-235)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * t_0) / l))));
	elseif (d <= -5e-310)
		tmp = Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(-0.125 * Float64((D ^ 2.0) * Float64((M_m ^ 2.0) * Float64(-1.0 / d)))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * t_0))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = ((M_m / d) * (D / 2.0)) ^ 2.0;
	tmp = 0.0;
	if (d <= -1.35e+251)
		tmp = -d * sqrt(((1.0 / h) / l));
	elseif (d <= -5.8e-235)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.5 * ((h * t_0) / l)));
	elseif (d <= -5e-310)
		tmp = sqrt((h / (l ^ 3.0))) * (-0.125 * ((D ^ 2.0) * ((M_m ^ 2.0) * (-1.0 / d))));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - ((h / l) * (0.5 * t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[Power[N[(N[(M$95$m / d), $MachinePrecision] * N[(D / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[d, -1.35e+251], N[((-d) * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5.8e-235], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * t$95$0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.125 * N[(N[Power[D, 2.0], $MachinePrecision] * N[(N[Power[M$95$m, 2.0], $MachinePrecision] * N[(-1.0 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.3500000000000001e251

   1. Initial program 53.2%

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

   3. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-/r* 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 92.0%

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

      6. neg-mul-1 92.0%

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

   5. Simplified 92.0%

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

   if -1.3500000000000001e251 < d < -5.80000000000000018e-235

   1. Initial program 78.0%

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

   3. Simplified 78.0%

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

      1. associate-*r/ 82.2%

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

      2. frac-times 82.3%

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

      3. *-commutative 82.3%

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

      4. times-frac 80.2%

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

   6. Applied egg-rr 80.2%

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

   if -5.80000000000000018e-235 < d < -4.999999999999985e-310

   1. Initial program 40.5%

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

   3. Taylor expanded in h around -inf 0.0%

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

      1. associate-*r* 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-/l* 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 63.2%

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

      6. associate-/l* 63.2%

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

   5. Simplified 63.2%

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

   if -4.999999999999985e-310 < d

   1. Initial program 64.4%

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

      1. pow1 64.4%

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

   4. Applied egg-rr 81.0%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.35 \cdot 10^{+251}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;d \leq -5.8 \cdot 10^{-235}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot \frac{-1}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.9% accurate, 0.8× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= h -5e-311)
   (*
    (*
     (/ (sqrt (- d)) (sqrt (- h)))
     (+ 1.0 (* (/ h l) (* (pow (* D (/ (/ M_m 2.0) d)) 2.0) -0.5))))
    (sqrt (/ d l)))
   (*
    (/ d (* (sqrt l) (sqrt h)))
    (- 1.0 (* (/ h l) (* 0.5 (pow (* (/ M_m d) (/ D 2.0)) 2.0)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (h <= -5e-311) {
		tmp = ((sqrt(-d) / sqrt(-h)) * (1.0 + ((h / l) * (pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5)))) * sqrt((d / l));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - ((h / l) * (0.5 * pow(((M_m / d) * (D / 2.0)), 2.0))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: tmp
    if (h <= (-5d-311)) then
        tmp = ((sqrt(-d) / sqrt(-h)) * (1.0d0 + ((h / l) * (((d_1 * ((m_m / 2.0d0) / d)) ** 2.0d0) * (-0.5d0))))) * sqrt((d / l))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 - ((h / l) * (0.5d0 * (((m_m / d) * (d_1 / 2.0d0)) ** 2.0d0))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (h <= -5e-311) {
		tmp = ((Math.sqrt(-d) / Math.sqrt(-h)) * (1.0 + ((h / l) * (Math.pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5)))) * Math.sqrt((d / l));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 - ((h / l) * (0.5 * Math.pow(((M_m / d) * (D / 2.0)), 2.0))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if h <= -5e-311:
		tmp = ((math.sqrt(-d) / math.sqrt(-h)) * (1.0 + ((h / l) * (math.pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5)))) * math.sqrt((d / l))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 - ((h / l) * (0.5 * math.pow(((M_m / d) * (D / 2.0)), 2.0))))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (h <= -5e-311)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D * Float64(Float64(M_m / 2.0) / d)) ^ 2.0) * -0.5)))) * sqrt(Float64(d / l)));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(M_m / d) * Float64(D / 2.0)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (h <= -5e-311)
		tmp = ((sqrt(-d) / sqrt(-h)) * (1.0 + ((h / l) * (((D * ((M_m / 2.0) / d)) ^ 2.0) * -0.5)))) * sqrt((d / l));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - ((h / l) * (0.5 * (((M_m / d) * (D / 2.0)) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[h, -5e-311], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D * N[(N[(M$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(N[(M$95$m / d), $MachinePrecision] * N[(D / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -5.00000000000023e-311

