Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.2% → 79.3%

Time: 30.3s

Alternatives: 18

Speedup: 1.0×

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: 66.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t_0 \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{\left(h \cdot {\left(\frac{M \cdot \left(0.5 \cdot D\right)}{d}\right)}^{2}\right) \cdot -0.5}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{+185}:\\ \;\;\;\;t_0 \cdot \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(1 + \frac{-0.5 \cdot \left(h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l))))
   (if (<= l -2e-310)
     (*
      t_0
      (*
       (/ (sqrt (- d)) (sqrt (- h)))
       (+ 1.0 (/ (* (* h (pow (/ (* M (* 0.5 D)) d) 2.0)) -0.5) l))))
     (if (<= l 2e+185)
       (*
        t_0
        (*
         (/ (sqrt d) (sqrt h))
         (+ 1.0 (/ (* -0.5 (* h (pow (* M (* 0.5 (/ D d))) 2.0))) l))))
       (/ d (* (sqrt h) (sqrt l)))))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / l));
	double tmp;
	if (l <= -2e-310) {
		tmp = t_0 * ((sqrt(-d) / sqrt(-h)) * (1.0 + (((h * pow(((M * (0.5 * D)) / d), 2.0)) * -0.5) / l)));
	} else if (l <= 2e+185) {
		tmp = t_0 * ((sqrt(d) / sqrt(h)) * (1.0 + ((-0.5 * (h * pow((M * (0.5 * (D / d))), 2.0))) / l)));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((d / l))
    if (l <= (-2d-310)) then
        tmp = t_0 * ((sqrt(-d) / sqrt(-h)) * (1.0d0 + (((h * (((m * (0.5d0 * d_1)) / d) ** 2.0d0)) * (-0.5d0)) / l)))
    else if (l <= 2d+185) then
        tmp = t_0 * ((sqrt(d) / sqrt(h)) * (1.0d0 + (((-0.5d0) * (h * ((m * (0.5d0 * (d_1 / d))) ** 2.0d0))) / l)))
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((d / l));
	double tmp;
	if (l <= -2e-310) {
		tmp = t_0 * ((Math.sqrt(-d) / Math.sqrt(-h)) * (1.0 + (((h * Math.pow(((M * (0.5 * D)) / d), 2.0)) * -0.5) / l)));
	} else if (l <= 2e+185) {
		tmp = t_0 * ((Math.sqrt(d) / Math.sqrt(h)) * (1.0 + ((-0.5 * (h * Math.pow((M * (0.5 * (D / d))), 2.0))) / l)));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.sqrt((d / l))
	tmp = 0
	if l <= -2e-310:
		tmp = t_0 * ((math.sqrt(-d) / math.sqrt(-h)) * (1.0 + (((h * math.pow(((M * (0.5 * D)) / d), 2.0)) * -0.5) / l)))
	elif l <= 2e+185:
		tmp = t_0 * ((math.sqrt(d) / math.sqrt(h)) * (1.0 + ((-0.5 * (h * math.pow((M * (0.5 * (D / d))), 2.0))) / l)))
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / l))
	tmp = 0.0
	if (l <= -2e-310)
		tmp = Float64(t_0 * Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(Float64(h * (Float64(Float64(M * Float64(0.5 * D)) / d) ^ 2.0)) * -0.5) / l))));
	elseif (l <= 2e+185)
		tmp = Float64(t_0 * Float64(Float64(sqrt(d) / sqrt(h)) * Float64(1.0 + Float64(Float64(-0.5 * Float64(h * (Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0))) / l))));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / l));
	tmp = 0.0;
	if (l <= -2e-310)
		tmp = t_0 * ((sqrt(-d) / sqrt(-h)) * (1.0 + (((h * (((M * (0.5 * D)) / d) ^ 2.0)) * -0.5) / l)));
	elseif (l <= 2e+185)
		tmp = t_0 * ((sqrt(d) / sqrt(h)) * (1.0 + ((-0.5 * (h * ((M * (0.5 * (D / d))) ^ 2.0))) / l)));
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -2e-310], N[(t$95$0 * N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(N[(h * N[Power[N[(N[(M * N[(0.5 * D), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2e+185], N[(t$95$0 * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(-0.5 * N[(h * N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.999999999999994e-310

   1. Initial program 63.5%

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

   3. Simplified 62.8%

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

      1. associate-*r* 62.8%

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

      2. *-commutative 62.8%

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

      3. associate-*r/ 67.8%

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

      4. associate-*l/ 67.8%

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

      5. div-inv 67.8%

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

      6. associate-*l* 67.8%

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

      7. metadata-eval 67.8%

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

   5. Applied egg-rr 67.8%

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

      1. associate-*r* 67.8%

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

      2. metadata-eval 67.8%

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

      3. div-inv 67.8%

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

      4. associate-*r/ 68.5%

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

      5. div-inv 68.5%

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

      6. metadata-eval 68.5%

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

      7. associate-*r* 68.5%

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

   7. Applied egg-rr 68.5%

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

      1. frac-2neg 68.5%

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

      2. sqrt-div 80.5%

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

   9. Applied egg-rr 80.5%

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

   if -1.999999999999994e-310 < l < 2e185

   1. Initial program 72.5%

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

   3. Simplified 70.7%

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

      1. associate-*r* 70.7%

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

      2. *-commutative 70.7%

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

      3. associate-*r/ 72.7%

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

      4. associate-*l/ 72.7%

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

      5. div-inv 72.7%

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

      6. associate-*l* 72.7%

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

      7. metadata-eval 72.7%

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

   5. Applied egg-rr 72.7%

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

      1. sqrt-div 85.0%

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

      2. div-inv 84.9%

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

   7. Applied egg-rr 84.9%

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

      1. associate-*r/ 85.0%

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

      2. *-rgt-identity 85.0%

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

   9. Simplified 85.0%

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

   if 2e185 < l

   1. Initial program 50.9%

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

   2. Taylor expanded in d around inf 62.0%

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

      1. expm1-log1p-u 61.7%

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

      2. expm1-udef 35.6%

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

      3. pow1/2 35.6%

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

      4. inv-pow 35.6%

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

      5. pow-pow 35.6%

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

      6. metadata-eval 35.6%

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

   4. Applied egg-rr 35.6%

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

      1. expm1-def 61.8%

         \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \]

      2. expm1-log1p 62.2%

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

   6. Simplified 62.2%

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

      1. add-sqr-sqrt 61.9%

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

      2. sqrt-unprod 62.2%

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

      3. pow-prod-up 62.0%

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

      4. metadata-eval 62.0%

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

      5. inv-pow 62.0%

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

      6. *-commutative 62.0%

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

      7. associate-/l/ 64.4%

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

      8. rem-cbrt-cube 45.6%

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

      9. unpow1/3 44.3%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.3333333333333333}}} \]

      10. expm1-log1p-u 43.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.3333333333333333}}\right)\right)} \]

      11. expm1-udef 27.7%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.3333333333333333}}\right)} - 1} \]

   8. Applied egg-rr 35.6%

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

      1. expm1-def 60.0%

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

      2. expm1-log1p 62.0%

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

   10. Simplified 62.0%

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

       1. *-commutative 62.0%

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

       2. sqrt-prod 82.5%

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

   12. Applied egg-rr 82.5%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{\left(h \cdot {\left(\frac{M \cdot \left(0.5 \cdot D\right)}{d}\right)}^{2}\right) \cdot -0.5}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{+185}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(1 + \frac{-0.5 \cdot \left(h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 2: 76.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := M \cdot \left(0.5 \cdot D\right)\\ t_3 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-85}:\\ \;\;\;\;t_3 \cdot \left(t_0 \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{d}{t_2}\right)}^{-2}\right)\right)\right)\\ \mathbf{elif}\;t_1 \leq 0 \lor \neg \left(t_1 \leq 2 \cdot 10^{+229}\right):\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \left(\left(1 + \frac{\left(h \cdot {\left(\frac{t_2}{d}\right)}^{2}\right) \cdot -0.5}{\ell}\right) \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h)))
        (t_1
         (*
          (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
          (- 1.0 (* (* 0.5 (pow (/ (* M D) (* d 2.0)) 2.0)) (/ h l)))))
        (t_2 (* M (* 0.5 D)))
        (t_3 (sqrt (/ d l))))
   (if (<= t_1 -1e-85)
     (* t_3 (* t_0 (+ 1.0 (* (/ h l) (* -0.5 (pow (/ d t_2) -2.0))))))
     (if (or (<= t_1 0.0) (not (<= t_1 2e+229)))
       (fabs (/ d (sqrt (* l h))))
       (* t_3 (* (+ 1.0 (/ (* (* h (pow (/ t_2 d) 2.0)) -0.5) l)) t_0))))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h));
	double t_1 = (pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((0.5 * pow(((M * D) / (d * 2.0)), 2.0)) * (h / l)));
	double t_2 = M * (0.5 * D);
	double t_3 = sqrt((d / l));
	double tmp;
	if (t_1 <= -1e-85) {
		tmp = t_3 * (t_0 * (1.0 + ((h / l) * (-0.5 * pow((d / t_2), -2.0)))));
	} else if ((t_1 <= 0.0) || !(t_1 <= 2e+229)) {
		tmp = fabs((d / sqrt((l * h))));
	} else {
		tmp = t_3 * ((1.0 + (((h * pow((t_2 / d), 2.0)) * -0.5) / l)) * t_0);
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = sqrt((d / h))
    t_1 = (((d / h) ** 0.5d0) * ((d / l) ** 0.5d0)) * (1.0d0 - ((0.5d0 * (((m * d_1) / (d * 2.0d0)) ** 2.0d0)) * (h / l)))
    t_2 = m * (0.5d0 * d_1)
    t_3 = sqrt((d / l))
    if (t_1 <= (-1d-85)) then
        tmp = t_3 * (t_0 * (1.0d0 + ((h / l) * ((-0.5d0) * ((d / t_2) ** (-2.0d0))))))
    else if ((t_1 <= 0.0d0) .or. (.not. (t_1 <= 2d+229))) then
        tmp = abs((d / sqrt((l * h))))
    else
        tmp = t_3 * ((1.0d0 + (((h * ((t_2 / d) ** 2.0d0)) * (-0.5d0)) / l)) * t_0)
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((d / h));
	double t_1 = (Math.pow((d / h), 0.5) * Math.pow((d / l), 0.5)) * (1.0 - ((0.5 * Math.pow(((M * D) / (d * 2.0)), 2.0)) * (h / l)));
	double t_2 = M * (0.5 * D);
	double t_3 = Math.sqrt((d / l));
	double tmp;
	if (t_1 <= -1e-85) {
		tmp = t_3 * (t_0 * (1.0 + ((h / l) * (-0.5 * Math.pow((d / t_2), -2.0)))));
	} else if ((t_1 <= 0.0) || !(t_1 <= 2e+229)) {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	} else {
		tmp = t_3 * ((1.0 + (((h * Math.pow((t_2 / d), 2.0)) * -0.5) / l)) * t_0);
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.sqrt((d / h))
	t_1 = (math.pow((d / h), 0.5) * math.pow((d / l), 0.5)) * (1.0 - ((0.5 * math.pow(((M * D) / (d * 2.0)), 2.0)) * (h / l)))
	t_2 = M * (0.5 * D)
	t_3 = math.sqrt((d / l))
	tmp = 0
	if t_1 <= -1e-85:
		tmp = t_3 * (t_0 * (1.0 + ((h / l) * (-0.5 * math.pow((d / t_2), -2.0)))))
	elif (t_1 <= 0.0) or not (t_1 <= 2e+229):
		tmp = math.fabs((d / math.sqrt((l * h))))
	else:
		tmp = t_3 * ((1.0 + (((h * math.pow((t_2 / d), 2.0)) * -0.5) / l)) * t_0)
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / h))
	t_1 = Float64(Float64((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5)) * Float64(1.0 - Float64(Float64(0.5 * (Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0)) * Float64(h / l))))
	t_2 = Float64(M * Float64(0.5 * D))
	t_3 = sqrt(Float64(d / l))
	tmp = 0.0
	if (t_1 <= -1e-85)
		tmp = Float64(t_3 * Float64(t_0 * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(d / t_2) ^ -2.0))))));
	elseif ((t_1 <= 0.0) || !(t_1 <= 2e+229))
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	else
		tmp = Float64(t_3 * Float64(Float64(1.0 + Float64(Float64(Float64(h * (Float64(t_2 / d) ^ 2.0)) * -0.5) / l)) * t_0));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / h));
	t_1 = (((d / h) ^ 0.5) * ((d / l) ^ 0.5)) * (1.0 - ((0.5 * (((M * D) / (d * 2.0)) ^ 2.0)) * (h / l)));
	t_2 = M * (0.5 * D);
	t_3 = sqrt((d / l));
	tmp = 0.0;
	if (t_1 <= -1e-85)
		tmp = t_3 * (t_0 * (1.0 + ((h / l) * (-0.5 * ((d / t_2) ^ -2.0)))));
	elseif ((t_1 <= 0.0) || ~((t_1 <= 2e+229)))
		tmp = abs((d / sqrt((l * h))));
	else
		tmp = t_3 * ((1.0 + (((h * ((t_2 / d) ^ 2.0)) * -0.5) / l)) * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(0.5 * N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(M * N[(0.5 * D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -1e-85], N[(t$95$3 * N[(t$95$0 * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(d / t$95$2), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$1, 0.0], N[Not[LessEqual[t$95$1, 2e+229]], $MachinePrecision]], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$3 * N[(N[(1.0 + N[(N[(N[(h * N[Power[N[(t$95$2 / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < -9.9999999999999998e-86