   1. Initial program 69.5%

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

   3. Simplified 67.5%

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

      1. frac-2neg 67.5%

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

      2. sqrt-div 78.5%

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

   6. Applied egg-rr 78.5%

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

   if -5.00000000000023e-311 < h

   1. Initial program 64.4%

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

      1. pow1 64.4%

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

   4. Applied egg-rr 81.0%

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

4. Final simplification 79.7%

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

Alternative 6: 78.9% accurate, 0.8× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= h -5e-311)
   (*
    (sqrt (/ d l))
    (*
     (/ (sqrt (- d)) (sqrt (- h)))
     (+ 1.0 (* (/ h l) (* -0.5 (pow (/ (/ (* D M_m) d) 2.0) 2.0))))))
   (*
    (/ d (* (sqrt l) (sqrt h)))
    (- 1.0 (* (/ h l) (* 0.5 (pow (* (/ M_m d) (/ D 2.0)) 2.0)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (h <= -5e-311) {
		tmp = sqrt((d / l)) * ((sqrt(-d) / sqrt(-h)) * (1.0 + ((h / l) * (-0.5 * pow((((D * M_m) / d) / 2.0), 2.0)))));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - ((h / l) * (0.5 * pow(((M_m / d) * (D / 2.0)), 2.0))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: tmp
    if (h <= (-5d-311)) then
        tmp = sqrt((d / l)) * ((sqrt(-d) / sqrt(-h)) * (1.0d0 + ((h / l) * ((-0.5d0) * ((((d_1 * m_m) / d) / 2.0d0) ** 2.0d0)))))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 - ((h / l) * (0.5d0 * (((m_m / d) * (d_1 / 2.0d0)) ** 2.0d0))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (h <= -5e-311) {
		tmp = Math.sqrt((d / l)) * ((Math.sqrt(-d) / Math.sqrt(-h)) * (1.0 + ((h / l) * (-0.5 * Math.pow((((D * M_m) / d) / 2.0), 2.0)))));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 - ((h / l) * (0.5 * Math.pow(((M_m / d) * (D / 2.0)), 2.0))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if h <= -5e-311:
		tmp = math.sqrt((d / l)) * ((math.sqrt(-d) / math.sqrt(-h)) * (1.0 + ((h / l) * (-0.5 * math.pow((((D * M_m) / d) / 2.0), 2.0)))))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 - ((h / l) * (0.5 * math.pow(((M_m / d) * (D / 2.0)), 2.0))))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (h <= -5e-311)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(Float64(Float64(D * M_m) / d) / 2.0) ^ 2.0))))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(M_m / d) * Float64(D / 2.0)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (h <= -5e-311)
		tmp = sqrt((d / l)) * ((sqrt(-d) / sqrt(-h)) * (1.0 + ((h / l) * (-0.5 * ((((D * M_m) / d) / 2.0) ^ 2.0)))));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - ((h / l) * (0.5 * (((M_m / d) * (D / 2.0)) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[h, -5e-311], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(N[(N[(D * M$95$m), $MachinePrecision] / d), $MachinePrecision] / 2.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(N[(M$95$m / d), $MachinePrecision] * N[(D / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -5.00000000000023e-311

   1. Initial program 69.5%

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

   3. Simplified 67.5%

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

      1. frac-2neg 67.5%

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

      2. sqrt-div 78.5%

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

   6. Applied egg-rr 78.5%

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

      1. associate-*r/ 80.5%

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

      2. *-un-lft-identity 80.5%

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

      3. times-frac 78.5%

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

      4. associate-/l/ 78.5%

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

      5. *-commutative 78.5%

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

      6. times-frac 80.5%

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

      7. *-commutative 80.5%

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

      8. *-un-lft-identity 80.5%

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

      9. frac-times 80.5%

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

      10. associate-*l/ 80.5%

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

      11. associate-*r/ 80.5%

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

   8. Applied egg-rr 80.5%

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

   if -5.00000000000023e-311 < h

   1. Initial program 64.4%

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

      1. pow1 64.4%

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

   4. Applied egg-rr 81.0%

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

4. Final simplification 80.7%

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

Alternative 7: 73.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1.7 \cdot 10^{+251}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M\_m}{d} \cdot \frac{D}{2}\right)}^{2}\right)\right) \cdot \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= d -1.7e+251)
   (* (- d) (sqrt (/ (/ 1.0 h) l)))
   (if (<= d -5e-310)
     (*
      (sqrt (/ d l))
      (*
       (+ 1.0 (* (/ h l) (* (pow (* D (/ (/ M_m 2.0) d)) 2.0) -0.5)))
       (sqrt (/ d h))))
     (*
      (- 1.0 (* (/ h l) (* 0.5 (pow (* (/ M_m d) (/ D 2.0)) 2.0))))
      (/ (/ d (sqrt l)) (sqrt h))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= -1.7e+251) {
		tmp = -d * sqrt(((1.0 / h) / l));
	} else if (d <= -5e-310) {
		tmp = sqrt((d / l)) * ((1.0 + ((h / l) * (pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5))) * sqrt((d / h)));
	} else {
		tmp = (1.0 - ((h / l) * (0.5 * pow(((M_m / d) * (D / 2.0)), 2.0)))) * ((d / sqrt(l)) / sqrt(h));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: tmp
    if (d <= (-1.7d+251)) then
        tmp = -d * sqrt(((1.0d0 / h) / l))
    else if (d <= (-5d-310)) then
        tmp = sqrt((d / l)) * ((1.0d0 + ((h / l) * (((d_1 * ((m_m / 2.0d0) / d)) ** 2.0d0) * (-0.5d0)))) * sqrt((d / h)))
    else
        tmp = (1.0d0 - ((h / l) * (0.5d0 * (((m_m / d) * (d_1 / 2.0d0)) ** 2.0d0)))) * ((d / sqrt(l)) / sqrt(h))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= -1.7e+251) {
		tmp = -d * Math.sqrt(((1.0 / h) / l));
	} else if (d <= -5e-310) {
		tmp = Math.sqrt((d / l)) * ((1.0 + ((h / l) * (Math.pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5))) * Math.sqrt((d / h)));
	} else {
		tmp = (1.0 - ((h / l) * (0.5 * Math.pow(((M_m / d) * (D / 2.0)), 2.0)))) * ((d / Math.sqrt(l)) / Math.sqrt(h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if d <= -1.7e+251:
		tmp = -d * math.sqrt(((1.0 / h) / l))
	elif d <= -5e-310:
		tmp = math.sqrt((d / l)) * ((1.0 + ((h / l) * (math.pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5))) * math.sqrt((d / h)))
	else:
		tmp = (1.0 - ((h / l) * (0.5 * math.pow(((M_m / d) * (D / 2.0)), 2.0)))) * ((d / math.sqrt(l)) / math.sqrt(h))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (d <= -1.7e+251)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / h) / l)));
	elseif (d <= -5e-310)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D * Float64(Float64(M_m / 2.0) / d)) ^ 2.0) * -0.5))) * sqrt(Float64(d / h))));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(M_m / d) * Float64(D / 2.0)) ^ 2.0)))) * Float64(Float64(d / sqrt(l)) / sqrt(h)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (d <= -1.7e+251)
		tmp = -d * sqrt(((1.0 / h) / l));
	elseif (d <= -5e-310)
		tmp = sqrt((d / l)) * ((1.0 + ((h / l) * (((D * ((M_m / 2.0) / d)) ^ 2.0) * -0.5))) * sqrt((d / h)));
	else
		tmp = (1.0 - ((h / l) * (0.5 * (((M_m / d) * (D / 2.0)) ^ 2.0)))) * ((d / sqrt(l)) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[d, -1.7e+251], N[((-d) * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D * N[(N[(M$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(N[(M$95$m / d), $MachinePrecision] * N[(D / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.70000000000000006e251

   1. Initial program 53.2%

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

   3. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-/r* 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 92.0%

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

      6. neg-mul-1 92.0%

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

   5. Simplified 92.0%

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

   if -1.70000000000000006e251 < d < -4.999999999999985e-310

   1. Initial program 71.8%

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

   3. Simplified 69.5%

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

   if -4.999999999999985e-310 < d

   1. Initial program 64.4%

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

      1. pow1 64.4%

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

   4. Applied egg-rr 81.0%

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

      1. unpow1 81.0%

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

      2. associate-/r* 79.9%

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

   6. Simplified 79.9%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.7 \cdot 10^{+251}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}\right)\right) \cdot \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := {\left(\frac{M\_m}{d} \cdot \frac{D}{2}\right)}^{2}\\ \mathbf{if}\;d \leq -2.3 \cdot 10^{+251}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot t\_0}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot t\_0\right)\right) \cdot \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (pow (* (/ M_m d) (/ D 2.0)) 2.0)))
   (if (<= d -2.3e+251)
     (* (- d) (sqrt (/ (/ 1.0 h) l)))
     (if (<= d -5e-310)
       (* (* (sqrt (/ d l)) (sqrt (/ d h))) (- 1.0 (* 0.5 (/ (* h t_0) l))))
       (* (- 1.0 (* (/ h l) (* 0.5 t_0))) (/ (/ d (sqrt l)) (sqrt h)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = pow(((M_m / d) * (D / 2.0)), 2.0);
	double tmp;
	if (d <= -2.3e+251) {
		tmp = -d * sqrt(((1.0 / h) / l));
	} else if (d <= -5e-310) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.5 * ((h * t_0) / l)));
	} else {
		tmp = (1.0 - ((h / l) * (0.5 * t_0))) * ((d / sqrt(l)) / sqrt(h));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((m_m / d) * (d_1 / 2.0d0)) ** 2.0d0
    if (d <= (-2.3d+251)) then
        tmp = -d * sqrt(((1.0d0 / h) / l))
    else if (d <= (-5d-310)) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - (0.5d0 * ((h * t_0) / l)))
    else
        tmp = (1.0d0 - ((h / l) * (0.5d0 * t_0))) * ((d / sqrt(l)) / sqrt(h))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = Math.pow(((M_m / d) * (D / 2.0)), 2.0);
	double tmp;
	if (d <= -2.3e+251) {
		tmp = -d * Math.sqrt(((1.0 / h) / l));
	} else if (d <= -5e-310) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (0.5 * ((h * t_0) / l)));
	} else {
		tmp = (1.0 - ((h / l) * (0.5 * t_0))) * ((d / Math.sqrt(l)) / Math.sqrt(h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = math.pow(((M_m / d) * (D / 2.0)), 2.0)
	tmp = 0
	if d <= -2.3e+251:
		tmp = -d * math.sqrt(((1.0 / h) / l))
	elif d <= -5e-310:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (0.5 * ((h * t_0) / l)))
	else:
		tmp = (1.0 - ((h / l) * (0.5 * t_0))) * ((d / math.sqrt(l)) / math.sqrt(h))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(Float64(M_m / d) * Float64(D / 2.0)) ^ 2.0
	tmp = 0.0
	if (d <= -2.3e+251)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / h) / l)));
	elseif (d <= -5e-310)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * t_0) / l))));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * t_0))) * Float64(Float64(d / sqrt(l)) / sqrt(h)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = ((M_m / d) * (D / 2.0)) ^ 2.0;
	tmp = 0.0;
	if (d <= -2.3e+251)
		tmp = -d * sqrt(((1.0 / h) / l));
	elseif (d <= -5e-310)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.5 * ((h * t_0) / l)));
	else
		tmp = (1.0 - ((h / l) * (0.5 * t_0))) * ((d / sqrt(l)) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[Power[N[(N[(M$95$m / d), $MachinePrecision] * N[(D / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[d, -2.3e+251], N[((-d) * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * t$95$0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.29999999999999988e251