   1. Initial program 89.2%

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

   3. Simplified 86.0%

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

      1. associate-*r/ 89.2%

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

      2. clear-num 89.2%

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

      3. div-inv 89.2%

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

      4. associate-*l* 89.2%

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

      5. metadata-eval 89.2%

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

   5. Applied egg-rr 89.2%

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

      1. expm1-log1p-u 0.0%

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

      2. expm1-udef 0.0%

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

      3. inv-pow 0.0%

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

      4. pow-pow 0.0%

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

      5. associate-/r* 0.0%

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

      6. metadata-eval 0.0%

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

   7. Applied egg-rr 0.0%

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

      1. expm1-def 0.0%

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

      2. expm1-log1p 84.9%

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

      3. *-commutative 84.9%

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

      4. associate-/r* 89.2%

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

      5. *-commutative 89.2%

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

   9. Simplified 89.2%

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

   if -9.9999999999999998e-86 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < 0.0 or 2e229 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l))))

   1. Initial program 22.4%

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

   2. Taylor expanded in d around inf 33.9%

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

      1. expm1-log1p-u 33.6%

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

      2. expm1-udef 26.3%

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

      3. pow1/2 26.3%

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

      4. inv-pow 26.3%

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

      5. pow-pow 26.3%

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

      6. metadata-eval 26.3%

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

   4. Applied egg-rr 26.3%

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

      1. expm1-def 33.6%

         \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \]

      2. expm1-log1p 33.9%

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

   6. Simplified 33.9%

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

      1. add-sqr-sqrt 30.9%

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

      2. sqrt-prod 37.3%

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

      3. unpow2 37.3%

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

      4. add-sqr-sqrt 37.2%

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

      5. sqrt-unprod 37.3%

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

      6. pow-prod-up 37.3%

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

      7. metadata-eval 37.3%

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

      8. inv-pow 37.3%

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

      9. *-commutative 37.3%

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

      10. associate-/l/ 37.3%

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

      11. sqrt-prod 32.6%

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

      12. associate-/l/ 32.7%

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

      13. *-commutative 32.7%

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

      14. div-inv 32.7%

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

      15. add-sqr-sqrt 32.7%

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

      16. rem-sqrt-square 32.7%

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

      17. sqrt-div 37.3%

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

      18. unpow2 37.3%

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

      19. sqrt-prod 30.9%

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

      20. add-sqr-sqrt 58.2%

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

   8. Applied egg-rr 58.2%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < 2e229

   1. Initial program 98.5%

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

   3. Simplified 98.5%

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

      1. associate-*r* 98.5%

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

      2. *-commutative 98.5%

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

      3. associate-*r/ 98.5%

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

      4. associate-*l/ 98.5%

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

      5. div-inv 98.5%

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

      6. associate-*l* 98.5%

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

      7. metadata-eval 98.5%

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

   5. Applied egg-rr 98.5%

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

      1. associate-*r* 98.5%

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

      2. metadata-eval 98.5%

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

      3. div-inv 98.5%

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

      4. associate-*r/ 98.5%

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

      5. div-inv 98.5%

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

      6. metadata-eval 98.5%

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

      7. associate-*r* 98.5%

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

   7. Applied egg-rr 98.5%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -1 \cdot 10^{-85}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{d}{M \cdot \left(0.5 \cdot D\right)}\right)}^{-2}\right)\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0 \lor \neg \left(\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2 \cdot 10^{+229}\right):\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \frac{\left(h \cdot {\left(\frac{M \cdot \left(0.5 \cdot D\right)}{d}\right)}^{2}\right) \cdot -0.5}{\ell}\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \end{array} \]

Alternative 3: 79.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := 1 + \frac{-0.5 \cdot \left(h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)}{\ell}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t_1 \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t_0\right)\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+184}:\\ \;\;\;\;t_1 \cdot \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (* -0.5 (* h (pow (* M (* 0.5 (/ D d))) 2.0))) l)))
        (t_1 (sqrt (/ d l))))
   (if (<= l -2e-310)
     (* t_1 (* (/ (sqrt (- d)) (sqrt (- h))) t_0))
     (if (<= l 1.35e+184)
       (* t_1 (* (/ (sqrt d) (sqrt h)) t_0))
       (/ d (* (sqrt h) (sqrt l)))))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 + ((-0.5 * (h * pow((M * (0.5 * (D / d))), 2.0))) / l);
	double t_1 = sqrt((d / l));
	double tmp;
	if (l <= -2e-310) {
		tmp = t_1 * ((sqrt(-d) / sqrt(-h)) * t_0);
	} else if (l <= 1.35e+184) {
		tmp = t_1 * ((sqrt(d) / sqrt(h)) * t_0);
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + (((-0.5d0) * (h * ((m * (0.5d0 * (d_1 / d))) ** 2.0d0))) / l)
    t_1 = sqrt((d / l))
    if (l <= (-2d-310)) then
        tmp = t_1 * ((sqrt(-d) / sqrt(-h)) * t_0)
    else if (l <= 1.35d+184) then
        tmp = t_1 * ((sqrt(d) / sqrt(h)) * t_0)
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 + ((-0.5 * (h * Math.pow((M * (0.5 * (D / d))), 2.0))) / l);
	double t_1 = Math.sqrt((d / l));
	double tmp;
	if (l <= -2e-310) {
		tmp = t_1 * ((Math.sqrt(-d) / Math.sqrt(-h)) * t_0);
	} else if (l <= 1.35e+184) {
		tmp = t_1 * ((Math.sqrt(d) / Math.sqrt(h)) * t_0);
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = 1.0 + ((-0.5 * (h * math.pow((M * (0.5 * (D / d))), 2.0))) / l)
	t_1 = math.sqrt((d / l))
	tmp = 0
	if l <= -2e-310:
		tmp = t_1 * ((math.sqrt(-d) / math.sqrt(-h)) * t_0)
	elif l <= 1.35e+184:
		tmp = t_1 * ((math.sqrt(d) / math.sqrt(h)) * t_0)
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(1.0 + Float64(Float64(-0.5 * Float64(h * (Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0))) / l))
	t_1 = sqrt(Float64(d / l))
	tmp = 0.0
	if (l <= -2e-310)
		tmp = Float64(t_1 * Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * t_0));
	elseif (l <= 1.35e+184)
		tmp = Float64(t_1 * Float64(Float64(sqrt(d) / sqrt(h)) * t_0));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = 1.0 + ((-0.5 * (h * ((M * (0.5 * (D / d))) ^ 2.0))) / l);
	t_1 = sqrt((d / l));
	tmp = 0.0;
	if (l <= -2e-310)
		tmp = t_1 * ((sqrt(-d) / sqrt(-h)) * t_0);
	elseif (l <= 1.35e+184)
		tmp = t_1 * ((sqrt(d) / sqrt(h)) * t_0);
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(1.0 + N[(N[(-0.5 * N[(h * N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -2e-310], N[(t$95$1 * N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.35e+184], N[(t$95$1 * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.999999999999994e-310

   1. Initial program 63.5%

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

   3. Simplified 62.8%

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

      1. associate-*r* 62.8%

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

      2. *-commutative 62.8%

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

      3. associate-*r/ 67.8%

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

      4. associate-*l/ 67.8%

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

      5. div-inv 67.8%

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

      6. associate-*l* 67.8%

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

      7. metadata-eval 67.8%

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

   5. Applied egg-rr 67.8%

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

      1. frac-2neg 68.5%

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

      2. sqrt-div 80.5%

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

   7. Applied egg-rr 79.8%

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

   if -1.999999999999994e-310 < l < 1.35e184

   1. Initial program 72.5%

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

   3. Simplified 70.7%

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

      1. associate-*r* 70.7%

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

      2. *-commutative 70.7%

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

      3. associate-*r/ 72.7%

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

      4. associate-*l/ 72.7%

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

      5. div-inv 72.7%

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

      6. associate-*l* 72.7%

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

      7. metadata-eval 72.7%

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

   5. Applied egg-rr 72.7%

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

      1. sqrt-div 85.0%

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

      2. div-inv 84.9%

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

   7. Applied egg-rr 84.9%

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

      1. associate-*r/ 85.0%

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

      2. *-rgt-identity 85.0%

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

   9. Simplified 85.0%

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

   if 1.35e184 < l

   1. Initial program 50.9%

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

   2. Taylor expanded in d around inf 62.0%

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

      1. expm1-log1p-u 61.7%

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

      2. expm1-udef 35.6%

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

      3. pow1/2 35.6%

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

      4. inv-pow 35.6%

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

      5. pow-pow 35.6%

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

      6. metadata-eval 35.6%

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

   4. Applied egg-rr 35.6%

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

      1. expm1-def 61.8%

         \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \]

      2. expm1-log1p 62.2%

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

   6. Simplified 62.2%

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

      1. add-sqr-sqrt 61.9%

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

      2. sqrt-unprod 62.2%

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

      3. pow-prod-up 62.0%

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

      4. metadata-eval 62.0%

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

      5. inv-pow 62.0%

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

      6. *-commutative 62.0%

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

      7. associate-/l/ 64.4%

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

      8. rem-cbrt-cube 45.6%

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

      9. unpow1/3 44.3%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.3333333333333333}}} \]

      10. expm1-log1p-u 43.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.3333333333333333}}\right)\right)} \]

      11. expm1-udef 27.7%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.3333333333333333}}\right)} - 1} \]

   8. Applied egg-rr 35.6%

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

      1. expm1-def 60.0%

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

      2. expm1-log1p 62.0%

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

   10. Simplified 62.0%

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

       1. *-commutative 62.0%

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

       2. sqrt-prod 82.5%

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

   12. Applied egg-rr 82.5%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{-0.5 \cdot \left(h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+184}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(1 + \frac{-0.5 \cdot \left(h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 4: 66.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.38 \cdot 10^{+119}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+133}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1.38e+119)
   (fabs (/ d (sqrt (* l h))))
   (if (<= l 2.6e+133)
     (*
      (sqrt (/ d h))
      (*
       (sqrt (/ d l))
       (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D d) (/ M 2.0)) 2.0))))))
     (/ d (* (sqrt h) (sqrt l))))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.38e+119) {
		tmp = fabs((d / sqrt((l * h))));
	} else if (l <= 2.6e+133) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * pow(((D / d) * (M / 2.0)), 2.0)))));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-1.38d+119)) then
        tmp = abs((d / sqrt((l * h))))
    else if (l <= 2.6d+133) then
        tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 - (0.5d0 * ((h / l) * (((d_1 / d) * (m / 2.0d0)) ** 2.0d0)))))
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.38e+119) {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	} else if (l <= 2.6e+133) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * Math.pow(((D / d) * (M / 2.0)), 2.0)))));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= -1.38e+119:
		tmp = math.fabs((d / math.sqrt((l * h))))
	elif l <= 2.6e+133:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * math.pow(((D / d) * (M / 2.0)), 2.0)))))
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1.38e+119)
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	elseif (l <= 2.6e+133)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D / d) * Float64(M / 2.0)) ^ 2.0))))));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -1.38e+119)
		tmp = abs((d / sqrt((l * h))));
	elseif (l <= 2.6e+133)
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * (((D / d) * (M / 2.0)) ^ 2.0)))));
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1.38e+119], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 2.6e+133], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.38000000000000001e119

   1. Initial program 42.1%

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

   2. Taylor expanded in d around inf 8.1%

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

      1. expm1-log1p-u 8.1%

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

      2. expm1-udef 8.4%

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

      3. pow1/2 8.4%

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

      4. inv-pow 8.4%

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

      5. pow-pow 8.4%

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

      6. metadata-eval 8.4%

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

   4. Applied egg-rr 8.4%

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

      1. expm1-def 8.1%

         \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \]