   1. Initial program 53.2%

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

   3. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-/r* 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 92.0%

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

      6. neg-mul-1 92.0%

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

   5. Simplified 92.0%

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

   if -2.29999999999999988e251 < d < -4.999999999999985e-310

   1. Initial program 71.8%

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

   3. Simplified 71.8%

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

      1. associate-*r/ 75.3%

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

      2. frac-times 75.3%

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

      3. *-commutative 75.3%

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

      4. times-frac 73.1%

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

   6. Applied egg-rr 73.1%

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

   if -4.999999999999985e-310 < d

   1. Initial program 64.4%

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

      1. pow1 64.4%

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

   4. Applied egg-rr 81.0%

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

      1. unpow1 81.0%

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

      2. associate-/r* 79.9%

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

   6. Simplified 79.9%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.3 \cdot 10^{+251}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}\right)\right) \cdot \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1.35 \cdot 10^{+157}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-299}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M\_m \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M\_m}{d} \cdot \frac{D}{2}\right)}^{2}\right)\right) \cdot \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= d -1.35e+157)
   (* (- d) (sqrt (/ (/ 1.0 l) h)))
   (if (<= d -2e-299)
     (*
      (sqrt (* (/ d l) (/ d h)))
      (+ 1.0 (* -0.5 (* (/ h l) (pow (* D (* M_m (/ 0.5 d))) 2.0)))))
     (*
      (- 1.0 (* (/ h l) (* 0.5 (pow (* (/ M_m d) (/ D 2.0)) 2.0))))
      (/ (/ d (sqrt l)) (sqrt h))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= -1.35e+157) {
		tmp = -d * sqrt(((1.0 / l) / h));
	} else if (d <= -2e-299) {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * ((h / l) * pow((D * (M_m * (0.5 / d))), 2.0))));
	} else {
		tmp = (1.0 - ((h / l) * (0.5 * pow(((M_m / d) * (D / 2.0)), 2.0)))) * ((d / sqrt(l)) / sqrt(h));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: tmp
    if (d <= (-1.35d+157)) then
        tmp = -d * sqrt(((1.0d0 / l) / h))
    else if (d <= (-2d-299)) then
        tmp = sqrt(((d / l) * (d / h))) * (1.0d0 + ((-0.5d0) * ((h / l) * ((d_1 * (m_m * (0.5d0 / d))) ** 2.0d0))))
    else
        tmp = (1.0d0 - ((h / l) * (0.5d0 * (((m_m / d) * (d_1 / 2.0d0)) ** 2.0d0)))) * ((d / sqrt(l)) / sqrt(h))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= -1.35e+157) {
		tmp = -d * Math.sqrt(((1.0 / l) / h));
	} else if (d <= -2e-299) {
		tmp = Math.sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * ((h / l) * Math.pow((D * (M_m * (0.5 / d))), 2.0))));
	} else {
		tmp = (1.0 - ((h / l) * (0.5 * Math.pow(((M_m / d) * (D / 2.0)), 2.0)))) * ((d / Math.sqrt(l)) / Math.sqrt(h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if d <= -1.35e+157:
		tmp = -d * math.sqrt(((1.0 / l) / h))
	elif d <= -2e-299:
		tmp = math.sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * ((h / l) * math.pow((D * (M_m * (0.5 / d))), 2.0))))
	else:
		tmp = (1.0 - ((h / l) * (0.5 * math.pow(((M_m / d) * (D / 2.0)), 2.0)))) * ((d / math.sqrt(l)) / math.sqrt(h))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (d <= -1.35e+157)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / l) / h)));
	elseif (d <= -2e-299)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(D * Float64(M_m * Float64(0.5 / d))) ^ 2.0)))));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(M_m / d) * Float64(D / 2.0)) ^ 2.0)))) * Float64(Float64(d / sqrt(l)) / sqrt(h)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (d <= -1.35e+157)
		tmp = -d * sqrt(((1.0 / l) / h));
	elseif (d <= -2e-299)
		tmp = sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * ((h / l) * ((D * (M_m * (0.5 / d))) ^ 2.0))));
	else
		tmp = (1.0 - ((h / l) * (0.5 * (((M_m / d) * (D / 2.0)) ^ 2.0)))) * ((d / sqrt(l)) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[d, -1.35e+157], N[((-d) * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -2e-299], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(N[(M$95$m / d), $MachinePrecision] * N[(D / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.35e157

   1. Initial program 68.3%

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

   3. Taylor expanded in d around inf 1.2%

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

      1. associate-/r* 1.2%

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

   5. Simplified 1.2%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-/r* 0.0%

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

      4. *-commutative 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 82.5%

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

      7. neg-mul-1 82.5%

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

   8. Simplified 82.5%

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

   if -1.35e157 < d < -1.99999999999999998e-299

   1. Initial program 69.9%

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

   3. Simplified 69.9%

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

      1. add-sqr-sqrt 69.9%

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

      2. pow2 69.9%

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

      3. sqrt-prod 69.9%

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

      4. sqrt-pow1 69.9%

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

      5. metadata-eval 69.9%

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

      6. pow1 69.9%

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

      7. frac-times 70.9%

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

      8. *-commutative 70.9%

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

      9. times-frac 68.3%

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

   6. Applied egg-rr 68.3%

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

   8. Simplified 67.3%

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

      1. pow1 67.3%

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

      2. sqrt-unprod 55.6%

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

      3. cancel-sign-sub-inv 55.6%

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

      4. metadata-eval 55.6%

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

      5. associate-*r* 56.5%

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

      6. unpow-prod-down 55.5%

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

      7. pow2 55.5%

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

      8. add-sqr-sqrt 55.5%

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

   10. Applied egg-rr 55.5%

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

       1. unpow1 55.5%

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

       2. *-commutative 55.5%

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

       3. *-commutative 55.5%

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

   12. Simplified 55.5%

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

   if -1.99999999999999998e-299 < d

   1. Initial program 64.4%

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

      1. pow1 64.4%

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

   4. Applied egg-rr 81.0%

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

      1. unpow1 81.0%

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

      2. associate-/r* 79.9%

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

   6. Simplified 79.9%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.35 \cdot 10^{+157}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-299}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}\right)\right) \cdot \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M\_m \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{if}\;\ell \leq -3.2 \cdot 10^{+167}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -3 \cdot 10^{+77}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{-63}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot \sqrt{\frac{1}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0
         (*
          (sqrt (* (/ d l) (/ d h)))
          (+ 1.0 (* -0.5 (* (/ h l) (pow (* D (* M_m (/ 0.5 d))) 2.0)))))))
   (if (<= l -3.2e+167)
     t_0
     (if (<= l -3e+77)
       (* (- d) (sqrt (/ (/ 1.0 h) l)))
       (if (<= l 5.5e-63) t_0 (* d (* (pow h -0.5) (sqrt (/ 1.0 l)))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * ((h / l) * pow((D * (M_m * (0.5 / d))), 2.0))));
	double tmp;
	if (l <= -3.2e+167) {
		tmp = t_0;
	} else if (l <= -3e+77) {
		tmp = -d * sqrt(((1.0 / h) / l));
	} else if (l <= 5.5e-63) {
		tmp = t_0;
	} else {
		tmp = d * (pow(h, -0.5) * sqrt((1.0 / l)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((d / l) * (d / h))) * (1.0d0 + ((-0.5d0) * ((h / l) * ((d_1 * (m_m * (0.5d0 / d))) ** 2.0d0))))
    if (l <= (-3.2d+167)) then
        tmp = t_0
    else if (l <= (-3d+77)) then
        tmp = -d * sqrt(((1.0d0 / h) / l))
    else if (l <= 5.5d-63) then
        tmp = t_0
    else
        tmp = d * ((h ** (-0.5d0)) * sqrt((1.0d0 / l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = Math.sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * ((h / l) * Math.pow((D * (M_m * (0.5 / d))), 2.0))));
	double tmp;
	if (l <= -3.2e+167) {
		tmp = t_0;
	} else if (l <= -3e+77) {
		tmp = -d * Math.sqrt(((1.0 / h) / l));
	} else if (l <= 5.5e-63) {
		tmp = t_0;
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.sqrt((1.0 / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = math.sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * ((h / l) * math.pow((D * (M_m * (0.5 / d))), 2.0))))
	tmp = 0
	if l <= -3.2e+167:
		tmp = t_0
	elif l <= -3e+77:
		tmp = -d * math.sqrt(((1.0 / h) / l))
	elif l <= 5.5e-63:
		tmp = t_0
	else:
		tmp = d * (math.pow(h, -0.5) * math.sqrt((1.0 / l)))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(D * Float64(M_m * Float64(0.5 / d))) ^ 2.0)))))
	tmp = 0.0
	if (l <= -3.2e+167)
		tmp = t_0;
	elseif (l <= -3e+77)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / h) / l)));
	elseif (l <= 5.5e-63)
		tmp = t_0;
	else
		tmp = Float64(d * Float64((h ^ -0.5) * sqrt(Float64(1.0 / l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * ((h / l) * ((D * (M_m * (0.5 / d))) ^ 2.0))));
	tmp = 0.0;
	if (l <= -3.2e+167)
		tmp = t_0;
	elseif (l <= -3e+77)
		tmp = -d * sqrt(((1.0 / h) / l));
	elseif (l <= 5.5e-63)
		tmp = t_0;
	else
		tmp = d * ((h ^ -0.5) * sqrt((1.0 / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -3.2e+167], t$95$0, If[LessEqual[l, -3e+77], N[((-d) * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.5e-63], t$95$0, N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.19999999999999981e167 or -2.9999999999999998e77 < l < 5.50000000000000043e-63

   1. Initial program 69.4%

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

   3. Simplified 69.4%

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

      1. add-sqr-sqrt 69.4%

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

      2. pow2 69.4%

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

      3. sqrt-prod 69.4%

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

      4. sqrt-pow1 69.4%

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

      5. metadata-eval 69.4%

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

      6. pow1 69.4%

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

      7. frac-times 70.0%

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

      8. *-commutative 70.0%

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

      9. times-frac 69.4%

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

   6. Applied egg-rr 69.4%

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

   8. Simplified 68.8%

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

      1. pow1 68.8%

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

      2. sqrt-unprod 59.1%

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

      3. cancel-sign-sub-inv 59.1%

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

      4. metadata-eval 59.1%

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

      5. associate-*r* 59.7%

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

      6. unpow-prod-down 59.0%

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

      7. pow2 59.0%

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

      8. add-sqr-sqrt 59.0%

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

   10. Applied egg-rr 59.0%

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

       1. unpow1 59.0%

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

       2. *-commutative 59.0%

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

       3. *-commutative 59.0%

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

   12. Simplified 59.0%

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

   if -3.19999999999999981e167 < l < -2.9999999999999998e77

   1. Initial program 70.8%

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

   3. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-/r* 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 76.7%