      2. expm1-log1p 8.1%

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

   6. Simplified 8.1%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-prod 39.8%

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

      3. unpow2 39.8%

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

      4. add-sqr-sqrt 39.5%

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

      5. sqrt-unprod 39.8%

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

      6. pow-prod-up 39.7%

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

      7. metadata-eval 39.7%

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

      8. inv-pow 39.7%

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

      9. *-commutative 39.7%

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

      10. associate-/l/ 39.6%

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

      11. sqrt-prod 34.7%

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

      12. associate-/l/ 34.8%

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

      13. *-commutative 34.8%

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

      14. div-inv 34.8%

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

      15. add-sqr-sqrt 34.8%

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

      16. rem-sqrt-square 34.8%

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

      17. sqrt-div 39.8%

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

      18. unpow2 39.8%

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

      19. sqrt-prod 0.0%

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

      20. add-sqr-sqrt 55.1%

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

   8. Applied egg-rr 55.1%

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

   if -1.38000000000000001e119 < l < 2.5999999999999998e133

   1. Initial program 73.0%

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

   3. Simplified 71.4%

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

   if 2.5999999999999998e133 < l

   1. Initial program 56.5%

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

   2. Taylor expanded in d around inf 65.6%

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

      1. expm1-log1p-u 65.2%

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

      2. expm1-udef 25.8%

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

      3. pow1/2 25.8%

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

      4. inv-pow 25.8%

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

      5. pow-pow 25.8%

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

      6. metadata-eval 25.8%

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

   4. Applied egg-rr 25.8%

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

      1. expm1-def 65.3%

         \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \]

      2. expm1-log1p 65.7%

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

   6. Simplified 65.7%

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

      1. add-sqr-sqrt 65.5%

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

      2. sqrt-unprod 65.7%

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

      3. pow-prod-up 65.6%

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

      4. metadata-eval 65.6%

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

      5. inv-pow 65.6%

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

      6. *-commutative 65.6%

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

      7. associate-/l/ 67.0%

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

      8. rem-cbrt-cube 34.7%

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

      9. unpow1/3 33.5%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.3333333333333333}}} \]

      10. expm1-log1p-u 32.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.3333333333333333}}\right)\right)} \]

      11. expm1-udef 20.8%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.3333333333333333}}\right)} - 1} \]

   8. Applied egg-rr 35.1%

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

      1. expm1-def 63.1%

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

      2. expm1-log1p 65.8%

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

   10. Simplified 65.8%

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

       1. *-commutative 65.8%

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

       2. sqrt-prod 81.0%

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

   12. Applied egg-rr 81.0%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.38 \cdot 10^{+119}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+133}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 5: 66.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.6 \cdot 10^{+117}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;\ell \leq 2.1 \cdot 10^{+133}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M \cdot \left(0.5 \cdot D\right)}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l -5.6e+117)
   (fabs (/ d (sqrt (* l h))))
   (if (<= l 2.1e+133)
     (*
      (sqrt (/ d h))
      (*
       (sqrt (/ d l))
       (- 1.0 (* 0.5 (* (pow (/ (* M (* 0.5 D)) d) 2.0) (/ h l))))))
     (/ d (* (sqrt h) (sqrt l))))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -5.6e+117) {
		tmp = fabs((d / sqrt((l * h))));
	} else if (l <= 2.1e+133) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (0.5 * (pow(((M * (0.5 * D)) / d), 2.0) * (h / l)))));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-5.6d+117)) then
        tmp = abs((d / sqrt((l * h))))
    else if (l <= 2.1d+133) then
        tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 - (0.5d0 * ((((m * (0.5d0 * d_1)) / d) ** 2.0d0) * (h / l)))))
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -5.6e+117) {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	} else if (l <= 2.1e+133) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 - (0.5 * (Math.pow(((M * (0.5 * D)) / d), 2.0) * (h / l)))));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= -5.6e+117:
		tmp = math.fabs((d / math.sqrt((l * h))))
	elif l <= 2.1e+133:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 - (0.5 * (math.pow(((M * (0.5 * D)) / d), 2.0) * (h / l)))))
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -5.6e+117)
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	elseif (l <= 2.1e+133)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(0.5 * Float64((Float64(Float64(M * Float64(0.5 * D)) / d) ^ 2.0) * Float64(h / l))))));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -5.6e+117)
		tmp = abs((d / sqrt((l * h))));
	elseif (l <= 2.1e+133)
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (0.5 * ((((M * (0.5 * D)) / d) ^ 2.0) * (h / l)))));
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, -5.6e+117], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 2.1e+133], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[Power[N[(N[(M * N[(0.5 * D), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.59999999999999995e117

   1. Initial program 42.1%

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

   2. Taylor expanded in d around inf 8.1%

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

      1. expm1-log1p-u 8.1%

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

      2. expm1-udef 8.4%

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

      3. pow1/2 8.4%

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

      4. inv-pow 8.4%

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

      5. pow-pow 8.4%

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

      6. metadata-eval 8.4%

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

   4. Applied egg-rr 8.4%

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

      1. expm1-def 8.1%

         \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \]

      2. expm1-log1p 8.1%

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

   6. Simplified 8.1%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-prod 39.8%

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

      3. unpow2 39.8%

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

      4. add-sqr-sqrt 39.5%

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

      5. sqrt-unprod 39.8%

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

      6. pow-prod-up 39.7%

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

      7. metadata-eval 39.7%

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

      8. inv-pow 39.7%

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

      9. *-commutative 39.7%

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

      10. associate-/l/ 39.6%

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

      11. sqrt-prod 34.7%

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

      12. associate-/l/ 34.8%

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

      13. *-commutative 34.8%

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

      14. div-inv 34.8%

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

      15. add-sqr-sqrt 34.8%

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

      16. rem-sqrt-square 34.8%

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

      17. sqrt-div 39.8%

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

      18. unpow2 39.8%

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

      19. sqrt-prod 0.0%

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

      20. add-sqr-sqrt 55.1%

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

   8. Applied egg-rr 55.1%

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

   if -5.59999999999999995e117 < l < 2.1e133

   1. Initial program 73.0%

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

   3. Simplified 71.4%

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

      1. associate-*r/ 73.0%

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

      2. div-inv 73.0%

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

      3. associate-*l* 73.0%

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

      4. metadata-eval 73.0%

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

   5. Applied egg-rr 73.0%

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

   if 2.1e133 < l

   1. Initial program 56.5%

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

   2. Taylor expanded in d around inf 65.6%

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

      1. expm1-log1p-u 65.2%

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

      2. expm1-udef 25.8%

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

      3. pow1/2 25.8%

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

      4. inv-pow 25.8%

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

      5. pow-pow 25.8%

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

      6. metadata-eval 25.8%

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

   4. Applied egg-rr 25.8%

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

      1. expm1-def 65.3%

         \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \]

      2. expm1-log1p 65.7%

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

   6. Simplified 65.7%

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

      1. add-sqr-sqrt 65.5%

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

      2. sqrt-unprod 65.7%

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

      3. pow-prod-up 65.6%

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

      4. metadata-eval 65.6%

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

      5. inv-pow 65.6%

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

      6. *-commutative 65.6%

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

      7. associate-/l/ 67.0%

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

      8. rem-cbrt-cube 34.7%

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

      9. unpow1/3 33.5%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.3333333333333333}}} \]

      10. expm1-log1p-u 32.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.3333333333333333}}\right)\right)} \]

      11. expm1-udef 20.8%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.3333333333333333}}\right)} - 1} \]

   8. Applied egg-rr 35.1%

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

      1. expm1-def 63.1%

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

      2. expm1-log1p 65.8%

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

   10. Simplified 65.8%

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

       1. *-commutative 65.8%

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

       2. sqrt-prod 81.0%

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

   12. Applied egg-rr 81.0%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.6 \cdot 10^{+117}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;\ell \leq 2.1 \cdot 10^{+133}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M \cdot \left(0.5 \cdot D\right)}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 6: 67.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.6 \cdot 10^{+118}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{+133}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{d}{M \cdot \left(0.5 \cdot D\right)}\right)}^{-2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l -2.6e+118)
   (fabs (/ d (sqrt (* l h))))
   (if (<= l 2.2e+133)
     (*
      (sqrt (/ d l))
      (*
       (sqrt (/ d h))
       (+ 1.0 (* (/ h l) (* -0.5 (pow (/ d (* M (* 0.5 D))) -2.0))))))
     (/ d (* (sqrt h) (sqrt l))))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -2.6e+118) {
		tmp = fabs((d / sqrt((l * h))));
	} else if (l <= 2.2e+133) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * pow((d / (M * (0.5 * D))), -2.0)))));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-2.6d+118)) then
        tmp = abs((d / sqrt((l * h))))
    else if (l <= 2.2d+133) then
        tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0d0 + ((h / l) * ((-0.5d0) * ((d / (m * (0.5d0 * d_1))) ** (-2.0d0))))))
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -2.6e+118) {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	} else if (l <= 2.2e+133) {
		tmp = Math.sqrt((d / l)) * (Math.sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * Math.pow((d / (M * (0.5 * D))), -2.0)))));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= -2.6e+118:
		tmp = math.fabs((d / math.sqrt((l * h))))
	elif l <= 2.2e+133:
		tmp = math.sqrt((d / l)) * (math.sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * math.pow((d / (M * (0.5 * D))), -2.0)))))
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -2.6e+118)
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	elseif (l <= 2.2e+133)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(d / Float64(M * Float64(0.5 * D))) ^ -2.0))))));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -2.6e+118)
		tmp = abs((d / sqrt((l * h))));
	elseif (l <= 2.2e+133)
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * ((d / (M * (0.5 * D))) ^ -2.0)))));
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, -2.6e+118], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 2.2e+133], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(d / N[(M * N[(0.5 * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.60000000000000016e118

   1. Initial program 42.1%

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

   2. Taylor expanded in d around inf 8.1%

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

      1. expm1-log1p-u 8.1%

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

      2. expm1-udef 8.4%

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

      3. pow1/2 8.4%

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

      4. inv-pow 8.4%

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

      5. pow-pow 8.4%

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

      6. metadata-eval 8.4%

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

   4. Applied egg-rr 8.4%

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

      1. expm1-def 8.1%

         \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \]

      2. expm1-log1p 8.1%

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

   6. Simplified 8.1%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-prod 39.8%

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

      3. unpow2 39.8%

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

      4. add-sqr-sqrt 39.5%

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

      5. sqrt-unprod 39.8%

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

      6. pow-prod-up 39.7%

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

      7. metadata-eval 39.7%

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

      8. inv-pow 39.7%

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

      9. *-commutative 39.7%

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

      10. associate-/l/ 39.6%

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

      11. sqrt-prod 34.7%

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

      12. associate-/l/ 34.8%

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

      13. *-commutative 34.8%

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

      14. div-inv 34.8%

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

      15. add-sqr-sqrt 34.8%

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

      16. rem-sqrt-square 34.8%

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

      17. sqrt-div 39.8%

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

      18. unpow2 39.8%

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

      19. sqrt-prod 0.0%

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

      20. add-sqr-sqrt 55.1%

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

   8. Applied egg-rr 55.1%

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

   if -2.60000000000000016e118 < l < 2.2e133

   1. Initial program 73.0%

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

   3. Simplified 71.4%

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

      1. associate-*r/ 73.0%

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

      2. clear-num 73.0%

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

      3. div-inv 73.0%

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

      4. associate-*l* 73.0%

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

      5. metadata-eval 73.0%

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

   5. Applied egg-rr 73.0%

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

      1. expm1-log1p-u 31.1%

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

      2. expm1-udef 31.1%

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

      3. inv-pow 31.1%

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

      4. pow-pow 31.1%

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

      5. associate-/r* 30.6%

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

      6. metadata-eval 30.6%

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

   7. Applied egg-rr 30.6%

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

      1. expm1-def 30.6%

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

      2. expm1-log1p 70.3%

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

      3. *-commutative 70.3%

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

      4. associate-/r* 73.0%

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

      5. *-commutative 73.0%

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

   9. Simplified 73.0%

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

   if 2.2e133 < l

   1. Initial program 56.5%

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

   2. Taylor expanded in d around inf 65.6%

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

      1. expm1-log1p-u 65.2%

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

      2. expm1-udef 25.8%

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

      3. pow1/2 25.8%

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

      4. inv-pow 25.8%

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

      5. pow-pow 25.8%

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

      6. metadata-eval 25.8%

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

   4. Applied egg-rr 25.8%

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

      1. expm1-def 65.3%

         \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \]

      2. expm1-log1p 65.7%

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

   6. Simplified 65.7%

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

      1. add-sqr-sqrt 65.5%

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

      2. sqrt-unprod 65.7%

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

      3. pow-prod-up 65.6%

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

      4. metadata-eval 65.6%

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

      5. inv-pow 65.6%

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

      6. *-commutative 65.6%

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

      7. associate-/l/ 67.0%

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

      8. rem-cbrt-cube 34.7%

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

      9. unpow1/3 33.5%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.3333333333333333}}} \]

      10. expm1-log1p-u 32.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.3333333333333333}}\right)\right)} \]

      11. expm1-udef 20.8%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.3333333333333333}}\right)} - 1} \]

   8. Applied egg-rr 35.1%

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

      1. expm1-def 63.1%

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

      2. expm1-log1p 65.8%

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

   10. Simplified 65.8%

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

       1. *-commutative 65.8%

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

       2. sqrt-prod 81.0%

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

   12. Applied egg-rr 81.0%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.6 \cdot 10^{+118}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{+133}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{d}{M \cdot \left(0.5 \cdot D\right)}\right)}^{-2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 7: 69.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5.5 \cdot 10^{+132}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \frac{-0.5 \cdot \left(h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)}{\ell}\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l 5.5e+132)
   (*
    (sqrt (/ d l))
    (*
     (+ 1.0 (/ (* -0.5 (* h (pow (* M (* 0.5 (/ D d))) 2.0))) l))
     (sqrt (/ d h))))
   (/ d (* (sqrt h) (sqrt l)))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 5.5e+132) {
		tmp = sqrt((d / l)) * ((1.0 + ((-0.5 * (h * pow((M * (0.5 * (D / d))), 2.0))) / l)) * sqrt((d / h)));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= 5.5d+132) then
        tmp = sqrt((d / l)) * ((1.0d0 + (((-0.5d0) * (h * ((m * (0.5d0 * (d_1 / d))) ** 2.0d0))) / l)) * sqrt((d / h)))
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 5.5e+132) {
		tmp = Math.sqrt((d / l)) * ((1.0 + ((-0.5 * (h * Math.pow((M * (0.5 * (D / d))), 2.0))) / l)) * Math.sqrt((d / h)));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= 5.5e+132:
		tmp = math.sqrt((d / l)) * ((1.0 + ((-0.5 * (h * math.pow((M * (0.5 * (D / d))), 2.0))) / l)) * math.sqrt((d / h)))
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 5.5e+132)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(Float64(1.0 + Float64(Float64(-0.5 * Float64(h * (Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0))) / l)) * sqrt(Float64(d / h))));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= 5.5e+132)
		tmp = sqrt((d / l)) * ((1.0 + ((-0.5 * (h * ((M * (0.5 * (D / d))) ^ 2.0))) / l)) * sqrt((d / h)));
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, 5.5e+132], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 + N[(N[(-0.5 * N[(h * N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.5e132

   1. Initial program 67.8%

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

   3. Simplified 66.5%

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

      1. associate-*r* 66.5%

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

      2. *-commutative 66.5%

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

      3. associate-*r/ 70.3%

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

      4. associate-*l/ 70.3%

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

      5. div-inv 70.3%

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

      6. associate-*l* 70.3%

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

      7. metadata-eval 70.3%

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

   5. Applied egg-rr 70.3%

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

   if 5.5e132 < l

   1. Initial program 56.5%

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

   2. Taylor expanded in d around inf 65.6%

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

      1. expm1-log1p-u 65.2%

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

      2. expm1-udef 25.8%

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

      3. pow1/2 25.8%

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

      4. inv-pow 25.8%

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

      5. pow-pow 25.8%

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

      6. metadata-eval 25.8%

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

   4. Applied egg-rr 25.8%

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

      1. expm1-def 65.3%

         \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \]

      2. expm1-log1p 65.7%

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

   6. Simplified 65.7%

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

      1. add-sqr-sqrt 65.5%

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

      2. sqrt-unprod 65.7%

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

      3. pow-prod-up 65.6%

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

      4. metadata-eval 65.6%

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

      5. inv-pow 65.6%

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

      6. *-commutative 65.6%

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

      7. associate-/l/ 67.0%

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

      8. rem-cbrt-cube 34.7%

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

      9. unpow1/3 33.5%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.3333333333333333}}} \]

      10. expm1-log1p-u 32.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.3333333333333333}}\right)\right)} \]

      11. expm1-udef 20.8%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.3333333333333333}}\right)} - 1} \]

   8. Applied egg-rr 35.1%

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

      1. expm1-def 63.1%

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

      2. expm1-log1p 65.8%

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

   10. Simplified 65.8%

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

       1. *-commutative 65.8%

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

       2. sqrt-prod 81.0%

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

   12. Applied egg-rr 81.0%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.5 \cdot 10^{+132}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \frac{-0.5 \cdot \left(h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)}{\ell}\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 8: 49.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := -0.125 \cdot \left(\sqrt{h \cdot {\ell}^{-3}} \cdot \frac{{\left(M \cdot D\right)}^{2}}{d}\right)\\ t_1 := \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;d \leq -1.25 \cdot 10^{+60}:\\ \;\;\;\;\left|t_1\right|\\ \mathbf{elif}\;d \leq -1.9 \cdot 10^{-292}:\\ \;\;\;\;t_1 \cdot \mathsf{fma}\left({\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}, \frac{0.5}{\frac{\ell}{h}}, 1\right)\\ \mathbf{elif}\;d \leq 10^{-86}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 3 \cdot 10^{-24}:\\ \;\;\;\;{\ell}^{-0.5} \cdot \left(d \cdot {h}^{-0.5}\right)\\ \mathbf{elif}\;d \leq 2.85 \cdot 10^{+137}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* -0.125 (* (sqrt (* h (pow l -3.0))) (/ (pow (* M D) 2.0) d))))
        (t_1 (/ d (sqrt (* l h)))))
   (if (<= d -1.25e+60)
     (fabs t_1)
     (if (<= d -1.9e-292)
       (* t_1 (fma (pow (* (/ D d) (* M 0.5)) 2.0) (/ 0.5 (/ l h)) 1.0))
       (if (<= d 1e-86)
         t_0
         (if (<= d 3e-24)
           (* (pow l -0.5) (* d (pow h -0.5)))
           (if (<= d 2.85e+137) t_0 (* d (* (pow l -0.5) (pow h -0.5))))))))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = -0.125 * (sqrt((h * pow(l, -3.0))) * (pow((M * D), 2.0) / d));
	double t_1 = d / sqrt((l * h));
	double tmp;
	if (d <= -1.25e+60) {
		tmp = fabs(t_1);
	} else if (d <= -1.9e-292) {
		tmp = t_1 * fma(pow(((D / d) * (M * 0.5)), 2.0), (0.5 / (l / h)), 1.0);
	} else if (d <= 1e-86) {
		tmp = t_0;
	} else if (d <= 3e-24) {
		tmp = pow(l, -0.5) * (d * pow(h, -0.5));
	} else if (d <= 2.85e+137) {
		tmp = t_0;
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(-0.125 * Float64(sqrt(Float64(h * (l ^ -3.0))) * Float64((Float64(M * D) ^ 2.0) / d)))
	t_1 = Float64(d / sqrt(Float64(l * h)))
	tmp = 0.0
	if (d <= -1.25e+60)
		tmp = abs(t_1);
	elseif (d <= -1.9e-292)
		tmp = Float64(t_1 * fma((Float64(Float64(D / d) * Float64(M * 0.5)) ^ 2.0), Float64(0.5 / Float64(l / h)), 1.0));
	elseif (d <= 1e-86)
		tmp = t_0;
	elseif (d <= 3e-24)
		tmp = Float64((l ^ -0.5) * Float64(d * (h ^ -0.5)));
	elseif (d <= 2.85e+137)
		tmp = t_0;
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(-0.125 * N[(N[Sqrt[N[(h * N[Power[l, -3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(M * D), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.25e+60], N[Abs[t$95$1], $MachinePrecision], If[LessEqual[d, -1.9e-292], N[(t$95$1 * N[(N[Power[N[(N[(D / d), $MachinePrecision] * N[(M * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 / N[(l / h), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1e-86], t$95$0, If[LessEqual[d, 3e-24], N[(N[Power[l, -0.5], $MachinePrecision] * N[(d * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.85e+137], t$95$0, N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -1.24999999999999994e60

   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. Taylor expanded in d around inf 5.5%

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

      1. expm1-log1p-u 5.5%

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

      2. expm1-udef 5.6%

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

      3. pow1/2 5.6%

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

      4. inv-pow 5.6%

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

      5. pow-pow 5.6%

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

      6. metadata-eval 5.6%

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

   4. Applied egg-rr 5.6%

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

      1. expm1-def 5.5%

         \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \]

      2. expm1-log1p 5.5%

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

   6. Simplified 5.5%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-prod 42.6%

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

      3. unpow2 42.6%

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

      4. add-sqr-sqrt 42.5%

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

      5. sqrt-unprod 42.6%

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

      6. pow-prod-up 42.6%

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

      7. metadata-eval 42.6%

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

      8. inv-pow 42.6%

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

      9. *-commutative 42.6%

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

      10. associate-/l/ 42.6%

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

      11. sqrt-prod 32.4%

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

      12. associate-/l/ 32.4%

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

      13. *-commutative 32.4%

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

      14. div-inv 32.4%

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

      15. add-sqr-sqrt 32.4%

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

      16. rem-sqrt-square 32.4%

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

      17. sqrt-div 42.6%

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

      18. unpow2 42.6%

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

      19. sqrt-prod 0.0%

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

      20. add-sqr-sqrt 65.6%

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

   8. Applied egg-rr 65.6%

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

   if -1.24999999999999994e60 < d < -1.9000000000000001e-292

   1. Initial program 64.2%

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

   3. Applied egg-rr 15.4%

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

      1. expm1-log1p-u 15.2%

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

      2. expm1-udef 6.9%

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

   5. Applied egg-rr 4.0%

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

      1. expm1-def 7.8%

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

      2. expm1-log1p 46.3%

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

      3. fma-udef 46.3%

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

      4. fma-udef 46.3%

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

      5. associate-/l* 46.3%

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

   7. Simplified 46.3%

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

   if -1.9000000000000001e-292 < d < 1.00000000000000008e-86 or 2.99999999999999995e-24 < d < 2.8499999999999999e137

   1. Initial program 60.5%

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

   2. Taylor expanded in d around 0 36.3%

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

      1. *-commutative 36.3%

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

      2. associate-*l* 36.3%

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

      3. associate-/l* 33.8%

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

      4. associate-/r/ 36.1%

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

   4. Simplified 36.1%

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

      1. expm1-log1p-u 4.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{{D}^{2}}{d} \cdot {M}^{2}\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)\right)\right)} \]

      2. expm1-udef 4.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\frac{{D}^{2}}{d} \cdot {M}^{2}\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)\right)} - 1} \]

      3. associate-*r* 4.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\left(\frac{{D}^{2}}{d} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot -0.125}\right)} - 1 \]

      4. associate-*l/ 4.7%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{d}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot -0.125\right)} - 1 \]

      5. pow-prod-down 4.9%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\frac{\color{blue}{{\left(D \cdot M\right)}^{2}}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot -0.125\right)} - 1 \]

      6. div-inv 4.9%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \sqrt{\color{blue}{h \cdot \frac{1}{{\ell}^{3}}}}\right) \cdot -0.125\right)} - 1 \]

      7. pow-flip 4.8%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \sqrt{h \cdot \color{blue}{{\ell}^{\left(-3\right)}}}\right) \cdot -0.125\right)} - 1 \]

      8. metadata-eval 4.8%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \sqrt{h \cdot {\ell}^{\color{blue}{-3}}}\right) \cdot -0.125\right)} - 1 \]

   6. Applied egg-rr 4.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \sqrt{h \cdot {\ell}^{-3}}\right) \cdot -0.125\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 4.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \sqrt{h \cdot {\ell}^{-3}}\right) \cdot -0.125\right)\right)} \]

      2. expm1-log1p 50.6%

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

      3. *-commutative 50.6%

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

      4. *-commutative 50.6%

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

   8. Simplified 50.6%

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

   if 1.00000000000000008e-86 < d < 2.99999999999999995e-24

   1. Initial program 69.8%

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

   2. Taylor expanded in d around inf 57.2%

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

      1. expm1-log1p-u 55.4%

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

      2. expm1-udef 23.8%

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

      3. pow1/2 23.8%

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

      4. inv-pow 23.8%

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

      5. pow-pow 23.8%

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

      6. metadata-eval 23.8%

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

   4. Applied egg-rr 23.8%

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

      1. expm1-def 55.4%

         \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \]