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

      6. neg-mul-1 76.7%

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

   5. Simplified 76.7%

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

   if 5.50000000000000043e-63 < l

   1. Initial program 61.0%

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

   3. Taylor expanded in d around inf 53.4%

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

      1. associate-/r* 53.4%

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

   5. Simplified 53.4%

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

      1. div-inv 53.4%

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

      2. sqrt-prod 65.9%

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

      3. inv-pow 65.9%

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

      4. sqrt-pow1 65.9%

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

      5. metadata-eval 65.9%

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

   7. Applied egg-rr 65.9%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.2 \cdot 10^{+167}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -3 \cdot 10^{+77}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{-63}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot \sqrt{\frac{1}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 46.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.6 \cdot 10^{+223}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l -4.6e+223)
   (* (sqrt (/ d l)) (sqrt (/ d h)))
   (if (<= l -5e-310)
     (* (- d) (sqrt (/ (/ 1.0 h) l)))
     (/ d (* (sqrt l) (sqrt h))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -4.6e+223) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (l <= -5e-310) {
		tmp = -d * sqrt(((1.0 / h) / l));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: tmp
    if (l <= (-4.6d+223)) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else if (l <= (-5d-310)) then
        tmp = -d * sqrt(((1.0d0 / h) / l))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -4.6e+223) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else if (l <= -5e-310) {
		tmp = -d * Math.sqrt(((1.0 / h) / l));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= -4.6e+223:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	elif l <= -5e-310:
		tmp = -d * math.sqrt(((1.0 / h) / l))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= -4.6e+223)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (l <= -5e-310)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / h) / l)));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= -4.6e+223)
		tmp = sqrt((d / l)) * sqrt((d / h));
	elseif (l <= -5e-310)
		tmp = -d * sqrt(((1.0 / h) / l));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, -4.6e+223], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-310], N[((-d) * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $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 < -4.60000000000000009e223

   1. Initial program 70.8%

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

   3. Simplified 70.8%

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

      1. frac-2neg 70.8%

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

      2. sqrt-div 75.2%

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

   6. Applied egg-rr 75.2%

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

   7. Taylor expanded in d around inf 57.4%

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

   if -4.60000000000000009e223 < l < -4.999999999999985e-310

   1. Initial program 69.3%

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

   3. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-/r* 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 46.8%

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

      6. neg-mul-1 46.8%

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

   5. Simplified 46.8%

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

   if -4.999999999999985e-310 < l

   1. Initial program 64.4%

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

   3. Taylor expanded in d around inf 45.4%

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

      1. associate-/r* 45.4%

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

   5. Simplified 45.4%

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

      1. pow1 45.4%

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

      2. associate-/l/ 45.4%

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

      3. sqrt-div 45.4%

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

      4. metadata-eval 45.4%

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

   7. Applied egg-rr 45.4%

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

      1. unpow1 45.4%

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

      2. associate-*r/ 45.5%

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

      3. *-rgt-identity 45.5%

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

      4. *-commutative 45.5%

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

   9. Simplified 45.5%

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

       1. pow1/2 45.5%

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

       2. *-commutative 45.5%

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

       3. unpow-prod-down 55.3%

          \[\leadsto \frac{d}{\color{blue}{{\ell}^{0.5} \cdot {h}^{0.5}}} \]

       4. pow1/2 55.3%

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell}} \cdot {h}^{0.5}} \]

       5. pow1/2 55.3%

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

   11. Applied egg-rr 55.3%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.6 \cdot 10^{+223}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.3 \cdot 10^{-308}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l 4.3e-308)
   (* (- d) (sqrt (/ (/ 1.0 h) l)))
   (/ d (* (sqrt l) (sqrt h)))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= 4.3e-308) {
		tmp = -d * sqrt(((1.0 / h) / l));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: tmp
    if (l <= 4.3d-308) then
        tmp = -d * sqrt(((1.0d0 / h) / l))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= 4.3e-308) {
		tmp = -d * Math.sqrt(((1.0 / h) / l));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= 4.3e-308:
		tmp = -d * math.sqrt(((1.0 / h) / l))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= 4.3e-308)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / h) / l)));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= 4.3e-308)
		tmp = -d * sqrt(((1.0 / h) / l));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, 4.3e-308], N[((-d) * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / 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 l < 4.3000000000000002e-308

   1. Initial program 69.5%

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

   3. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-/r* 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 45.0%