      2. expm1-log1p 57.2%

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

   6. Simplified 57.2%

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

      1. add-sqr-sqrt 56.9%

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

      2. sqrt-unprod 57.2%

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

      3. pow-prod-up 57.2%

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

      4. metadata-eval 57.2%

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

      5. inv-pow 57.2%

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

      6. *-commutative 57.2%

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

      7. associate-/l/ 57.2%

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

      8. rem-cbrt-cube 25.4%

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

      9. unpow1/3 23.9%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.3333333333333333}}} \]

      10. expm1-log1p-u 23.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.3333333333333333}}\right)\right)} \]

      11. expm1-udef 4.2%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.3333333333333333}}\right)} - 1} \]

   8. Applied egg-rr 24.4%

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

      1. expm1-def 56.1%

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

      2. expm1-log1p 57.4%

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

   10. Simplified 57.4%

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

       1. clear-num 57.1%

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

       2. associate-/r/ 57.2%

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

       3. pow1/2 57.2%

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

       4. pow-flip 57.2%

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

       5. *-commutative 57.2%

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

       6. metadata-eval 57.2%

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

       7. pow-prod-down 77.6%

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

       8. associate-*l* 77.7%

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

   12. Applied egg-rr 77.7%

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

   if 2.8499999999999999e137 < d

   1. Initial program 78.7%

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

   2. Taylor expanded in d around inf 68.5%

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

      1. expm1-log1p-u 67.7%

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

      2. expm1-udef 44.8%

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

      3. pow1/2 44.8%

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

      4. inv-pow 44.8%

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

      5. pow-pow 44.8%

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

      6. metadata-eval 44.8%

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

   4. Applied egg-rr 44.8%

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

      1. expm1-def 67.7%

         \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \]

      2. expm1-log1p 68.5%

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

   6. Simplified 68.5%

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

      1. *-commutative 68.5%

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

      2. unpow-prod-down 77.1%

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

   8. Applied egg-rr 77.1%

      \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.25 \cdot 10^{+60}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;d \leq -1.9 \cdot 10^{-292}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left({\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}, \frac{0.5}{\frac{\ell}{h}}, 1\right)\\ \mathbf{elif}\;d \leq 10^{-86}:\\ \;\;\;\;-0.125 \cdot \left(\sqrt{h \cdot {\ell}^{-3}} \cdot \frac{{\left(M \cdot D\right)}^{2}}{d}\right)\\ \mathbf{elif}\;d \leq 3 \cdot 10^{-24}:\\ \;\;\;\;{\ell}^{-0.5} \cdot \left(d \cdot {h}^{-0.5}\right)\\ \mathbf{elif}\;d \leq 2.85 \cdot 10^{+137}:\\ \;\;\;\;-0.125 \cdot \left(\sqrt{h \cdot {\ell}^{-3}} \cdot \frac{{\left(M \cdot D\right)}^{2}}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 9: 44.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 5.9 \cdot 10^{-28}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;M \leq 1700000:\\ \;\;\;\;d \cdot \mathsf{log1p}\left(\mathsf{expm1}\left({\left(\ell \cdot h\right)}^{-0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\sqrt{h \cdot {\ell}^{-3}} \cdot \frac{{\left(M \cdot D\right)}^{2}}{d}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= M 5.9e-28)
   (fabs (/ d (sqrt (* l h))))
   (if (<= M 1700000.0)
     (* d (log1p (expm1 (pow (* l h) -0.5))))
     (* -0.125 (* (sqrt (* h (pow l -3.0))) (/ (pow (* M D) 2.0) d))))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (M <= 5.9e-28) {
		tmp = fabs((d / sqrt((l * h))));
	} else if (M <= 1700000.0) {
		tmp = d * log1p(expm1(pow((l * h), -0.5)));
	} else {
		tmp = -0.125 * (sqrt((h * pow(l, -3.0))) * (pow((M * D), 2.0) / d));
	}
	return tmp;
}

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (M <= 5.9e-28) {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	} else if (M <= 1700000.0) {
		tmp = d * Math.log1p(Math.expm1(Math.pow((l * h), -0.5)));
	} else {
		tmp = -0.125 * (Math.sqrt((h * Math.pow(l, -3.0))) * (Math.pow((M * D), 2.0) / d));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if M <= 5.9e-28:
		tmp = math.fabs((d / math.sqrt((l * h))))
	elif M <= 1700000.0:
		tmp = d * math.log1p(math.expm1(math.pow((l * h), -0.5)))
	else:
		tmp = -0.125 * (math.sqrt((h * math.pow(l, -3.0))) * (math.pow((M * D), 2.0) / d))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (M <= 5.9e-28)
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	elseif (M <= 1700000.0)
		tmp = Float64(d * log1p(expm1((Float64(l * h) ^ -0.5))));
	else
		tmp = Float64(-0.125 * Float64(sqrt(Float64(h * (l ^ -3.0))) * Float64((Float64(M * D) ^ 2.0) / d)));
	end
	return tmp
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[M, 5.9e-28], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[M, 1700000.0], N[(d * N[Log[1 + N[(Exp[N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-0.125 * N[(N[Sqrt[N[(h * N[Power[l, -3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(M * D), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if M < 5.9000000000000002e-28

   1. Initial program 66.7%

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

   2. Taylor expanded in d around inf 30.9%

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

      1. expm1-log1p-u 30.4%

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

      2. expm1-udef 20.7%

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

      3. pow1/2 20.7%

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

      4. inv-pow 20.7%

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

      5. pow-pow 20.6%

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

      6. metadata-eval 20.6%

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

   4. Applied egg-rr 20.6%

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

      1. expm1-def 30.3%

         \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \]

      2. expm1-log1p 30.8%

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

   6. Simplified 30.8%

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

      1. add-sqr-sqrt 26.4%

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

      2. sqrt-prod 32.0%

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

      3. unpow2 32.0%

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

      4. add-sqr-sqrt 31.8%

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

      5. sqrt-unprod 31.7%

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

      6. pow-prod-up 31.7%

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

      7. metadata-eval 31.7%

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

      8. inv-pow 31.7%

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

      9. *-commutative 31.7%

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

      10. associate-/l/ 31.7%

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

      11. sqrt-prod 25.3%

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

      12. associate-/l/ 25.3%

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

      13. *-commutative 25.3%

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

      14. div-inv 25.6%

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

      15. add-sqr-sqrt 25.6%

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

      16. rem-sqrt-square 25.6%

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

      17. sqrt-div 32.0%

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

      18. unpow2 32.0%

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

      19. sqrt-prod 26.4%

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

      20. add-sqr-sqrt 51.5%

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

   8. Applied egg-rr 51.5%

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

   if 5.9000000000000002e-28 < M < 1.7e6

   1. Initial program 74.5%

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

   2. Taylor expanded in d around inf 26.3%

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

      1. expm1-log1p-u 26.3%

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

      2. expm1-udef 3.0%

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

      3. pow1/2 3.0%

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

      4. inv-pow 3.0%

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

      5. pow-pow 3.0%

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

      6. metadata-eval 3.0%

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

   4. Applied egg-rr 3.0%

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

      1. expm1-def 26.3%

         \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \]

      2. expm1-log1p 26.3%

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

   6. Simplified 26.3%

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

      1. log1p-expm1-u 38.3%

         \[\leadsto d \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \]

   8. Applied egg-rr 38.3%

      \[\leadsto d \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \]

   if 1.7e6 < M

   1. Initial program 63.4%

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

   2. Taylor expanded in d around 0 26.5%

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

      1. *-commutative 26.5%

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

      2. associate-*l* 26.5%

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

      3. associate-/l* 24.8%

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

      4. associate-/r/ 26.3%

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

   4. Simplified 26.3%

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

      1. expm1-log1p-u 3.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{{D}^{2}}{d} \cdot {M}^{2}\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)\right)\right)} \]

      2. expm1-udef 3.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\frac{{D}^{2}}{d} \cdot {M}^{2}\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)\right)} - 1} \]

      3. associate-*r* 3.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\left(\frac{{D}^{2}}{d} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot -0.125}\right)} - 1 \]

      4. associate-*l/ 3.8%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{d}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot -0.125\right)} - 1 \]

      5. pow-prod-down 3.9%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\frac{\color{blue}{{\left(D \cdot M\right)}^{2}}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot -0.125\right)} - 1 \]

      6. div-inv 3.9%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \sqrt{\color{blue}{h \cdot \frac{1}{{\ell}^{3}}}}\right) \cdot -0.125\right)} - 1 \]

      7. pow-flip 3.8%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \sqrt{h \cdot \color{blue}{{\ell}^{\left(-3\right)}}}\right) \cdot -0.125\right)} - 1 \]

      8. metadata-eval 3.8%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \sqrt{h \cdot {\ell}^{\color{blue}{-3}}}\right) \cdot -0.125\right)} - 1 \]

   6. Applied egg-rr 3.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \sqrt{h \cdot {\ell}^{-3}}\right) \cdot -0.125\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 3.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \sqrt{h \cdot {\ell}^{-3}}\right) \cdot -0.125\right)\right)} \]

      2. expm1-log1p 30.3%

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

      3. *-commutative 30.3%

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

      4. *-commutative 30.3%

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

   8. Simplified 30.3%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 5.9 \cdot 10^{-28}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;M \leq 1700000:\\ \;\;\;\;d \cdot \mathsf{log1p}\left(\mathsf{expm1}\left({\left(\ell \cdot h\right)}^{-0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\sqrt{h \cdot {\ell}^{-3}} \cdot \frac{{\left(M \cdot D\right)}^{2}}{d}\right)\\ \end{array} \]

Alternative 10: 47.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.6 \cdot 10^{-94}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\log \left(e^{\sqrt{\ell \cdot h}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l -6.6e-94)
   (* (- d) (pow (* l h) -0.5))
   (if (<= l -2e-310)
     (/ d (log (exp (sqrt (* l h)))))
     (/ d (* (sqrt h) (sqrt l))))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -6.6e-94) {
		tmp = -d * pow((l * h), -0.5);
	} else if (l <= -2e-310) {
		tmp = d / log(exp(sqrt((l * h))));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-6.6d-94)) then
        tmp = -d * ((l * h) ** (-0.5d0))
    else if (l <= (-2d-310)) then
        tmp = d / log(exp(sqrt((l * h))))
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -6.6e-94) {
		tmp = -d * Math.pow((l * h), -0.5);
	} else if (l <= -2e-310) {
		tmp = d / Math.log(Math.exp(Math.sqrt((l * h))));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= -6.6e-94:
		tmp = -d * math.pow((l * h), -0.5)
	elif l <= -2e-310:
		tmp = d / math.log(math.exp(math.sqrt((l * h))))
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -6.6e-94)
		tmp = Float64(Float64(-d) * (Float64(l * h) ^ -0.5));
	elseif (l <= -2e-310)
		tmp = Float64(d / log(exp(sqrt(Float64(l * h)))));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -6.6e-94)
		tmp = -d * ((l * h) ^ -0.5);
	elseif (l <= -2e-310)
		tmp = d / log(exp(sqrt((l * h))));
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, -6.6e-94], N[((-d) * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-310], N[(d / N[Log[N[Exp[N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.6000000000000003e-94

   1. Initial program 58.6%

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

   3. Applied egg-rr 24.3%

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

   4. Taylor expanded in d around -inf 52.1%

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

      1. mul-1-neg 52.1%

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

      2. unpow-1 52.1%

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

      3. metadata-eval 52.1%

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

      4. pow-sqr 52.1%

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

      5. rem-sqrt-square 53.1%

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

      6. metadata-eval 53.1%

         \[\leadsto -d \cdot \left|{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.25\right)}}\right| \]

      7. pow-sqr 52.8%

         \[\leadsto -d \cdot \left|\color{blue}{{\left(h \cdot \ell\right)}^{-0.25} \cdot {\left(h \cdot \ell\right)}^{-0.25}}\right| \]

      8. fabs-sqr 52.8%

         \[\leadsto -d \cdot \color{blue}{\left({\left(h \cdot \ell\right)}^{-0.25} \cdot {\left(h \cdot \ell\right)}^{-0.25}\right)} \]

      9. pow-sqr 53.1%

         \[\leadsto -d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(2 \cdot -0.25\right)}} \]

      10. metadata-eval 53.1%

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

      11. distribute-rgt-neg-in 53.1%

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

   6. Simplified 53.1%

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

   if -6.6000000000000003e-94 < l < -1.999999999999994e-310

   1. Initial program 72.1%

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

   2. Taylor expanded in d around inf 22.0%

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

      1. expm1-log1p-u 22.0%

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

      2. expm1-udef 21.9%

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

      3. pow1/2 21.9%

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

      4. inv-pow 21.9%

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

      5. pow-pow 17.7%

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

      6. metadata-eval 17.7%

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

   4. Applied egg-rr 17.7%

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

      1. expm1-def 17.8%

         \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \]

      2. expm1-log1p 17.8%

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

   6. Simplified 17.8%

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

      1. add-sqr-sqrt 17.8%

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

      2. sqrt-unprod 22.0%

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

      3. pow-prod-up 22.0%

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

      4. metadata-eval 22.0%

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

      5. inv-pow 22.0%

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

      6. *-commutative 22.0%

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

      7. associate-/l/ 22.0%

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

      8. rem-cbrt-cube 30.1%

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

      9. unpow1/3 30.1%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.3333333333333333}}} \]

      10. expm1-log1p-u 0.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.3333333333333333}}\right)\right)} \]

      11. expm1-udef 0.2%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.3333333333333333}}\right)} - 1} \]

   8. Applied egg-rr 0.4%

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

      1. expm1-def 0.6%

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

      2. expm1-log1p 17.8%

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

   10. Simplified 17.8%

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

       1. add-log-exp 42.6%

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

   12. Applied egg-rr 42.6%

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

   if -1.999999999999994e-310 < l

   1. Initial program 68.7%

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

   2. Taylor expanded in d around inf 44.4%

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

      1. expm1-log1p-u 43.5%

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

      2. expm1-udef 28.1%

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

      3. pow1/2 28.1%

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

      4. inv-pow 28.1%

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

      5. pow-pow 28.7%

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

      6. metadata-eval 28.7%

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

   4. Applied egg-rr 28.7%

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

      1. expm1-def 44.1%

         \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \]

      2. expm1-log1p 45.0%

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

   6. Simplified 45.0%

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

      1. add-sqr-sqrt 44.9%

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

      2. sqrt-unprod 44.5%

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

      3. pow-prod-up 44.4%

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

      4. metadata-eval 44.4%

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

      5. inv-pow 44.4%

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

      6. *-commutative 44.4%

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

      7. associate-/l/ 44.8%

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

      8. rem-cbrt-cube 27.6%

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

      9. unpow1/3 26.8%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.3333333333333333}}} \]

      10. expm1-log1p-u 26.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.3333333333333333}}\right)\right)} \]

      11. expm1-udef 21.0%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.3333333333333333}}\right)} - 1} \]

   8. Applied egg-rr 31.7%

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

      1. expm1-def 43.1%

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

      2. expm1-log1p 45.1%

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

   10. Simplified 45.1%

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

       1. *-commutative 45.1%

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

       2. sqrt-prod 50.9%

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

   12. Applied egg-rr 50.9%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.6 \cdot 10^{-94}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\log \left(e^{\sqrt{\ell \cdot h}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 11: 42.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 7.2 \cdot 10^{-26}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;d \cdot \log \left(e^{{\left(\ell \cdot h\right)}^{-0.5}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= M 7.2e-26)
   (fabs (/ d (sqrt (* l h))))
   (* d (log (exp (pow (* l h) -0.5))))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (M <= 7.2e-26) {
		tmp = fabs((d / sqrt((l * h))));
	} else {
		tmp = d * log(exp(pow((l * h), -0.5)));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m <= 7.2d-26) then
        tmp = abs((d / sqrt((l * h))))
    else
        tmp = d * log(exp(((l * h) ** (-0.5d0))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (M <= 7.2e-26) {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	} else {
		tmp = d * Math.log(Math.exp(Math.pow((l * h), -0.5)));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if M <= 7.2e-26:
		tmp = math.fabs((d / math.sqrt((l * h))))
	else:
		tmp = d * math.log(math.exp(math.pow((l * h), -0.5)))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (M <= 7.2e-26)
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	else
		tmp = Float64(d * log(exp((Float64(l * h) ^ -0.5))));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (M <= 7.2e-26)
		tmp = abs((d / sqrt((l * h))));
	else
		tmp = d * log(exp(((l * h) ^ -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[M, 7.2e-26], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(d * N[Log[N[Exp[N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 7.2000000000000003e-26

   1. Initial program 66.3%

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

   2. Taylor expanded in d around inf 31.3%

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

      1. expm1-log1p-u 30.7%

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

      2. expm1-udef 20.6%

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

      3. pow1/2 20.6%

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

      4. inv-pow 20.6%

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

      5. pow-pow 20.5%

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

      6. metadata-eval 20.5%

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

   4. Applied egg-rr 20.5%

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

      1. expm1-def 30.7%

         \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \]

      2. expm1-log1p 31.2%

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

   6. Simplified 31.2%

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

      1. add-sqr-sqrt 26.8%

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

      2. sqrt-prod 32.3%

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

      3. unpow2 32.3%

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

      4. add-sqr-sqrt 32.2%

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

      5. sqrt-unprod 32.1%

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

      6. pow-prod-up 32.0%

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

      7. metadata-eval 32.0%

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

      8. inv-pow 32.0%

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

      9. *-commutative 32.0%

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

      10. associate-/l/ 32.0%

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

      11. sqrt-prod 25.7%

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

      12. associate-/l/ 25.7%

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

      13. *-commutative 25.7%

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

      14. div-inv 26.0%

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

      15. add-sqr-sqrt 26.0%

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

      16. rem-sqrt-square 26.0%

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

      17. sqrt-div 32.4%

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

      18. unpow2 32.4%

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

      19. sqrt-prod 26.8%

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

      20. add-sqr-sqrt 51.8%

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

   8. Applied egg-rr 51.8%

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

   if 7.2000000000000003e-26 < M

   1. Initial program 65.6%

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

   2. Taylor expanded in d around inf 19.2%

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

      1. add-log-exp 25.9%

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

      2. pow1/2 25.9%

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

      3. inv-pow 25.9%

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

      4. pow-pow 25.9%

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

      5. metadata-eval 25.9%

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

   4. Applied egg-rr 25.9%

      \[\leadsto d \cdot \color{blue}{\log \left(e^{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 7.2 \cdot 10^{-26}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;d \cdot \log \left(e^{{\left(\ell \cdot h\right)}^{-0.5}}\right)\\ \end{array} \]

Alternative 12: 43.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 2.5 \cdot 10^{-31}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;d \cdot \mathsf{log1p}\left(\mathsf{expm1}\left({\left(\ell \cdot h\right)}^{-0.5}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= M 2.5e-31)
   (fabs (/ d (sqrt (* l h))))
   (* d (log1p (expm1 (pow (* l h) -0.5))))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (M <= 2.5e-31) {
		tmp = fabs((d / sqrt((l * h))));
	} else {
		tmp = d * log1p(expm1(pow((l * h), -0.5)));
	}
	return tmp;
}

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (M <= 2.5e-31) {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	} else {
		tmp = d * Math.log1p(Math.expm1(Math.pow((l * h), -0.5)));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if M <= 2.5e-31:
		tmp = math.fabs((d / math.sqrt((l * h))))
	else:
		tmp = d * math.log1p(math.expm1(math.pow((l * h), -0.5)))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (M <= 2.5e-31)
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	else
		tmp = Float64(d * log1p(expm1((Float64(l * h) ^ -0.5))));
	end
	return tmp
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[M, 2.5e-31], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(d * N[Log[1 + N[(Exp[N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.5e-31

   1. Initial program 66.5%

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

   2. Taylor expanded in d around inf 31.0%

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

      1. expm1-log1p-u 30.5%

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

      2. expm1-udef 20.8%

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

      3. pow1/2 20.8%

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

      4. inv-pow 20.8%

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

      5. pow-pow 20.7%

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

      6. metadata-eval 20.7%

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

   4. Applied egg-rr 20.7%

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

      1. expm1-def 30.4%

         \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \]

      2. expm1-log1p 31.0%

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

   6. Simplified 31.0%

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

      1. add-sqr-sqrt 26.5%

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

      2. sqrt-prod 32.2%

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

      3. unpow2 32.2%

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

      4. add-sqr-sqrt 32.0%

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

      5. sqrt-unprod 31.9%

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

      6. pow-prod-up 31.8%

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

      7. metadata-eval 31.8%

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

      8. inv-pow 31.8%

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

      9. *-commutative 31.8%

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

      10. associate-/l/ 31.8%

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

      11. sqrt-prod 25.5%

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

      12. associate-/l/ 25.5%

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

      13. *-commutative 25.5%

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

      14. div-inv 25.7%

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

      15. add-sqr-sqrt 25.7%

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

      16. rem-sqrt-square 25.7%

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

      17. sqrt-div 32.2%

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

      18. unpow2 32.2%

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

      19. sqrt-prod 26.5%

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

      20. add-sqr-sqrt 51.8%

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

   8. Applied egg-rr 51.8%

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

   if 2.5e-31 < M

   1. Initial program 65.2%

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

   2. Taylor expanded in d around inf 20.1%

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

      1. expm1-log1p-u 19.8%

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

      2. expm1-udef 17.0%

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

      3. pow1/2 17.0%

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

      4. inv-pow 17.0%

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

      5. pow-pow 15.7%

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

      6. metadata-eval 15.7%

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

   4. Applied egg-rr 15.7%

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

      1. expm1-def 18.4%

         \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \]

      2. expm1-log1p 18.8%

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

   6. Simplified 18.8%

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

      1. log1p-expm1-u 28.1%

         \[\leadsto d \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \]

   8. Applied egg-rr 28.1%

      \[\leadsto d \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.5 \cdot 10^{-31}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;d \cdot \mathsf{log1p}\left(\mathsf{expm1}\left({\left(\ell \cdot h\right)}^{-0.5}\right)\right)\\ \end{array} \]

Alternative 13: 42.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;M \leq 4.1 \cdot 10^{-23}:\\ \;\;\;\;\left|t_0\right|\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{t_0}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ d (sqrt (* l h)))))
   (if (<= M 4.1e-23) (fabs t_0) (log (exp t_0)))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = d / sqrt((l * h));
	double tmp;
	if (M <= 4.1e-23) {
		tmp = fabs(t_0);
	} else {
		tmp = log(exp(t_0));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d / sqrt((l * h))
    if (m <= 4.1d-23) then
        tmp = abs(t_0)
    else
        tmp = log(exp(t_0))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = d / Math.sqrt((l * h));
	double tmp;
	if (M <= 4.1e-23) {
		tmp = Math.abs(t_0);
	} else {
		tmp = Math.log(Math.exp(t_0));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = d / math.sqrt((l * h))
	tmp = 0
	if M <= 4.1e-23:
		tmp = math.fabs(t_0)
	else:
		tmp = math.log(math.exp(t_0))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(d / sqrt(Float64(l * h)))
	tmp = 0.0
	if (M <= 4.1e-23)
		tmp = abs(t_0);
	else
		tmp = log(exp(t_0));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = d / sqrt((l * h));
	tmp = 0.0;
	if (M <= 4.1e-23)
		tmp = abs(t_0);
	else
		tmp = log(exp(t_0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, 4.1e-23], N[Abs[t$95$0], $MachinePrecision], N[Log[N[Exp[t$95$0], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < 4.10000000000000029e-23

   1. Initial program 66.7%

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

   2. Taylor expanded in d around inf 31.5%

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

      1. expm1-log1p-u 30.9%

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

      2. expm1-udef 20.4%

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

      3. pow1/2 20.4%

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

      4. inv-pow 20.4%

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

      5. pow-pow 20.3%

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

      6. metadata-eval 20.3%

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

   4. Applied egg-rr 20.3%

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

      1. expm1-def 30.9%

         \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \]