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

      6. neg-mul-1 45.0%

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

   5. Simplified 45.0%

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

   if 4.3000000000000002e-308 < l

   1. Initial program 64.4%

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

   3. Taylor expanded in d around inf 45.4%

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

      1. associate-/r* 45.4%

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

   5. Simplified 45.4%

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

      1. pow1 45.4%

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

      2. associate-/l/ 45.4%

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

      3. sqrt-div 45.4%

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

      4. metadata-eval 45.4%

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

   7. Applied egg-rr 45.4%

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

      1. unpow1 45.4%

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

      2. associate-*r/ 45.5%

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

      3. *-rgt-identity 45.5%

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

      4. *-commutative 45.5%

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

   9. Simplified 45.5%

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

       1. pow1/2 45.5%

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

       2. *-commutative 45.5%

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

       3. unpow-prod-down 55.3%

          \[\leadsto \frac{d}{\color{blue}{{\ell}^{0.5} \cdot {h}^{0.5}}} \]

       4. pow1/2 55.3%

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell}} \cdot {h}^{0.5}} \]

       5. pow1/2 55.3%

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

   11. Applied egg-rr 55.3%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.3 \cdot 10^{-308}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 42.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 6.2 \cdot 10^{-167}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= d 6.2e-167) (* (- d) (sqrt (/ (/ 1.0 h) l))) (/ d (sqrt (* l h)))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= 6.2e-167) {
		tmp = -d * sqrt(((1.0 / h) / l));
	} else {
		tmp = d / sqrt((l * h));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: tmp
    if (d <= 6.2d-167) then
        tmp = -d * sqrt(((1.0d0 / h) / l))
    else
        tmp = d / sqrt((l * h))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= 6.2e-167) {
		tmp = -d * Math.sqrt(((1.0 / h) / l));
	} else {
		tmp = d / Math.sqrt((l * h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if d <= 6.2e-167:
		tmp = -d * math.sqrt(((1.0 / h) / l))
	else:
		tmp = d / math.sqrt((l * h))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (d <= 6.2e-167)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / h) / l)));
	else
		tmp = Float64(d / sqrt(Float64(l * h)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (d <= 6.2e-167)
		tmp = -d * sqrt(((1.0 / h) / l));
	else
		tmp = d / sqrt((l * h));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[d, 6.2e-167], N[((-d) * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 6.2e-167

   1. Initial program 66.1%

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

   3. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-/r* 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 42.1%

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

      6. neg-mul-1 42.1%

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

   5. Simplified 42.1%

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

   if 6.2e-167 < d

   1. Initial program 68.4%

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

   3. Taylor expanded in d around inf 51.1%

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

      1. associate-/r* 51.2%

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

   5. Simplified 51.2%

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

      1. pow1 51.2%

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

      2. associate-/l/ 51.1%

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

      3. sqrt-div 51.1%

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

      4. metadata-eval 51.1%

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

   7. Applied egg-rr 51.1%

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

      1. unpow1 51.1%

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

      2. associate-*r/ 51.2%

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

      3. *-rgt-identity 51.2%

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

      4. *-commutative 51.2%

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

   9. Simplified 51.2%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 6.2 \cdot 10^{-167}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 42.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 1.9 \cdot 10^{-167}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= d 1.9e-167) (* (- d) (pow (* l h) -0.5)) (/ d (sqrt (* l h)))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= 1.9e-167) {
		tmp = -d * pow((l * h), -0.5);
	} else {
		tmp = d / sqrt((l * h));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: tmp
    if (d <= 1.9d-167) then
        tmp = -d * ((l * h) ** (-0.5d0))
    else
        tmp = d / sqrt((l * h))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= 1.9e-167) {
		tmp = -d * Math.pow((l * h), -0.5);
	} else {
		tmp = d / Math.sqrt((l * h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if d <= 1.9e-167:
		tmp = -d * math.pow((l * h), -0.5)
	else:
		tmp = d / math.sqrt((l * h))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (d <= 1.9e-167)
		tmp = Float64(Float64(-d) * (Float64(l * h) ^ -0.5));
	else
		tmp = Float64(d / sqrt(Float64(l * h)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (d <= 1.9e-167)
		tmp = -d * ((l * h) ^ -0.5);
	else
		tmp = d / sqrt((l * h));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[d, 1.9e-167], N[((-d) * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.89999999999999984e-167