      2. expm1-log1p 31.4%

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

   6. Simplified 31.4%

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

      1. add-sqr-sqrt 27.0%

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

      2. sqrt-prod 32.5%

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

      3. unpow2 32.5%

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

      4. add-sqr-sqrt 32.4%

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

      5. sqrt-unprod 32.2%

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

      6. pow-prod-up 32.2%

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

      7. metadata-eval 32.2%

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

      8. inv-pow 32.2%

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

      9. *-commutative 32.2%

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

      10. associate-/l/ 32.2%

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

      11. sqrt-prod 25.5%

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

      12. associate-/l/ 25.5%

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

      13. *-commutative 25.5%

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

      14. div-inv 25.8%

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

      15. add-sqr-sqrt 25.8%

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

      16. rem-sqrt-square 25.8%

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

      17. sqrt-div 32.5%

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

      18. unpow2 32.5%

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

      19. sqrt-prod 27.1%

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

      20. add-sqr-sqrt 51.7%

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

   8. Applied egg-rr 51.7%

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

   if 4.10000000000000029e-23 < M

   1. Initial program 64.6%

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

   2. Taylor expanded in d around inf 18.3%

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

      1. expm1-log1p-u 18.0%

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

      2. expm1-udef 17.8%

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

      3. pow1/2 17.8%

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

      4. inv-pow 17.8%

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

      5. pow-pow 16.4%

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

      6. metadata-eval 16.4%

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

   4. Applied egg-rr 16.4%

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

      1. expm1-def 16.5%

         \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \]

      2. expm1-log1p 16.9%

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

   6. Simplified 16.9%

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

      1. add-sqr-sqrt 10.7%

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

      2. sqrt-prod 19.1%

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

      3. unpow2 19.1%

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

      4. add-sqr-sqrt 19.0%

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

      5. sqrt-unprod 19.0%

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

      6. pow-prod-up 19.1%

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

      7. metadata-eval 19.1%

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

      8. inv-pow 19.1%

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

      9. *-commutative 19.1%

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

      10. associate-/l/ 19.1%

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

      11. sqrt-prod 16.1%

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

      12. associate-/l/ 16.1%

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

      13. *-commutative 16.1%

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

      14. div-inv 16.1%

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

      15. add-log-exp 7.5%

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

      16. sqrt-div 7.5%

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

      17. unpow2 7.5%

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

      18. sqrt-prod 6.4%

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

      19. add-sqr-sqrt 23.7%

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

   8. Applied egg-rr 23.7%

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

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4.1 \cdot 10^{-23}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{d}{\sqrt{\ell \cdot h}}}\right)\\ \end{array} \]

Alternative 14: 42.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;M \leq 1.42 \cdot 10^{-28}:\\ \;\;\;\;\left|t_0\right|\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(t_0\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ d (sqrt (* l h)))))
   (if (<= M 1.42e-28) (fabs t_0) (log1p (expm1 t_0)))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = d / sqrt((l * h));
	double tmp;
	if (M <= 1.42e-28) {
		tmp = fabs(t_0);
	} else {
		tmp = log1p(expm1(t_0));
	}
	return tmp;
}

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = d / Math.sqrt((l * h));
	double tmp;
	if (M <= 1.42e-28) {
		tmp = Math.abs(t_0);
	} else {
		tmp = Math.log1p(Math.expm1(t_0));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = d / math.sqrt((l * h))
	tmp = 0
	if M <= 1.42e-28:
		tmp = math.fabs(t_0)
	else:
		tmp = math.log1p(math.expm1(t_0))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(d / sqrt(Float64(l * h)))
	tmp = 0.0
	if (M <= 1.42e-28)
		tmp = abs(t_0);
	else
		tmp = log1p(expm1(t_0));
	end
	return tmp
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, 1.42e-28], N[Abs[t$95$0], $MachinePrecision], N[Log[1 + N[(Exp[t$95$0] - 1), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < 1.42000000000000001e-28

   1. Initial program 66.7%

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

   2. Taylor expanded in d around inf 30.9%

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

      1. expm1-log1p-u 30.4%

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

      2. expm1-udef 20.7%

         \[\leadsto d \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{h \cdot \ell}}\right)} - 1\right)} \]

      3. pow1/2 20.7%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right)} - 1\right) \]

      4. inv-pow 20.7%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left({\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right)} - 1\right) \]

      5. pow-pow 20.6%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 0.5\right)}}\right)} - 1\right) \]

      6. metadata-eval 20.6%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right)} - 1\right) \]

   4. Applied egg-rr 20.6%

      \[\leadsto d \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-def 30.3%

         \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \]

      2. expm1-log1p 30.8%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   6. Simplified 30.8%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 26.4%

         \[\leadsto \color{blue}{\left(\sqrt{d} \cdot \sqrt{d}\right)} \cdot {\left(h \cdot \ell\right)}^{-0.5} \]

      2. sqrt-prod 32.0%

         \[\leadsto \color{blue}{\sqrt{d \cdot d}} \cdot {\left(h \cdot \ell\right)}^{-0.5} \]

      3. unpow2 32.0%

         \[\leadsto \sqrt{\color{blue}{{d}^{2}}} \cdot {\left(h \cdot \ell\right)}^{-0.5} \]

      4. add-sqr-sqrt 31.8%

         \[\leadsto \sqrt{{d}^{2}} \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

      5. sqrt-unprod 31.7%

         \[\leadsto \sqrt{{d}^{2}} \cdot \color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

      6. pow-prod-up 31.7%

         \[\leadsto \sqrt{{d}^{2}} \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(-0.5 + -0.5\right)}}} \]

      7. metadata-eval 31.7%

         \[\leadsto \sqrt{{d}^{2}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{-1}}} \]

      8. inv-pow 31.7%

         \[\leadsto \sqrt{{d}^{2}} \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      9. *-commutative 31.7%

         \[\leadsto \sqrt{{d}^{2}} \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      10. associate-/l/ 31.7%

          \[\leadsto \sqrt{{d}^{2}} \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

      11. sqrt-prod 25.3%

          \[\leadsto \color{blue}{\sqrt{{d}^{2} \cdot \frac{\frac{1}{h}}{\ell}}} \]

      12. associate-/l/ 25.3%

          \[\leadsto \sqrt{{d}^{2} \cdot \color{blue}{\frac{1}{\ell \cdot h}}} \]

      13. *-commutative 25.3%

          \[\leadsto \sqrt{{d}^{2} \cdot \frac{1}{\color{blue}{h \cdot \ell}}} \]

      14. div-inv 25.6%

          \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{h \cdot \ell}}} \]

      15. add-sqr-sqrt 25.6%

          \[\leadsto \sqrt{\color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}} \cdot \sqrt{\frac{{d}^{2}}{h \cdot \ell}}}} \]

      16. rem-sqrt-square 25.6%

          \[\leadsto \color{blue}{\left|\sqrt{\frac{{d}^{2}}{h \cdot \ell}}\right|} \]

      17. sqrt-div 32.0%

          \[\leadsto \left|\color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}}}\right| \]

      18. unpow2 32.0%

          \[\leadsto \left|\frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}}\right| \]

      19. sqrt-prod 26.4%

          \[\leadsto \left|\frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}}\right| \]

      20. add-sqr-sqrt 51.5%

          \[\leadsto \left|\frac{\color{blue}{d}}{\sqrt{h \cdot \ell}}\right| \]

   8. Applied egg-rr 51.5%

      \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \]

   if 1.42000000000000001e-28 < M

   1. Initial program 64.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 20.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   3. Step-by-step derivation

      1. expm1-log1p-u 20.0%

         \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \]

      2. expm1-udef 17.2%

         \[\leadsto d \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{h \cdot \ell}}\right)} - 1\right)} \]

      3. pow1/2 17.2%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right)} - 1\right) \]

      4. inv-pow 17.2%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left({\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right)} - 1\right) \]

      5. pow-pow 15.8%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 0.5\right)}}\right)} - 1\right) \]

      6. metadata-eval 15.8%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right)} - 1\right) \]

   4. Applied egg-rr 15.8%

      \[\leadsto d \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-def 18.7%

         \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \]

      2. expm1-log1p 19.0%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   6. Simplified 19.0%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 18.9%

         \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

      2. sqrt-unprod 20.4%

         \[\leadsto d \cdot \color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

      3. pow-prod-up 20.4%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(-0.5 + -0.5\right)}}} \]

      4. metadata-eval 20.4%

         \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{-1}}} \]

      5. inv-pow 20.4%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      6. *-commutative 20.4%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      7. associate-/l/ 20.4%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

      8. rem-cbrt-cube 23.0%

         \[\leadsto d \cdot \sqrt{\color{blue}{\sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}}}} \]

      9. unpow1/3 22.6%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.3333333333333333}}} \]

      10. expm1-log1p-u 10.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.3333333333333333}}\right)\right)} \]

      11. expm1-udef 7.9%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.3333333333333333}}\right)} - 1} \]

   8. Applied egg-rr 9.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{h \cdot \ell}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 13.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{h \cdot \ell}}\right)\right)} \]

      2. expm1-log1p 19.0%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

   10. Simplified 19.0%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. log1p-expm1-u 27.0%

          \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{d}{\sqrt{h \cdot \ell}}\right)\right)} \]

   12. Applied egg-rr 27.0%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{d}{\sqrt{h \cdot \ell}}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.42 \cdot 10^{-28}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)\right)\\ \end{array} \]

Alternative 15: 41.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;D \leq 2.1 \cdot 10^{+231}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= D 2.1e+231) (fabs (/ d (sqrt (* l h)))) (* d (sqrt (/ (/ 1.0 h) l)))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (D <= 2.1e+231) {
		tmp = fabs((d / sqrt((l * h))));
	} else {
		tmp = d * sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d_1 <= 2.1d+231) then
        tmp = abs((d / sqrt((l * h))))
    else
        tmp = d * sqrt(((1.0d0 / h) / l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (D <= 2.1e+231) {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	} else {
		tmp = d * Math.sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if D <= 2.1e+231:
		tmp = math.fabs((d / math.sqrt((l * h))))
	else:
		tmp = d * math.sqrt(((1.0 / h) / l))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (D <= 2.1e+231)
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / h) / l)));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (D <= 2.1e+231)
		tmp = abs((d / sqrt((l * h))));
	else
		tmp = d * sqrt(((1.0 / h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[D, 2.1e+231], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 2.09999999999999984e231

   1. Initial program 64.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 28.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   3. Step-by-step derivation

      1. expm1-log1p-u 28.3%

         \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \]

      2. expm1-udef 20.0%

         \[\leadsto d \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{h \cdot \ell}}\right)} - 1\right)} \]

      3. pow1/2 20.0%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right)} - 1\right) \]

      4. inv-pow 20.0%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left({\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right)} - 1\right) \]

      5. pow-pow 19.5%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 0.5\right)}}\right)} - 1\right) \]

      6. metadata-eval 19.5%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right)} - 1\right) \]

   4. Applied egg-rr 19.5%

      \[\leadsto d \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-def 27.8%

         \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \]

      2. expm1-log1p 28.3%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   6. Simplified 28.3%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 24.1%

         \[\leadsto \color{blue}{\left(\sqrt{d} \cdot \sqrt{d}\right)} \cdot {\left(h \cdot \ell\right)}^{-0.5} \]

      2. sqrt-prod 30.8%

         \[\leadsto \color{blue}{\sqrt{d \cdot d}} \cdot {\left(h \cdot \ell\right)}^{-0.5} \]

      3. unpow2 30.8%

         \[\leadsto \sqrt{\color{blue}{{d}^{2}}} \cdot {\left(h \cdot \ell\right)}^{-0.5} \]

      4. add-sqr-sqrt 30.6%

         \[\leadsto \sqrt{{d}^{2}} \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

      5. sqrt-unprod 30.6%

         \[\leadsto \sqrt{{d}^{2}} \cdot \color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

      6. pow-prod-up 30.5%

         \[\leadsto \sqrt{{d}^{2}} \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(-0.5 + -0.5\right)}}} \]

      7. metadata-eval 30.5%

         \[\leadsto \sqrt{{d}^{2}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{-1}}} \]

      8. inv-pow 30.5%

         \[\leadsto \sqrt{{d}^{2}} \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      9. *-commutative 30.5%

         \[\leadsto \sqrt{{d}^{2}} \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      10. associate-/l/ 30.5%

          \[\leadsto \sqrt{{d}^{2}} \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

      11. sqrt-prod 24.4%

          \[\leadsto \color{blue}{\sqrt{{d}^{2} \cdot \frac{\frac{1}{h}}{\ell}}} \]

      12. associate-/l/ 24.4%

          \[\leadsto \sqrt{{d}^{2} \cdot \color{blue}{\frac{1}{\ell \cdot h}}} \]

      13. *-commutative 24.4%

          \[\leadsto \sqrt{{d}^{2} \cdot \frac{1}{\color{blue}{h \cdot \ell}}} \]

      14. div-inv 24.6%

          \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{h \cdot \ell}}} \]

      15. add-sqr-sqrt 24.6%

          \[\leadsto \sqrt{\color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}} \cdot \sqrt{\frac{{d}^{2}}{h \cdot \ell}}}} \]

      16. rem-sqrt-square 24.6%

          \[\leadsto \color{blue}{\left|\sqrt{\frac{{d}^{2}}{h \cdot \ell}}\right|} \]

      17. sqrt-div 30.8%

          \[\leadsto \left|\color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}}}\right| \]

      18. unpow2 30.8%

          \[\leadsto \left|\frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}}\right| \]

      19. sqrt-prod 24.2%

          \[\leadsto \left|\frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}}\right| \]

      20. add-sqr-sqrt 46.7%

          \[\leadsto \left|\frac{\color{blue}{d}}{\sqrt{h \cdot \ell}}\right| \]

   8. Applied egg-rr 46.7%

      \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \]

   if 2.09999999999999984e231 < D

   1. Initial program 86.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 15.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   3. Step-by-step derivation

      1. add-cbrt-cube 28.2%

         \[\leadsto d \cdot \sqrt{\color{blue}{\sqrt[3]{\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}}}} \]

      2. pow1/3 28.2%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}\right)}^{0.3333333333333333}}} \]

      3. pow3 28.2%

         \[\leadsto d \cdot \sqrt{{\color{blue}{\left({\left(\frac{1}{h \cdot \ell}\right)}^{3}\right)}}^{0.3333333333333333}} \]

      4. associate-/r* 28.2%

         \[\leadsto d \cdot \sqrt{{\left({\color{blue}{\left(\frac{\frac{1}{h}}{\ell}\right)}}^{3}\right)}^{0.3333333333333333}} \]

   4. Applied egg-rr 28.2%

      \[\leadsto d \cdot \sqrt{\color{blue}{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.3333333333333333}}} \]

   5. Taylor expanded in h around 0 15.5%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. associate-/r* 15.5%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   7. Simplified 15.5%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 2.1 \cdot 10^{+231}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]

Alternative 16: 45.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 2.3 \cdot 10^{-236}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= d 2.3e-236) (* (- d) (pow (* l h) -0.5)) (/ d (* (sqrt h) (sqrt l)))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 2.3e-236) {
		tmp = -d * pow((l * h), -0.5);
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= 2.3d-236) then
        tmp = -d * ((l * h) ** (-0.5d0))
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 2.3e-236) {
		tmp = -d * Math.pow((l * h), -0.5);
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if d <= 2.3e-236:
		tmp = -d * math.pow((l * h), -0.5)
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= 2.3e-236)
		tmp = Float64(Float64(-d) * (Float64(l * h) ^ -0.5));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= 2.3e-236)
		tmp = -d * ((l * h) ^ -0.5);
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[d, 2.3e-236], N[((-d) * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 2.30000000000000006e-236

   1. Initial program 58.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 18.8%

      \[\leadsto \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell} \cdot {\left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)}^{2}}} \]

   4. Taylor expanded in d around -inf 40.3%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 40.3%

         \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. unpow-1 40.3%

         \[\leadsto -d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

      3. metadata-eval 40.3%

         \[\leadsto -d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      4. pow-sqr 40.3%

         \[\leadsto -d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

      5. rem-sqrt-square 40.8%

         \[\leadsto -d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

      6. metadata-eval 40.8%

         \[\leadsto -d \cdot \left|{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.25\right)}}\right| \]

      7. pow-sqr 40.7%

         \[\leadsto -d \cdot \left|\color{blue}{{\left(h \cdot \ell\right)}^{-0.25} \cdot {\left(h \cdot \ell\right)}^{-0.25}}\right| \]

      8. fabs-sqr 40.7%

         \[\leadsto -d \cdot \color{blue}{\left({\left(h \cdot \ell\right)}^{-0.25} \cdot {\left(h \cdot \ell\right)}^{-0.25}\right)} \]

      9. pow-sqr 40.8%

         \[\leadsto -d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(2 \cdot -0.25\right)}} \]

      10. metadata-eval 40.8%

          \[\leadsto -d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \]

      11. distribute-rgt-neg-in 40.8%

          \[\leadsto \color{blue}{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \]

   6. Simplified 40.8%

      \[\leadsto \color{blue}{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \]

   if 2.30000000000000006e-236 < d

   1. Initial program 76.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 48.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   3. Step-by-step derivation

      1. expm1-log1p-u 47.2%

         \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \]

      2. expm1-udef 29.5%

         \[\leadsto d \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{h \cdot \ell}}\right)} - 1\right)} \]

      3. pow1/2 29.5%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right)} - 1\right) \]

      4. inv-pow 29.5%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left({\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right)} - 1\right) \]

      5. pow-pow 30.2%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 0.5\right)}}\right)} - 1\right) \]

      6. metadata-eval 30.2%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right)} - 1\right) \]

   4. Applied egg-rr 30.2%

      \[\leadsto d \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-def 47.9%

         \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \]

      2. expm1-log1p 49.0%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   6. Simplified 49.0%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 48.8%

         \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

      2. sqrt-unprod 48.3%

         \[\leadsto d \cdot \color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

      3. pow-prod-up 48.2%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(-0.5 + -0.5\right)}}} \]

      4. metadata-eval 48.2%

         \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{-1}}} \]

      5. inv-pow 48.2%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      6. *-commutative 48.2%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      7. associate-/l/ 48.7%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

      8. rem-cbrt-cube 28.9%

         \[\leadsto d \cdot \sqrt{\color{blue}{\sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}}}} \]

      9. unpow1/3 28.0%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.3333333333333333}}} \]

      10. expm1-log1p-u 27.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.3333333333333333}}\right)\right)} \]

      11. expm1-udef 22.2%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.3333333333333333}}\right)} - 1} \]

   8. Applied egg-rr 34.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{h \cdot \ell}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 46.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{h \cdot \ell}}\right)\right)} \]

      2. expm1-log1p 49.0%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

   10. Simplified 49.0%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. *-commutative 49.0%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       2. sqrt-prod 55.7%

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

   12. Applied egg-rr 55.7%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 2.3 \cdot 10^{-236}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 17: 40.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -6 \cdot 10^{-283}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= h -6e-283) (* (- d) (pow (* l h) -0.5)) (/ d (sqrt (* l h)))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -6e-283) {
		tmp = -d * pow((l * h), -0.5);
	} else {
		tmp = d / sqrt((l * h));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (h <= (-6d-283)) then
        tmp = -d * ((l * h) ** (-0.5d0))
    else
        tmp = d / sqrt((l * h))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -6e-283) {
		tmp = -d * Math.pow((l * h), -0.5);
	} else {
		tmp = d / Math.sqrt((l * h));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if h <= -6e-283:
		tmp = -d * math.pow((l * h), -0.5)
	else:
		tmp = d / math.sqrt((l * h))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= -6e-283)
		tmp = Float64(Float64(-d) * (Float64(l * h) ^ -0.5));
	else
		tmp = Float64(d / sqrt(Float64(l * h)));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= -6e-283)
		tmp = -d * ((l * h) ^ -0.5);
	else
		tmp = d / sqrt((l * h));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[h, -6e-283], 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 h < -5.99999999999999992e-283

   1. Initial program 64.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 21.7%

      \[\leadsto \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell} \cdot {\left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)}^{2}}} \]

   4. Taylor expanded in d around -inf 44.7%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 44.7%

         \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. unpow-1 44.7%

         \[\leadsto -d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

      3. metadata-eval 44.7%

         \[\leadsto -d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      4. pow-sqr 44.8%

         \[\leadsto -d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

      5. rem-sqrt-square 45.4%

         \[\leadsto -d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

      6. metadata-eval 45.4%

         \[\leadsto -d \cdot \left|{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.25\right)}}\right| \]

      7. pow-sqr 45.2%

         \[\leadsto -d \cdot \left|\color{blue}{{\left(h \cdot \ell\right)}^{-0.25} \cdot {\left(h \cdot \ell\right)}^{-0.25}}\right| \]

      8. fabs-sqr 45.2%

         \[\leadsto -d \cdot \color{blue}{\left({\left(h \cdot \ell\right)}^{-0.25} \cdot {\left(h \cdot \ell\right)}^{-0.25}\right)} \]

      9. pow-sqr 45.4%

         \[\leadsto -d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(2 \cdot -0.25\right)}} \]

      10. metadata-eval 45.4%

          \[\leadsto -d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \]

      11. distribute-rgt-neg-in 45.4%

          \[\leadsto \color{blue}{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \]

   6. Simplified 45.4%

      \[\leadsto \color{blue}{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \]

   if -5.99999999999999992e-283 < h

   1. Initial program 67.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 43.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   3. Step-by-step derivation

      1. expm1-log1p-u 43.0%

         \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \]

      2. expm1-udef 28.4%

         \[\leadsto d \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{h \cdot \ell}}\right)} - 1\right)} \]

      3. pow1/2 28.4%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right)} - 1\right) \]

      4. inv-pow 28.4%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left({\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right)} - 1\right) \]

      5. pow-pow 28.9%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 0.5\right)}}\right)} - 1\right) \]

      6. metadata-eval 28.9%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right)} - 1\right) \]

   4. Applied egg-rr 28.9%

      \[\leadsto d \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-def 43.6%

         \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \]

      2. expm1-log1p 44.5%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   6. Simplified 44.5%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 44.3%

         \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

      2. sqrt-unprod 43.9%

         \[\leadsto d \cdot \color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

      3. pow-prod-up 43.8%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(-0.5 + -0.5\right)}}} \]

      4. metadata-eval 43.8%

         \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{-1}}} \]

      5. inv-pow 43.8%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      6. *-commutative 43.8%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      7. associate-/l/ 44.3%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

      8. rem-cbrt-cube 28.5%

         \[\leadsto d \cdot \sqrt{\color{blue}{\sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}}}} \]

      9. unpow1/3 27.7%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.3333333333333333}}} \]

      10. expm1-log1p-u 25.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.3333333333333333}}\right)\right)} \]

      11. expm1-udef 19.9%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.3333333333333333}}\right)} - 1} \]

   8. Applied egg-rr 30.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{h \cdot \ell}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 41.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{h \cdot \ell}}\right)\right)} \]

      2. expm1-log1p 44.5%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

   10. Simplified 44.5%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -6 \cdot 10^{-283}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]

Alternative 18: 26.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \frac{d}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D) :precision binary64 (/ d (sqrt (* l h))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	return d / sqrt((l * h));
}

Fortran

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = d / sqrt((l * h))
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	return d / Math.sqrt((l * h));
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	return d / math.sqrt((l * h))

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	return Float64(d / sqrt(Float64(l * h)))
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp = code(d, h, l, M, D)
	tmp = d / sqrt((l * h));
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 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. Taylor expanded in d around inf 28.0%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
3. Step-by-step derivation

   1. expm1-log1p-u 27.5%

      \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \]

   2. expm1-udef 19.7%

      \[\leadsto d \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{h \cdot \ell}}\right)} - 1\right)} \]

   3. pow1/2 19.7%

      \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right)} - 1\right) \]

   4. inv-pow 19.7%

      \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left({\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right)} - 1\right) \]

   5. pow-pow 19.3%

      \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 0.5\right)}}\right)} - 1\right) \]

   6. metadata-eval 19.3%

      \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right)} - 1\right) \]

4. Applied egg-rr 19.3%

   \[\leadsto d \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)} - 1\right)} \]
5. Step-by-step derivation

   1. expm1-def 27.1%

      \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \]

   2. expm1-log1p 27.6%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

6. Simplified 27.6%

   \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
7. Step-by-step derivation

   1. add-sqr-sqrt 27.5%

      \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

   2. sqrt-unprod 28.0%

      \[\leadsto d \cdot \color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

   3. pow-prod-up 28.0%

      \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(-0.5 + -0.5\right)}}} \]

   4. metadata-eval 28.0%

      \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{-1}}} \]

   5. inv-pow 28.0%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

   6. *-commutative 28.0%

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. associate-/l/ 28.2%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   8. rem-cbrt-cube 22.4%

      \[\leadsto d \cdot \sqrt{\color{blue}{\sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}}}} \]

   9. unpow1/3 22.0%

      \[\leadsto d \cdot \sqrt{\color{blue}{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.3333333333333333}}} \]

   10. expm1-log1p-u 14.7%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.3333333333333333}}\right)\right)} \]

   11. expm1-udef 12.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.3333333333333333}}\right)} - 1} \]

8. Applied egg-rr 17.5%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{h \cdot \ell}}\right)} - 1} \]
9. Step-by-step derivation

   1. expm1-def 23.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{h \cdot \ell}}\right)\right)} \]

   2. expm1-log1p 27.6%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

10. Simplified 27.6%

    \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

11. Final simplification 27.6%

    \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

Reproduce

herbie shell --seed 2023306 
(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)))))