   1. Initial program 66.1%

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

   3. Taylor expanded in d around inf 8.9%

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

      1. associate-/r* 8.8%

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

   5. Simplified 8.8%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow1/2 0.0%

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

      4. rem-exp-log 0.0%

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

      5. exp-neg 0.0%

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

      6. exp-prod 0.0%

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

      7. distribute-lft-neg-out 0.0%

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

      8. distribute-rgt-neg-in 0.0%

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

      9. metadata-eval 0.0%

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

      10. exp-to-pow 0.0%

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

      11. *-commutative 0.0%

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

      12. *-commutative 0.0%

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

      13. unpow2 0.0%

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

      14. rem-square-sqrt 41.0%

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

      15. neg-mul-1 41.0%

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

   8. Simplified 41.0%

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

   if 1.89999999999999984e-167 < d

   1. Initial program 68.4%

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

   3. Taylor expanded in d around inf 51.1%

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

      1. associate-/r* 51.2%

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

   5. Simplified 51.2%

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

      1. pow1 51.2%

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

      2. associate-/l/ 51.1%

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

      3. sqrt-div 51.1%

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

      4. metadata-eval 51.1%

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

   7. Applied egg-rr 51.1%

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

      1. unpow1 51.1%

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

      2. associate-*r/ 51.2%

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

      3. *-rgt-identity 51.2%

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

      4. *-commutative 51.2%

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

   9. Simplified 51.2%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.9 \cdot 10^{-167}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 42.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;d \leq 2 \cdot 10^{-167}:\\ \;\;\;\;\frac{d}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{t\_0}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (sqrt (* l h)))) (if (<= d 2e-167) (/ d (- t_0)) (/ d t_0))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = sqrt((l * h));
	double tmp;
	if (d <= 2e-167) {
		tmp = d / -t_0;
	} else {
		tmp = d / t_0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((l * h))
    if (d <= 2d-167) then
        tmp = d / -t_0
    else
        tmp = d / t_0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = Math.sqrt((l * h));
	double tmp;
	if (d <= 2e-167) {
		tmp = d / -t_0;
	} else {
		tmp = d / t_0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = math.sqrt((l * h))
	tmp = 0
	if d <= 2e-167:
		tmp = d / -t_0
	else:
		tmp = d / t_0
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = sqrt(Float64(l * h))
	tmp = 0.0
	if (d <= 2e-167)
		tmp = Float64(d / Float64(-t_0));
	else
		tmp = Float64(d / t_0);
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = sqrt((l * h));
	tmp = 0.0;
	if (d <= 2e-167)
		tmp = d / -t_0;
	else
		tmp = d / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, 2e-167], N[(d / (-t$95$0)), $MachinePrecision], N[(d / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < 2e-167

   1. Initial program 66.1%

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

   3. Simplified 66.1%

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

      1. add-sqr-sqrt 66.1%

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

      2. pow2 66.1%

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

      3. sqrt-prod 66.1%

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

      4. sqrt-pow1 66.3%

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

      5. metadata-eval 66.3%

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

      6. pow1 66.3%

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

      7. frac-times 67.0%

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

      8. *-commutative 67.0%

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

      9. times-frac 65.2%

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

   6. Applied egg-rr 65.2%

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

   8. Simplified 64.6%

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

      1. clear-num 64.6%

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

      2. sqrt-div 64.9%

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

      3. metadata-eval 64.9%

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

   10. Applied egg-rr 64.9%

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

   11. Taylor expanded in d around -inf 41.0%

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

       1. mul-1-neg 41.0%

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

       2. *-commutative 41.0%

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

       3. unpow1/2 41.0%

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

       4. rem-exp-log 39.2%

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

       5. exp-neg 39.2%

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

       6. exp-prod 39.2%

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

       7. distribute-lft-neg-out 39.2%

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

       8. rec-exp 39.2%

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

       9. exp-to-pow 41.0%

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

       10. unpow1/2 41.0%

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

       11. /-rgt-identity 41.0%

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

       12. times-frac 41.0%

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

       13. *-lft-identity 41.0%

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

       14. *-rgt-identity 41.0%

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

   13. Simplified 41.0%

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

   if 2e-167 < d

   1. Initial program 68.4%

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

   3. Taylor expanded in d around inf 51.1%

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

      1. associate-/r* 51.2%

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

   5. Simplified 51.2%

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

      1. pow1 51.2%

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

      2. associate-/l/ 51.1%

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

      3. sqrt-div 51.1%

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

      4. metadata-eval 51.1%

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

   7. Applied egg-rr 51.1%

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

      1. unpow1 51.1%

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

      2. associate-*r/ 51.2%

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

      3. *-rgt-identity 51.2%

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

      4. *-commutative 51.2%

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

   9. Simplified 51.2%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 2 \cdot 10^{-167}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 26.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \frac{d}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D) :precision binary64 (/ d (sqrt (* l h))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	return d / sqrt((l * h));
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    code = d / sqrt((l * h))
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	return d / Math.sqrt((l * h));
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	return d / math.sqrt((l * h))

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	return Float64(d / sqrt(Float64(l * h)))
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp = code(d, h, l, M_m, D)
	tmp = d / sqrt((l * h));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.0%

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

3. Taylor expanded in d around inf 25.9%

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

   1. associate-/r* 25.9%

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

5. Simplified 25.9%

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

   1. pow1 25.9%

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

   2. associate-/l/ 25.9%

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

   3. sqrt-div 25.5%

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

   4. metadata-eval 25.5%

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

7. Applied egg-rr 25.5%

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

   1. unpow1 25.5%

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

   2. associate-*r/ 25.6%

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

   3. *-rgt-identity 25.6%

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

   4. *-commutative 25.6%

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

9. Simplified 25.6%

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

10. Final simplification 25.6%

    \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]
11. Add Preprocessing

Alternative 17: 4.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ d \cdot \sqrt{0} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D) :precision binary64 (* d (sqrt 0.0)))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	return d * sqrt(0.0);
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    code = d * sqrt(0.0d0)
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	return d * Math.sqrt(0.0);
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	return d * math.sqrt(0.0)

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	return Float64(d * sqrt(0.0))
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp = code(d, h, l, M_m, D)
	tmp = d * sqrt(0.0);
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := N[(d * N[Sqrt[0.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.0%

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

3. Taylor expanded in d around inf 25.9%

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

   1. associate-/r* 25.9%

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

5. Simplified 25.9%

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

   1. expm1-log1p-u 25.3%

      \[\leadsto d \cdot \sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{h}}{\ell}\right)\right)}} \]

   2. expm1-undefine 16.2%

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

   3. associate-/l/ 16.2%

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

7. Applied egg-rr 16.2%

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

8. Taylor expanded in l around inf 5.7%

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

9. Final simplification 5.7%

   \[\leadsto d \cdot \sqrt{0} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024051 
(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)))))